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<!DOCTYPE html><html lang="zh-CN"><head><meta charset="UTF-8"><meta name="viewport" content="width=device-width, initial-scale=1.0"><meta name="description" content="以一个含两个已知点、三个未知点和七条观测边的水准网为例,梳理间接平差的观测方程、法方程、精度评定与程序实现边界,并给出可复算的系数矩阵和计算结果。"><meta name="author" content="biss"><meta name="keywords" content="学习, 测量平差, 测量平差, 最小二乘, C#, 矩阵"><link rel="canonical" href="https://blog.biss.click/posts/e67166f3/"><link rel="alternate" type="application/rss+xml" title="Bi's Blog RSS" href="https://blog.biss.click/rss.xml"><meta property="og:site_name" content="Bi's Blog"><meta property="og:locale" content="zh_CN"><meta property="og:type" content="article"><meta property="og:title" content="从间接平差到程序:一个水准网最小二乘解算案例 | Bi's Blog"><meta property="og:description" content="以一个含两个已知点、三个未知点和七条观测边的水准网为例,梳理间接平差的观测方程、法方程、精度评定与程序实现边界,并给出可复算的系数矩阵和计算结果。"><meta property="og:url" content="https://blog.biss.click/posts/e67166f3/"><meta property="og:image" content="https://free.picui.cn/free/2025/08/10/689845496a283.png"><meta property="og:image:alt" content="从间接平差到程序:一个水准网最小二乘解算案例"><meta name="twitter:card" content="summary_large_image"><meta name="twitter:title" content="从间接平差到程序:一个水准网最小二乘解算案例 | Bi's Blog"><meta name="twitter:description" content="以一个含两个已知点、三个未知点和七条观测边的水准网为例,梳理间接平差的观测方程、法方程、精度评定与程序实现边界,并给出可复算的系数矩阵和计算结果。"><meta name="twitter:image" content="https://free.picui.cn/free/2025/08/10/689845496a283.png"><meta property="article:published_time" content="2026-08-27T21:00:00.000Z"><meta property="article:modified_time" content="2026-08-27T21:00:00.000Z"><meta property="article:author" content="biss"><meta property="article:tag" content="学习"><meta property="article:tag" content="测量平差"><meta property="article:tag" content="测量平差"><meta property="article:tag" content="最小二乘"><meta property="article:tag" content="C#"><meta property="article:tag" content="矩阵"><script type="application/ld+json">{"@context":"https://schema.org","@type":"BlogPosting","headline":"从间接平差到程序:一个水准网最小二乘解算案例","description":"以一个含两个已知点、三个未知点和七条观测边的水准网为例,梳理间接平差的观测方程、法方程、精度评定与程序实现边界,并给出可复算的系数矩阵和计算结果。","image":["https://free.picui.cn/free/2025/08/10/689845496a283.png"],"datePublished":"2026-08-27T21:00:00.000Z","dateModified":"2026-08-27T21:00:00.000Z","wordCount":2386,"inLanguage":"zh-CN","mainEntityOfPage":{"@type":"WebPage","@id":"https://blog.biss.click/posts/e67166f3/"},"author":{"@type":"Person","name":"biss","url":"https://github.com/bishshi"},"publisher":{"@type":"Person","name":"biss","image":"https://free.picui.cn/free/2025/08/10/689845496a283.png"},"keywords":"学习, 测量平差, 测量平差, 最小二乘, C#, 矩阵","articleSection":"学习"}</script><link rel="icon" href="/images/Bi.ico"><link rel="manifest" href="/manifest.webmanifest"><link rel="apple-touch-icon" href="/images/apple-touch-icon.png"><meta name="theme-color" content="#f7f5ef" media="(prefers-color-scheme: light)"><meta name="theme-color" content="#0b1014" media="(prefers-color-scheme: dark)"><script defer src="https://umami.biss.click/script.js" data-website-id="3d97b310-b241-4fc6-bc9c-68e317fc7d42"></script><script>window.SITE_CONFIG = {"site":{"url":"https://blog.biss.click","name":"Bi's Blog","description":"Personal blog by biss.","language":"zh-CN","domain":"blog.biss.click","slogan":"好奇心转化为代码的空间。","startTime":"2025-01-01T00:00:00+08:00","copyrightStartYear":2025,"license":"内容采用 CC BY-NC-SA 4.0 协议"},"author":{"name":"biss","email":"bishsh2006@gmail.com","github":"https://github.com/bishshi","avatar":"https://free.picui.cn/free/2025/08/10/689845496a283.png"},"comments":{"envId":"https://comment.biss.click","region":"","script":"https://cdn.jsdelivr.net/npm/twikoo@1.7.7/dist/twikoo.min.js"},"analytics":{"umami":{"script":"https://umami.biss.click/script.js","websiteId":"3d97b310-b241-4fc6-bc9c-68e317fc7d42","statsEndpoint":"https://blogumami.biss.click"}},"talks":{"apiUrl":"https://mm.biss.click/api/echo/page","pageSize":30,"homeLimit":6,"cacheKey":"talksCache","cacheTimeKey":"talksCacheTime","cacheDurationMs":1800000,"fallbackAvatar":"https://free.picui.cn/free/2025/08/10/689845496a283.png"},"search":{"apiKey":"2uxQzk7eD2GBCljdSmIiuU3ause7UK9n","server":{"host":"typesense.biss.click","port":"443","protocol":"https"},"indexName":"blogs","searchParams":{"query_by":"title,content","highlight_full_fields":"title,content","per_page":8,"num_typos":1,"typo_tokens_threshold":1,"prefix":true},"ui":{"maxRetries":10,"retryDelay":100,"animationDuration":300,"maxInitRetries":30,"placeholder":"输入关键词寻找故事..."}},"externalLinks":{"waitSeconds":10,"whitelist":["blog.biss.click","biss.click","github.com","localhost","127.0.0.1"]}};</script><script>
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class="control-console-info" aria-label="站点信息"><section><span>当前站点</span><strong>blog.biss.click</strong></section><section><span>站点作者</span><strong>biss</strong></section><section><span>运行起点</span><strong>2025-01-01</strong></section></div></section></div><main><div class="reading-progress" aria-hidden="true"><span data-reading-progress></span></div><aside class="reading-resume" data-reading-resume hidden aria-live="polite"><div class="reading-resume-icon" aria-hidden="true"><i class="fa-solid fa-book-open"></i></div><div><strong>继续上次阅读?</strong><p>你上次读到了 <span data-reading-resume-percent>--</span>。</p></div><div class="reading-resume-actions"><button type="button" data-reading-continue>继续阅读</button><button type="button" data-reading-restart>从头开始</button></div></aside><section class="post-layout"><article class="article post-article"><header class="article-header"><div class="article-header-content"><div class="article-date-line"><time datetime="2026-08-27T21:00:00.000Z">发布于 2026/08/28</time></div><h1>从间接平差到程序:一个水准网最小二乘解算案例</h1><div class="meta-list"><a href="/categories/study/">学习</a><a href="/categories/category/">测量平差</a><a href="/tags/%E6%B5%8B%E9%87%8F%E5%B9%B3%E5%B7%AE/">#测量平差</a><a href="/tags/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98/">#最小二乘</a><a href="/tags/C%23/">#C#</a><a href="/tags/%E7%9F%A9%E9%98%B5/">#矩阵</a></div></div></header><section class="article-summary-card" aria-label="文章导读"><div class="article-summary-header"><span class="article-summary-brand"><span class="article-summary-icon" aria-hidden="true">↳</span>文章导读</span><span class="article-summary-model">摘要由 Qwen 生成</span></div><p class="article-summary-content"><span class="article-summary-text">以一个含两个已知点、三个未知点和七条观测边的水准网为例,梳理间接平差的观测方程、法方程、精度评定与程序实现边界,并给出可复算的系数矩阵和计算结果。</span></p></section><aside class="butterfly-note butterfly-note--warning article-outdated-note" aria-label="文章时效性提示" data-freshness-at="2026-08-27T21:00:00.000Z" data-has-new-version="false" hidden><i class="fa-solid fa-triangle-exclamation" aria-hidden="true"></i><div class="butterfly-note__content"><p>注意:本文最后维护于 2026/08/28,距今已超过一年,部分内容可能已经过时,请结合最新资料判断。</p></div></aside><div class="prose" data-toc="[{"depth":2,"text":"问题:已知两点,求三个未知点","id":"问题已知两点求三个未知点"},{"depth":2,"text":"从观测方程到法方程","id":"从观测方程到法方程"},{"depth":2,"text":"解算后还能得到什么","id":"解算后还能得到什么"},{"depth":2,"text":"程序实现:让流程可操作,而不只是可计算","id":"程序实现让流程可操作而不只是可计算"},{"depth":2,"text":"小结","id":"小结"}]"><p>记录一下课程设计。</p>
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<h2 id="问题已知两点求三个未知点">问题:已知两点,求三个未知点</h2>
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<p>设已知点为 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-1-TEX-I-1D434"></use></g></g></g></svg></mjx-container>、<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 759 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D435" xlink:href="#MJX-2-TEX-I-1D435"></use></g></g></g></svg></mjx-container>,未知点为 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.885ex" role="img" focusable="false" viewBox="0 -683 1078.6 833" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-3-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-3-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-3-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-3-TEX-N-31"></use></g></g></g></g></svg></mjx-container>、<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.885ex" role="img" focusable="false" viewBox="0 -683 1078.6 833" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-4-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-4-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-4-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-4-TEX-N-32"></use></g></g></g></g></svg></mjx-container>、<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.92ex" role="img" focusable="false" viewBox="0 -683 1078.6 848.6" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-5-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-5-TEX-N-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-5-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="33" xlink:href="#MJX-5-TEX-N-33"></use></g></g></g></g></svg></mjx-container>。已知高程分别为:</p>
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<p>观测得到七条高差及对应路线长度。路线越长,观测通常越不稳定,因此这里采用与距离成反比的权:</p>
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<mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -1.909ex;" xmlns="http://www.w3.org/2000/svg" width="7.691ex" height="4.945ex" role="img" focusable="false" viewBox="0 -1342 3399.5 2185.8" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-7-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-7-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-7-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-7-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-7-TEX-I-1D460" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-7-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-7-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(1107.7,0)"><use data-c="3D" xlink:href="#MJX-7-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(2163.5,0)"><g data-mml-node="mn" transform="translate(368,676)"><use data-c="31" xlink:href="#MJX-7-TEX-N-31"></use></g><g data-mml-node="msub" transform="translate(220,-686)"><g data-mml-node="mi"><use data-c="1D460" xlink:href="#MJX-7-TEX-I-1D460"></use></g><g data-mml-node="mi" transform="translate(502,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-7-TEX-I-1D456"></use></g></g><rect width="996" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container>
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<p>其中 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.357ex;" xmlns="http://www.w3.org/2000/svg" width="1.801ex" height="1.357ex" role="img" focusable="false" viewBox="0 -442 796 599.8" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-8-TEX-I-1D460" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path id="MJX-8-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D460" xlink:href="#MJX-8-TEX-I-1D460"></use></g><g data-mml-node="mi" transform="translate(502,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-8-TEX-I-1D456"></use></g></g></g></g></svg></mjx-container> 的单位为 km,<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.439ex;" xmlns="http://www.w3.org/2000/svg" width="1.878ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 830 636" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-9-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-9-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-9-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-9-TEX-I-1D456"></use></g></g></g></g></svg></mjx-container> 为第 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="0.781ex" height="1.52ex" role="img" focusable="false" viewBox="0 -661 345 672" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-10-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-10-TEX-I-1D456"></use></g></g></g></svg></mjx-container> 条观测的权。</p>
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<table><thead><tr><th>边号</th><th>起点</th><th>终点</th><th align="right">观测高差 / m</th><th align="right">距离 / km</th><th align="right">权</th></tr></thead><tbody><tr><td>1</td><td>A</td><td><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.885ex" role="img" focusable="false" viewBox="0 -683 1078.6 833" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-11-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-11-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-11-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-11-TEX-N-31"></use></g></g></g></g></svg></mjx-container></td><td align="right">1.359000</td><td align="right">1.1</td><td align="right">0.909091</td></tr><tr><td>2</td><td>A</td><td><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.885ex" role="img" focusable="false" viewBox="0 -683 1078.6 833" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-12-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-12-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-12-TEX-N-32"></use></g></g></g></g></svg></mjx-container></td><td align="right">2.009000</td><td align="right">1.7</td><td align="right">0.588235</td></tr><tr><td>3</td><td>B</td><td><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.885ex" role="img" focusable="false" viewBox="0 -683 1078.6 833" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-13-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-13-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-13-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-13-TEX-N-31"></use></g></g></g></g></svg></mjx-container></td><td align="right">0.363000</td><td align="right">2.3</td><td align="right">0.434783</td></tr><tr><td>4</td><td>B</td><td><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.885ex" role="img" focusable="false" viewBox="0 -683 1078.6 833" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-14-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-14-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-14-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-14-TEX-N-32"></use></g></g></g></g></svg></mjx-container></td><td align="right">1.012000</td><td align="right">2.7</td><td align="right">0.370370</td></tr><tr><td>5</td><td><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.885ex" role="img" focusable="false" viewBox="0 -683 1078.6 833" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-15-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-15-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-15-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-15-TEX-N-31"></use></g></g></g></g></svg></mjx-container></td><td><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.885ex" role="img" focusable="false" viewBox="0 -683 1078.6 833" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-16-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-16-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-16-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-16-TEX-N-32"></use></g></g></g></g></svg></mjx-container></td><td align="right">0.657000</td><td align="right">2.4</td><td align="right">0.416667</td></tr><tr><td>6</td><td><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.885ex" role="img" focusable="false" viewBox="0 -683 1078.6 833" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-17-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-17-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-17-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-17-TEX-N-31"></use></g></g></g></g></svg></mjx-container></td><td><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.92ex" role="img" focusable="false" viewBox="0 -683 1078.6 848.6" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-18-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-18-TEX-N-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-18-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="33" xlink:href="#MJX-18-TEX-N-33"></use></g></g></g></g></svg></mjx-container></td><td align="right">0.238000</td><td align="right">1.4</td><td align="right">0.714286</td></tr><tr><td>7</td><td><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.92ex" role="img" focusable="false" viewBox="0 -683 1078.6 848.6" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-19-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-19-TEX-N-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-19-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="33" xlink:href="#MJX-19-TEX-N-33"></use></g></g></g></g></svg></mjx-container></td><td>B</td><td align="right">-0.595000</td><td align="right">2.6</td><td align="right">0.384615</td></tr></tbody></table>
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<p>单独沿任一路线推高程,会因观测误差得到彼此不完全一致的答案;平差的任务就是在全部观测共同约束下,找到一组最合理的未知高程。</p>
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<h2 id="从观测方程到法方程">从观测方程到法方程</h2>
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<p>一般地,设观测值为 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.357ex;" xmlns="http://www.w3.org/2000/svg" width="2.28ex" height="1.902ex" role="img" focusable="false" viewBox="0 -683 1008 840.8" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-20-TEX-I-1D43F" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path><path id="MJX-20-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D43F" xlink:href="#MJX-20-TEX-I-1D43F"></use></g><g data-mml-node="mi" transform="translate(714,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-20-TEX-I-1D456"></use></g></g></g></g></svg></mjx-container>,观测值改正数为 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.357ex;" xmlns="http://www.w3.org/2000/svg" width="1.837ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 812 600.8" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-21-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path><path id="MJX-21-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D463" xlink:href="#MJX-21-TEX-I-1D463"></use></g><g data-mml-node="mi" transform="translate(518,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-21-TEX-I-1D456"></use></g></g></g></g></svg></mjx-container>,待定参数为 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.666ex;" xmlns="http://www.w3.org/2000/svg" width="2.72ex" height="2.211ex" role="img" focusable="false" viewBox="0 -683 1202.3 977.2" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-22-TEX-I-1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path id="MJX-22-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D44B" xlink:href="#MJX-22-TEX-I-1D44B"></use></g><g data-mml-node="mi" transform="translate(861,-150) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-22-TEX-I-1D457"></use></g></g></g></g></svg></mjx-container>,则平差后的观测值满足:</p>
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<mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="27.916ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 12338.9 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-23-TEX-I-1D43F" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path><path id="MJX-23-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-23-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-23-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path><path id="MJX-23-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-23-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-23-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-23-TEX-I-1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path id="MJX-23-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-23-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-23-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-23-TEX-N-2026" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60ZM525 60Q525 84 542 102T585 120Q609 120 627 104T646 61Q646 36 629 18T586 0T543 17T525 60ZM972 60Q972 84 989 102T1032 120Q1056 120 1074 104T1093 61Q1093 36 1076 18T1033 0T990 17T972 60Z"></path><path id="MJX-23-TEX-I-1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path id="MJX-23-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D43F" xlink:href="#MJX-23-TEX-I-1D43F"></use></g><g data-mml-node="mi" transform="translate(714,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-23-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(1230.2,0)"><use data-c="2B" xlink:href="#MJX-23-TEX-N-2B"></use></g><g data-mml-node="msub" transform="translate(2230.4,0)"><g data-mml-node="mi"><use data-c="1D463" xlink:href="#MJX-23-TEX-I-1D463"></use></g><g data-mml-node="mi" transform="translate(518,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-23-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(3320.1,0)"><use data-c="3D" xlink:href="#MJX-23-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(4375.9,0)"><g data-mml-node="mi"><use data-c="1D453" xlink:href="#MJX-23-TEX-I-1D453"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-23-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(5192.9,0)"><use data-c="28" xlink:href="#MJX-23-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(5581.9,0)"><g data-mml-node="mi"><use data-c="1D44B" xlink:href="#MJX-23-TEX-I-1D44B"></use></g><g data-mml-node="mn" transform="translate(861,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-23-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(6846.4,0)"><use data-c="2C" xlink:href="#MJX-23-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(7291.1,0)"><g data-mml-node="mi"><use data-c="1D44B" xlink:href="#MJX-23-TEX-I-1D44B"></use></g><g data-mml-node="mn" transform="translate(861,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-23-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(8555.6,0)"><use data-c="2C" xlink:href="#MJX-23-TEX-N-2C"></use></g><g data-mml-node="mo" transform="translate(9000.3,0)"><use data-c="2026" xlink:href="#MJX-23-TEX-N-2026"></use></g><g data-mml-node="mo" transform="translate(10339,0)"><use data-c="2C" xlink:href="#MJX-23-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(10783.6,0)"><g data-mml-node="mi"><use data-c="1D44B" xlink:href="#MJX-23-TEX-I-1D44B"></use></g><g data-mml-node="mi" transform="translate(861,-150) scale(0.707)"><use data-c="1D461" xlink:href="#MJX-23-TEX-I-1D461"></use></g></g><g data-mml-node="mo" transform="translate(11949.9,0)"><use data-c="29" xlink:href="#MJX-23-TEX-N-29"></use></g></g></g></svg></mjx-container>
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<p>对非线性问题,可在近似值 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.991ex;" xmlns="http://www.w3.org/2000/svg" width="3.031ex" height="2.892ex" role="img" focusable="false" viewBox="0 -840.1 1339.8 1278.1" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-24-TEX-I-1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path id="MJX-24-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-24-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msubsup"><g data-mml-node="mi"><use data-c="1D44B" xlink:href="#MJX-24-TEX-I-1D44B"></use></g><g data-mml-node="mn" transform="translate(936.2,369.2) scale(0.707)"><use data-c="30" xlink:href="#MJX-24-TEX-N-30"></use></g><g data-mml-node="mi" transform="translate(861,-293.8) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-24-TEX-I-1D457"></use></g></g></g></g></svg></mjx-container> 附近展开:</p>
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<mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.892ex;" xmlns="http://www.w3.org/2000/svg" width="13.675ex" height="2.892ex" role="img" focusable="false" viewBox="0 -883.9 6044.4 1278.1" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-25-TEX-I-1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path id="MJX-25-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path><path id="MJX-25-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-25-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-25-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-25-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-25-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D44B" xlink:href="#MJX-25-TEX-I-1D44B"></use></g><g data-mml-node="mi" transform="translate(861,-150) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-25-TEX-I-1D457"></use></g></g><g data-mml-node="mo" transform="translate(1480.1,0)"><use data-c="3D" xlink:href="#MJX-25-TEX-N-3D"></use></g><g data-mml-node="msubsup" transform="translate(2535.9,0)"><g data-mml-node="mi"><use data-c="1D44B" xlink:href="#MJX-25-TEX-I-1D44B"></use></g><g data-mml-node="mn" transform="translate(936.2,413) scale(0.707)"><use data-c="30" xlink:href="#MJX-25-TEX-N-30"></use></g><g data-mml-node="mi" transform="translate(861,-250) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-25-TEX-I-1D457"></use></g></g><g data-mml-node="mo" transform="translate(4097.9,0)"><use data-c="2B" xlink:href="#MJX-25-TEX-N-2B"></use></g><g data-mml-node="msub" transform="translate(5098.1,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-25-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(313.8,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-25-TEX-N-5E"></use></g></g></g><g data-mml-node="mi" transform="translate(605,-150) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-25-TEX-I-1D457"></use></g></g></g></g></svg></mjx-container>
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<p>忽略二阶及以上小量后,得到线性化观测方程。写成矩阵形式就是:</p>
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<mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="11.842ex" height="1.756ex" role="img" focusable="false" viewBox="0 -694 5234 776" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-26-TEX-I-1D449" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path id="MJX-26-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-26-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path><path id="MJX-26-TEX-I-1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path id="MJX-26-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-26-TEX-I-1D459" d="M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D449" xlink:href="#MJX-26-TEX-I-1D449"></use></g><g data-mml-node="mo" transform="translate(1046.8,0)"><use data-c="3D" xlink:href="#MJX-26-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(2102.6,0)"><use data-c="1D435" xlink:href="#MJX-26-TEX-I-1D435"></use></g><g data-mml-node="mi" transform="translate(2861.6,0)"><use data-c="1D44B" xlink:href="#MJX-26-TEX-I-1D44B"></use></g><g data-mml-node="mo" transform="translate(3935.8,0)"><use data-c="2212" xlink:href="#MJX-26-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(4936,0)"><use data-c="1D459" xlink:href="#MJX-26-TEX-I-1D459"></use></g></g></g></svg></mjx-container>
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<p>这里 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="1.74ex" height="1.595ex" role="img" focusable="false" viewBox="0 -683 769 705" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-27-TEX-I-1D449" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D449" xlink:href="#MJX-27-TEX-I-1D449"></use></g></g></g></svg></mjx-container> 是观测改正数向量,<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 759 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-28-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D435" xlink:href="#MJX-28-TEX-I-1D435"></use></g></g></g></svg></mjx-container> 是系数矩阵,<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.928ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 852 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-29-TEX-I-1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D44B" xlink:href="#MJX-29-TEX-I-1D44B"></use></g></g></g></svg></mjx-container> 是参数改正数向量,<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="0.674ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 298 705" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-30-TEX-I-1D459" d="M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D459" xlink:href="#MJX-30-TEX-I-1D459"></use></g></g></g></svg></mjx-container> 是常数项向量。</p>
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<p>间接平差要求加权改正数平方和最小:</p>
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<mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="13.416ex" height="2.203ex" role="img" focusable="false" viewBox="0 -891.7 5929.7 973.7" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-31-TEX-I-1D449" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path id="MJX-31-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 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619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-31-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-31-TEX-N-6D" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path id="MJX-31-TEX-N-69" d="M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z"></path><path id="MJX-31-TEX-N-6E" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D449" xlink:href="#MJX-31-TEX-I-1D449"></use></g><g data-mml-node="mi" transform="translate(861.3,413) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-31-TEX-I-1D447"></use></g></g><g data-mml-node="mi" transform="translate(1409.1,0)"><use data-c="1D443" xlink:href="#MJX-31-TEX-I-1D443"></use></g><g data-mml-node="mi" transform="translate(2160.1,0)"><use data-c="1D449" xlink:href="#MJX-31-TEX-I-1D449"></use></g><g data-mml-node="mo" transform="translate(3206.9,0)"><use data-c="3D" xlink:href="#MJX-31-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(4262.7,0)"><use data-c="6D" xlink:href="#MJX-31-TEX-N-6D"></use><use data-c="69" xlink:href="#MJX-31-TEX-N-69" transform="translate(833,0)"></use><use data-c="6E" xlink:href="#MJX-31-TEX-N-6E" transform="translate(1111,0)"></use></g></g></g></svg></mjx-container>
|
||
<p>将 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="11.842ex" height="1.756ex" role="img" focusable="false" viewBox="0 -694 5234 776" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-32-TEX-I-1D449" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path id="MJX-32-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-32-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path><path id="MJX-32-TEX-I-1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path id="MJX-32-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-32-TEX-I-1D459" d="M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D449" xlink:href="#MJX-32-TEX-I-1D449"></use></g><g data-mml-node="mo" transform="translate(1046.8,0)"><use data-c="3D" xlink:href="#MJX-32-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(2102.6,0)"><use data-c="1D435" xlink:href="#MJX-32-TEX-I-1D435"></use></g><g data-mml-node="mi" transform="translate(2861.6,0)"><use data-c="1D44B" xlink:href="#MJX-32-TEX-I-1D44B"></use></g><g data-mml-node="mo" transform="translate(3935.8,0)"><use data-c="2212" xlink:href="#MJX-32-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(4936,0)"><use data-c="1D459" xlink:href="#MJX-32-TEX-I-1D459"></use></g></g></g></svg></mjx-container> 代入并对 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.928ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 852 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-33-TEX-I-1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D44B" xlink:href="#MJX-33-TEX-I-1D44B"></use></g></g></g></svg></mjx-container> 求极值,可得:</p>
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||
<mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="16.797ex" height="2.203ex" role="img" focusable="false" viewBox="0 -891.7 7424.2 973.7" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-34-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path><path id="MJX-34-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-34-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-34-TEX-I-1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path id="MJX-34-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-34-TEX-I-1D459" d="M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D435" xlink:href="#MJX-34-TEX-I-1D435"></use></g><g data-mml-node="mi" transform="translate(792,413) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-34-TEX-I-1D447"></use></g></g><g data-mml-node="mi" transform="translate(1339.8,0)"><use data-c="1D443" xlink:href="#MJX-34-TEX-I-1D443"></use></g><g data-mml-node="mi" transform="translate(2090.8,0)"><use data-c="1D435" xlink:href="#MJX-34-TEX-I-1D435"></use></g><g data-mml-node="mi" transform="translate(2849.8,0)"><use data-c="1D44B" xlink:href="#MJX-34-TEX-I-1D44B"></use></g><g data-mml-node="mo" transform="translate(3979.6,0)"><use data-c="3D" xlink:href="#MJX-34-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(5035.4,0)"><g data-mml-node="mi"><use data-c="1D435" xlink:href="#MJX-34-TEX-I-1D435"></use></g><g data-mml-node="mi" transform="translate(792,413) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-34-TEX-I-1D447"></use></g></g><g data-mml-node="mi" transform="translate(6375.2,0)"><use data-c="1D443" xlink:href="#MJX-34-TEX-I-1D443"></use></g><g data-mml-node="mi" transform="translate(7126.2,0)"><use data-c="1D459" xlink:href="#MJX-34-TEX-I-1D459"></use></g></g></g></svg></mjx-container>
|
||
<p>记</p>
|
||
<mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.439ex;" xmlns="http://www.w3.org/2000/svg" width="30.221ex" height="2.456ex" role="img" focusable="false" viewBox="0 -891.7 13357.8 1085.7" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-35-TEX-I-1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z"></path><path id="MJX-35-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path><path id="MJX-35-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-35-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-35-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-35-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-35-TEX-I-1D44A" d="M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 377T629 138L631 141Q633 144 637 151T649 170T666 200T690 241T720 295T759 362Q863 546 877 572T892 604Q892 619 873 628T831 637Q817 637 817 647Q817 650 819 660Q823 676 825 679T839 682Q842 682 856 682T895 682T949 681Q1015 681 1034 683Q1048 683 1048 672Q1048 666 1045 655T1038 640T1028 637Q1006 637 988 631T958 617T939 600T927 584L923 578L754 282Q586 -14 585 -15Q579 -22 561 -22Q546 -22 542 -17Q539 -14 523 229T506 480L494 462Q472 425 366 239Q222 -13 220 -15T215 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<p>则法方程与参数改正数分别为:</p>
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<p>在这个案例中,未知参数的顺序取为 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="11.093ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4903 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-38-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-38-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-38-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-38-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-38-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-38-TEX-N-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path id="MJX-38-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-38-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(389,0)"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-38-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-38-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1467.6,0)"><use data-c="2C" xlink:href="#MJX-38-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1912.2,0)"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-38-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-38-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(2990.8,0)"><use data-c="2C" xlink:href="#MJX-38-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(3435.4,0)"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-38-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="33" xlink:href="#MJX-38-TEX-N-33"></use></g></g><g data-mml-node="mo" transform="translate(4514,0)"><use data-c="29" xlink:href="#MJX-38-TEX-N-29"></use></g></g></g></svg></mjx-container>,对应的系数矩阵为:</p>
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<mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -10.068ex;" xmlns="http://www.w3.org/2000/svg" width="19.191ex" height="21.267ex" role="img" focusable="false" viewBox="0 -4950 8482.6 9400" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-39-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path><path id="MJX-39-TEX-N-3D" d="M56 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transform="translate(4167,0)"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-39-TEX-N-30"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,1400)"><g data-mml-node="mtd" transform="translate(389,0)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-39-TEX-N-31"></use></g></g><g data-mml-node="mtd" transform="translate(2278,0)"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-39-TEX-N-30"></use></g></g><g data-mml-node="mtd" transform="translate(4167,0)"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-39-TEX-N-30"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,0)"><g data-mml-node="mtd" transform="translate(389,0)"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-39-TEX-N-30"></use></g></g><g data-mml-node="mtd" transform="translate(2278,0)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-39-TEX-N-31"></use></g></g><g data-mml-node="mtd" transform="translate(4167,0)"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-39-TEX-N-30"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-1400)"><g data-mml-node="mtd"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-39-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJX-39-TEX-N-31"></use></g></g><g data-mml-node="mtd" transform="translate(2278,0)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-39-TEX-N-31"></use></g></g><g data-mml-node="mtd" transform="translate(4167,0)"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-39-TEX-N-30"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-2800)"><g data-mml-node="mtd"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-39-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJX-39-TEX-N-31"></use></g></g><g data-mml-node="mtd" transform="translate(2278,0)"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-39-TEX-N-30"></use></g></g><g data-mml-node="mtd" transform="translate(4167,0)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-39-TEX-N-31"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-4200)"><g data-mml-node="mtd" transform="translate(389,0)"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-39-TEX-N-30"></use></g></g><g data-mml-node="mtd" transform="translate(2278,0)"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-39-TEX-N-30"></use></g></g><g data-mml-node="mtd" transform="translate(3778,0)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-39-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJX-39-TEX-N-31"></use></g></g></g></g><g data-mml-node="mo" transform="translate(5723,0)"><use data-c="23A4" xlink:href="#MJX-39-TEX-S4-23A4" transform="translate(0,3796)"></use><use data-c="23A6" xlink:href="#MJX-39-TEX-S4-23A6" transform="translate(0,-3806)"></use><svg width="667" height="6002" y="-2751" x="0" viewBox="0 1500.5 667 6002"><use data-c="23A5" xlink:href="#MJX-39-TEX-S4-23A5" transform="scale(1,14.955)"></use></svg></g></g></g></g></svg></mjx-container>
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<p>每一行只表达一条观测边对未知点高程的影响。例如,第五条边从 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.885ex" role="img" focusable="false" viewBox="0 -683 1078.6 833" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-40-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-40-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-40-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-40-TEX-N-31"></use></g></g></g></g></svg></mjx-container> 指向 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.885ex" role="img" focusable="false" viewBox="0 -683 1078.6 833" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-41-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-41-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-41-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-41-TEX-N-32"></use></g></g></g></g></svg></mjx-container>,所以该行是 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="8.926ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3945.3 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-42-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-42-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-42-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-42-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-42-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-42-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-42-TEX-N-28"></use></g><g data-mml-node="mo" transform="translate(389,0)"><use data-c="2212" xlink:href="#MJX-42-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1167,0)"><use data-c="31" xlink:href="#MJX-42-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(1667,0)"><use data-c="2C" xlink:href="#MJX-42-TEX-N-2C"></use></g><g data-mml-node="mn" transform="translate(2111.7,0)"><use data-c="31" xlink:href="#MJX-42-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(2611.7,0)"><use data-c="2C" xlink:href="#MJX-42-TEX-N-2C"></use></g><g data-mml-node="mn" transform="translate(3056.3,0)"><use data-c="30" xlink:href="#MJX-42-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(3556.3,0)"><use data-c="29" xlink:href="#MJX-42-TEX-N-29"></use></g></g></g></svg></mjx-container>;第七条边从 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.92ex" role="img" focusable="false" viewBox="0 -683 1078.6 848.6" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-43-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-43-TEX-N-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-43-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="33" xlink:href="#MJX-43-TEX-N-33"></use></g></g></g></g></svg></mjx-container> 指向已知点 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 759 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-44-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D435" xlink:href="#MJX-44-TEX-I-1D435"></use></g></g></g></svg></mjx-container>,因而只保留 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.44ex" height="1.92ex" role="img" focusable="false" viewBox="0 -683 1078.6 848.6" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-45-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-45-TEX-N-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-45-TEX-I-1D443"></use></g><g data-mml-node="mn" transform="translate(675,-150) scale(0.707)"><use data-c="33" xlink:href="#MJX-45-TEX-N-33"></use></g></g></g></g></svg></mjx-container> 的系数 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="2.891ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 1278 748" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-46-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-46-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-46-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJX-46-TEX-N-31"></use></g></g></g></svg></mjx-container>。</p>
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<h2 id="解算后还能得到什么">解算后还能得到什么</h2>
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<p>求出 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.928ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 852 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-47-TEX-I-1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D44B" xlink:href="#MJX-47-TEX-I-1D44B"></use></g></g></g></svg></mjx-container> 后,程序还应继续完成结果回代,而不是只停在一个参数向量:</p>
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<mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="12.669ex" height="2.563ex" role="img" focusable="false" viewBox="0 -1051 5599.8 1133" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-48-TEX-I-1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path id="MJX-48-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path><path id="MJX-48-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-48-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-48-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D44B" xlink:href="#MJX-48-TEX-I-1D44B"></use></g><g data-mml-node="mo" transform="translate(515.3,257) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-48-TEX-N-5E"></use></g></g></g><g data-mml-node="mo" transform="translate(1129.8,0)"><use data-c="3D" xlink:href="#MJX-48-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(2185.6,0)"><g data-mml-node="mi"><use data-c="1D44B" xlink:href="#MJX-48-TEX-I-1D44B"></use></g><g data-mml-node="mn" transform="translate(936.2,413) scale(0.707)"><use data-c="30" xlink:href="#MJX-48-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(3747.5,0)"><use data-c="2B" xlink:href="#MJX-48-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(4747.8,0)"><use data-c="1D44B" xlink:href="#MJX-48-TEX-I-1D44B"></use></g></g></g></svg></mjx-container>
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<mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="11.842ex" height="1.756ex" role="img" focusable="false" viewBox="0 -694 5234 776" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-49-TEX-I-1D449" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path id="MJX-49-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-49-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path><path id="MJX-49-TEX-I-1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path id="MJX-49-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-49-TEX-I-1D459" d="M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D449" xlink:href="#MJX-49-TEX-I-1D449"></use></g><g data-mml-node="mo" transform="translate(1046.8,0)"><use data-c="3D" xlink:href="#MJX-49-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(2102.6,0)"><use data-c="1D435" xlink:href="#MJX-49-TEX-I-1D435"></use></g><g data-mml-node="mi" transform="translate(2861.6,0)"><use data-c="1D44B" xlink:href="#MJX-49-TEX-I-1D44B"></use></g><g data-mml-node="mo" transform="translate(3935.8,0)"><use data-c="2212" xlink:href="#MJX-49-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(4936,0)"><use data-c="1D459" xlink:href="#MJX-49-TEX-I-1D459"></use></g></g></g></svg></mjx-container>
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<p>这三步分别给出未知点平差值、观测改正数和观测值平差值。对于本例,最终高程为:</p>
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<p>对应的改正数很小:</p>
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<table><thead><tr><th>边号</th><th align="right">改正数 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.097ex" height="1.027ex" role="img" focusable="false" viewBox="0 -443 485 454" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-52-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D463" xlink:href="#MJX-52-TEX-I-1D463"></use></g></g></g></svg></mjx-container> / m</th><th align="right">平差后高差 / m</th></tr></thead><tbody><tr><td>1</td><td align="right">-0.000243</td><td align="right">1.358757</td></tr><tr><td>2</td><td align="right">0.002855</td><td align="right">2.011855</td></tr><tr><td>3</td><td align="right">-0.004243</td><td align="right">0.358757</td></tr><tr><td>4</td><td align="right">-0.000145</td><td align="right">1.011855</td></tr><tr><td>5</td><td align="right">-0.003902</td><td align="right">0.653098</td></tr><tr><td>6</td><td align="right">-0.000615</td><td align="right">0.237385</td></tr><tr><td>7</td><td align="right">-0.001142</td><td align="right">-0.596142</td></tr></tbody></table>
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<p>精度评定也必须伴随结果一起输出。若观测数为 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.357ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 600 453" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-53-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-53-TEX-I-1D45B"></use></g></g></g></svg></mjx-container>、未知参数数为 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="0.817ex" height="1.441ex" role="img" focusable="false" viewBox="0 -626 361 637" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-54-TEX-I-1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-54-TEX-I-1D461"></use></g></g></g></svg></mjx-container>,则多余观测数为:</p>
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<mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="8.977ex" height="1.602ex" role="img" focusable="false" viewBox="0 -626 3968 708" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-55-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-55-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-55-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-55-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-55-TEX-I-1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-55-TEX-I-1D45F"></use></g><g data-mml-node="mo" transform="translate(728.8,0)"><use data-c="3D" xlink:href="#MJX-55-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(1784.6,0)"><use data-c="1D45B" xlink:href="#MJX-55-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(2606.8,0)"><use data-c="2212" xlink:href="#MJX-55-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(3607,0)"><use data-c="1D461" xlink:href="#MJX-55-TEX-I-1D461"></use></g></g></g></svg></mjx-container>
|
||
<p>单位权中误差为:</p>
|
||
<mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -2.369ex;" xmlns="http://www.w3.org/2000/svg" width="15.227ex" height="6.923ex" role="img" focusable="false" viewBox="0 -2013.1 6730.2 3060" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-56-TEX-I-1D70E" d="M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z"></path><path id="MJX-56-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path><path id="MJX-56-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-56-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-56-TEX-S4-221A" d="M983 1739Q988 1750 1001 1750Q1008 1750 1013 1745T1020 1733Q1020 1726 742 244T460 -1241Q458 -1250 439 -1250H436Q424 -1250 424 -1248L410 -1166Q395 -1083 367 -920T312 -601L201 44L137 -83L111 -57L187 96L264 247Q265 246 369 -357Q470 -958 473 -963L727 384Q979 1729 983 1739Z"></path><path id="MJX-56-TEX-I-1D449" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path id="MJX-56-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-56-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-56-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-56-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-56-TEX-I-1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D70E" xlink:href="#MJX-56-TEX-I-1D70E"></use></g><g data-mml-node="mo" transform="translate(285.5,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-56-TEX-N-5E"></use></g></g></g><g data-mml-node="mn" transform="translate(604,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-56-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(1285.3,0)"><use data-c="3D" xlink:href="#MJX-56-TEX-N-3D"></use></g><g data-mml-node="msqrt" transform="translate(2341.1,0)"><g transform="translate(1020,0)"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D449" xlink:href="#MJX-56-TEX-I-1D449"></use></g><g data-mml-node="mi" transform="translate(861.3,289) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-56-TEX-I-1D447"></use></g></g><g data-mml-node="mi" transform="translate(1409.1,0)"><use data-c="1D443" xlink:href="#MJX-56-TEX-I-1D443"></use></g><g data-mml-node="mi" transform="translate(2160.1,0)"><use data-c="1D449" xlink:href="#MJX-56-TEX-I-1D449"></use></g></g><g data-mml-node="mrow" transform="translate(592.8,-686)"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-56-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(822.2,0)"><use data-c="2212" xlink:href="#MJX-56-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(1822.4,0)"><use data-c="1D461" xlink:href="#MJX-56-TEX-I-1D461"></use></g></g><rect width="3129.1" height="60" x="120" y="220"></rect></g></g><g data-mml-node="mo" transform="translate(0,203.1)"><use data-c="221A" xlink:href="#MJX-56-TEX-S4-221A"></use></g><rect width="3369.1" height="60" x="1020" y="1893.1"></rect></g></g></g></svg></mjx-container>
|
||
<p>未知参数的协因数阵与方差阵为:</p>
|
||
<mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.881ex;" xmlns="http://www.w3.org/2000/svg" width="18.161ex" height="2.899ex" role="img" focusable="false" viewBox="0 -891.7 8027 1281.3" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-57-TEX-I-1D444" d="M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z"></path><path id="MJX-57-TEX-I-1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path id="MJX-57-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path><path id="MJX-57-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-57-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-57-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path><path id="MJX-57-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-57-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-57-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-57-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 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<mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.881ex;" xmlns="http://www.w3.org/2000/svg" width="14.787ex" height="2.881ex" role="img" focusable="false" viewBox="0 -883.9 6535.9 1273.5" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-58-TEX-I-1D437" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z"></path><path id="MJX-58-TEX-I-1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path id="MJX-58-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path><path id="MJX-58-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-58-TEX-I-1D70E" d="M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z"></path><path id="MJX-58-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-58-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-58-TEX-I-1D444" d="M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D437" xlink:href="#MJX-58-TEX-I-1D437"></use></g><g data-mml-node="TeXAtom" transform="translate(861,-389.6) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D44B" xlink:href="#MJX-58-TEX-I-1D44B"></use></g><g data-mml-node="mo" transform="translate(515.3,257) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-58-TEX-N-5E"></use></g></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(852,0)"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D44B" 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<p>本例计算出的单位权中误差约为 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="10.998ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 4861 688" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-59-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-59-TEX-N-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path id="MJX-59-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-59-TEX-N-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path id="MJX-59-TEX-N-A0" d=""></path><path id="MJX-59-TEX-N-6D" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-59-TEX-N-30"></use><use data-c="2E" xlink:href="#MJX-59-TEX-N-2E" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-59-TEX-N-30" transform="translate(778,0)"></use><use data-c="30" xlink:href="#MJX-59-TEX-N-30" transform="translate(1278,0)"></use><use data-c="32" xlink:href="#MJX-59-TEX-N-32" transform="translate(1778,0)"></use><use data-c="32" xlink:href="#MJX-59-TEX-N-32" transform="translate(2278,0)"></use><use data-c="32" xlink:href="#MJX-59-TEX-N-32" transform="translate(2778,0)"></use><use data-c="35" xlink:href="#MJX-59-TEX-N-35" transform="translate(3278,0)"></use></g><g data-mml-node="mtext" transform="translate(3778,0)"><use data-c="A0" xlink:href="#MJX-59-TEX-N-A0"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(4028,0)"><g data-mml-node="mi"><use data-c="6D" xlink:href="#MJX-59-TEX-N-6D"></use></g></g></g></g></svg></mjx-container>。它不是某一个点的误差,而是这套观测在单位权意义下的整体精度指标。</p>
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<h2 id="程序实现让流程可操作而不只是可计算">程序实现:让流程可操作,而不只是可计算</h2>
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<p>将平差公式写成程序时,我把责任分开,而不是把所有逻辑堆在窗体事件里:</p>
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<table><thead><tr><th>模块</th><th>责任</th></tr></thead><tbody><tr><td><code>KnownPoint</code>、<code>Observation</code></td><td>保存已知点、观测边与距离等输入数据</td></tr><tr><td><code>DataCheck</code></td><td>检查空点名、重复点名、高差格式、正距离、与已知点的连通性、观测数量等</td></tr><tr><td><code>MatrixOperations</code></td><td>用二维数组实现加减乘、转置和求逆</td></tr><tr><td><code>LeastSquares</code></td><td>生成 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 759 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-60-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D435" xlink:href="#MJX-60-TEX-I-1D435"></use></g></g></g></svg></mjx-container>、<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.699ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 751 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-61-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-61-TEX-I-1D443"></use></g></g></g></svg></mjx-container>、<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="0.674ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 298 705" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-62-TEX-I-1D459" d="M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D459" xlink:href="#MJX-62-TEX-I-1D459"></use></g></g></g></svg></mjx-container>,解法方程并回代计算</td></tr><tr><td><code>DataStore</code></td><td>暂存已导入或手动录入的数据</td></tr><tr><td><code>Report</code></td><td>用 <code>StringBuilder</code> 组织平差报告,供预览或导出</td></tr><tr><td><code>LogService</code></td><td>记录导入、检查、计算与输出过程</td></tr></tbody></table>
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<p>整个交互流程可以概括为:</p>
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<pre class="astro-code github-dark" style="background-color:#24292e;color:#e1e4e8; overflow-x: auto; white-space: pre-wrap; word-wrap: break-word;" tabindex="0" data-language="text"><code><span class="line"><span>导入/手动输入 → 数据检查 → 保存数据 → 平差计算 → 查看结果 → 输出报告</span></span>
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<span class="line"><span> ↑</span></span>
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<span class="line"><span> 检查失败则提示并返回输入</span></span></code></pre>
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<p>这里数据检查并非附属功能。若水准网中存在孤立点、观测边没有连到已知点,或者距离为零,后续矩阵即使“算出结果”也没有测量意义。因此,先验证网络可用性,再进入最小二乘计算,是程序层面对应理论前提的一道关口。</p>
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<h2 id="小结">小结</h2>
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<p>这个案例把水准网间接平差压缩成了一条清晰的链路:观测数据决定权阵与常数项,网络拓扑决定系数矩阵,最小二乘条件导出法方程,回代和精度评定则让结果从“一个答案”变成可解释的测量成果。</p>
|
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<p>对我来说,自己实现矩阵运算和法方程求解的价值不在于替代现成库,而在于真正看清 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="6.448ex" height="1.904ex" role="img" focusable="false" viewBox="0 -841.7 2849.8 841.7" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-63-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path><path 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667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-64-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 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</style></div><section class="post-copyright-card" aria-label="文章版权信息"><div class="post-copyright-mark" aria-hidden="true"><span>CC</span></div><div class="post-copyright-main"><h2>从间接平差到程序:一个水准网最小二乘解算案例</h2><a class="post-copyright-link" href="/posts/e67166f3/">/posts/e67166f3/</a><dl class="post-copyright-meta"><div><dt>作者</dt><dd>biss</dd></div><div><dt>发布于</dt><dd><time datetime="2026-08-27T21:00:00.000Z">2026/08/28</time></dd></div><div><dt>许可协议</dt><dd><a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" target="_blank" rel="noreferrer">CC BY-NC-SA 4.0</a></dd></div></dl></div><div class="post-copyright-footer"><div class="post-copyright-tags" aria-label="文章标签"><a href="/tags/%E6%B5%8B%E9%87%8F%E5%B9%B3%E5%B7%AE/">测量平差</a><a href="/tags/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98/">最小二乘</a><a href="/tags/C%23/">C#</a><a href="/tags/%E7%9F%A9%E9%98%B5/">矩阵</a></div><div class="post-copyright-actions" aria-label="文章操作"><button class="post-copyright-action is-primary post-share-button" type="button" data-share-title="从间接平差到程序:一个水准网最小二乘解算案例" data-share-url="https://blog.biss.click/posts/e67166f3/" title="分享文章"><svg aria-hidden="true" viewBox="0 0 24 24"><path d="M4 12v7a1 1 0 0 0 1 1h14a1 1 0 0 0 1-1v-7"></path><path d="M12 16V4"></path><path d="m7 9 5-5 5 5"></path></svg><span>分享</span></button><a class="post-copyright-action" href="/rss.xml" title="订阅 RSS"><svg aria-hidden="true" viewBox="0 0 24 24"><path d="M5 5a14 14 0 0 1 14 14"></path><path d="M5 11a8 8 0 0 1 8 8"></path><path d="M6 18h.01"></path></svg><span>订阅</span></a><a class="post-copyright-action" href="/about/" title="打赏作者"><svg aria-hidden="true" viewBox="0 0 24 24"><path d="M7 8h10"></path><path d="M8 8V5h8v3"></path><path d="M6 8h12l-1.5 11h-9L6 8Z"></path><path d="M10 12h4"></path></svg><span>打赏</span></a><a class="post-copyright-icon" href="https://twitter.com/intent/tweet?url=https%3A%2F%2Fblog.biss.click%2Fposts%2Fe67166f3%2F&text=%E4%BB%8E%E9%97%B4%E6%8E%A5%E5%B9%B3%E5%B7%AE%E5%88%B0%E7%A8%8B%E5%BA%8F%EF%BC%9A%E4%B8%80%E4%B8%AA%E6%B0%B4%E5%87%86%E7%BD%91%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98%E8%A7%A3%E7%AE%97%E6%A1%88%E4%BE%8B" target="_blank" rel="noreferrer" title="分享到 X"><span aria-hidden="true">x</span></a><a class="post-copyright-icon" href="https://service.weibo.com/share/share.php?url=https%3A%2F%2Fblog.biss.click%2Fposts%2Fe67166f3%2F&title=%E4%BB%8E%E9%97%B4%E6%8E%A5%E5%B9%B3%E5%B7%AE%E5%88%B0%E7%A8%8B%E5%BA%8F%EF%BC%9A%E4%B8%80%E4%B8%AA%E6%B0%B4%E5%87%86%E7%BD%91%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98%E8%A7%A3%E7%AE%97%E6%A1%88%E4%BE%8B" target="_blank" rel="noreferrer" title="分享到微博"><span aria-hidden="true">微</span></a></div></div></section><script>
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|
||
closeMenu();
|
||
|
||
if (action === 'back') history.back();
|
||
if (action === 'forward') history.forward();
|
||
if (action === 'reload') location.reload();
|
||
if (action === 'copy-selection' && lastSelectionText) copyText(lastSelectionText);
|
||
if (action === 'search-selection') searchSelection();
|
||
if (action === 'quote-selection') quoteToComments();
|
||
if (action === 'copy-image') copyImage();
|
||
if (action === 'open-image') openImage();
|
||
if (action === 'copy-image-url') copyText(new URL(getImageUrl(), window.location.href).toString());
|
||
if (action === 'random-post' && typeof window.randomPost === 'function') window.randomPost(actionTarget);
|
||
if (action === 'comments') jumpToComments();
|
||
if (action === 'print') window.print();
|
||
});
|
||
|
||
document.addEventListener('keydown', (event) => {
|
||
if (event.key === 'Escape') closeMenu();
|
||
if (event.key === 'Tab' && menu.classList.contains('is-open')) {
|
||
const items = [...menu.querySelectorAll(focusableSelector)].filter((item) => !item.closest('[hidden]'));
|
||
if (items.length === 0) return;
|
||
const first = items[0];
|
||
const last = items[items.length - 1];
|
||
|
||
if (event.shiftKey && document.activeElement === first) {
|
||
event.preventDefault();
|
||
last.focus();
|
||
} else if (!event.shiftKey && document.activeElement === last) {
|
||
event.preventDefault();
|
||
first.focus();
|
||
}
|
||
}
|
||
});
|
||
|
||
window.addEventListener('scroll', closeMenu, { passive: true });
|
||
window.addEventListener('resize', closeMenu);
|
||
})();
|
||
</script><aside class="pwa-toast" data-pwa-toast hidden role="status" aria-live="polite"><div class="pwa-toast-icon" aria-hidden="true"><i class="fa-solid fa-download"></i></div><div class="pwa-toast-copy"><strong data-pwa-title>PWA 提示</strong><p data-pwa-message></p></div><div class="pwa-toast-actions"><button class="pwa-toast-primary" type="button" data-pwa-primary>确定</button><button class="pwa-toast-close" type="button" data-pwa-close aria-label="关闭提示">稍后</button></div></aside><script src="https://cdn.jsdmirror.com/npm/instantsearch.js@4.56.0"></script><script src="https://cdn.jsdmirror.com/npm/typesense-instantsearch-adapter@2.7.0/dist/typesense-instantsearch-adapter.min.js"></script><script src="https://cdn.jsdmirror.com/npm/@fancyapps/ui@5.0/dist/fancybox/fancybox.umd.js"></script><script src="/js/random.js"></script><script src="/js/search/typesense-search.js"></script><script src="/js/external-links.js"></script><script src="/js/pwa.js"></script><script>
|
||
(() => {
|
||
const nav = document.querySelector('.nav');
|
||
const toggle = document.querySelector('[data-nav-toggle]');
|
||
const menu = document.querySelector('[data-nav-menu]');
|
||
const backdrop = document.querySelector('[data-nav-backdrop]');
|
||
if (!nav || !toggle || !menu) return;
|
||
|
||
function setMenu(open) {
|
||
nav.classList.toggle('is-open', open);
|
||
toggle.setAttribute('aria-expanded', String(open));
|
||
toggle.setAttribute('aria-label', open ? '关闭菜单' : '打开菜单');
|
||
menu.setAttribute('aria-hidden', String(!open && window.innerWidth <= 720));
|
||
document.body.classList.toggle('has-nav-drawer', open);
|
||
}
|
||
|
||
toggle.addEventListener('click', () => {
|
||
setMenu(!nav.classList.contains('is-open'));
|
||
});
|
||
|
||
backdrop?.addEventListener('click', () => setMenu(false));
|
||
|
||
menu.addEventListener('click', (event) => {
|
||
if (event.target instanceof Element && event.target.closest('a')) setMenu(false);
|
||
});
|
||
|
||
document.addEventListener('click', (event) => {
|
||
if (!nav.classList.contains('is-open')) return;
|
||
if (event.target instanceof Node && nav.contains(event.target)) return;
|
||
setMenu(false);
|
||
});
|
||
|
||
document.addEventListener('keydown', (event) => {
|
||
if (event.key === 'Escape' && nav.classList.contains('is-open')) setMenu(false);
|
||
});
|
||
|
||
window.addEventListener('resize', () => {
|
||
if (window.innerWidth > 720) setMenu(false);
|
||
else menu.setAttribute('aria-hidden', String(!nav.classList.contains('is-open')));
|
||
});
|
||
|
||
menu.setAttribute('aria-hidden', String(window.innerWidth <= 720));
|
||
})();
|
||
|
||
(() => {
|
||
const modal = document.querySelector('[data-console-modal]');
|
||
const panel = modal?.querySelector('.control-console-panel');
|
||
const openButtons = document.querySelectorAll('[data-console-open]');
|
||
const closeButtons = modal?.querySelectorAll('[data-console-close]') || [];
|
||
if (!modal || !panel || openButtons.length === 0) return;
|
||
|
||
function setConsole(open) {
|
||
modal.classList.toggle('is-open', open);
|
||
modal.setAttribute('aria-hidden', String(!open));
|
||
document.body.classList.toggle('has-control-console', open);
|
||
if (open) panel.focus();
|
||
}
|
||
|
||
openButtons.forEach((button) => {
|
||
button.addEventListener('click', () => setConsole(true));
|
||
});
|
||
|
||
closeButtons.forEach((button) => {
|
||
button.addEventListener('click', () => setConsole(false));
|
||
});
|
||
|
||
document.addEventListener('keydown', (event) => {
|
||
if (event.key === 'Escape' && modal.classList.contains('is-open')) setConsole(false);
|
||
});
|
||
})();
|
||
|
||
(() => {
|
||
const storageKey = 'site-theme';
|
||
const root = document.documentElement;
|
||
const button = document.querySelector('[data-theme-toggle]');
|
||
const media = window.matchMedia('(prefers-color-scheme: dark)');
|
||
const getSystemTheme = () => (media.matches ? 'dark' : 'light');
|
||
const getSavedTheme = () => {
|
||
try {
|
||
const theme = localStorage.getItem(storageKey);
|
||
return theme === 'dark' || theme === 'light' ? theme : null;
|
||
} catch {
|
||
return null;
|
||
}
|
||
};
|
||
const setSavedTheme = (theme) => {
|
||
try {
|
||
if (theme === getSystemTheme()) localStorage.removeItem(storageKey);
|
||
else localStorage.setItem(storageKey, theme);
|
||
} catch {
|
||
// localStorage can be unavailable in strict privacy modes.
|
||
}
|
||
};
|
||
const getTheme = () => getSavedTheme() || getSystemTheme();
|
||
|
||
function applyTheme(theme) {
|
||
root.dataset.theme = theme;
|
||
root.dataset.themeMode = getSavedTheme() ? 'manual' : 'auto';
|
||
button?.setAttribute('aria-label', theme === 'dark' ? '切换到日间模式' : '切换到夜间模式');
|
||
button?.setAttribute(
|
||
'title',
|
||
getSavedTheme()
|
||
? (theme === 'dark' ? '切换到日间模式' : '切换到夜间模式')
|
||
: `跟随系统:${theme === 'dark' ? '夜间' : '日间'}模式`
|
||
);
|
||
}
|
||
|
||
applyTheme(getTheme());
|
||
|
||
button?.addEventListener('click', () => {
|
||
const nextTheme = root.dataset.theme === 'dark' ? 'light' : 'dark';
|
||
setSavedTheme(nextTheme);
|
||
applyTheme(nextTheme);
|
||
});
|
||
|
||
const handleSystemThemeChange = () => {
|
||
if (!getSavedTheme()) applyTheme(getSystemTheme());
|
||
};
|
||
|
||
if (typeof media.addEventListener === 'function') {
|
||
media.addEventListener('change', handleSystemThemeChange);
|
||
} else if (typeof media.addListener === 'function') {
|
||
media.addListener(handleSystemThemeChange);
|
||
}
|
||
})();
|
||
|
||
(() => {
|
||
const runtime = document.getElementById('site-runtime');
|
||
if (!runtime) return;
|
||
|
||
const start = new Date(window.SITE_CONFIG.site.startTime).getTime();
|
||
const pad = (value) => String(value).padStart(2, '0');
|
||
|
||
function updateRuntime() {
|
||
const diff = Math.max(0, Date.now() - start);
|
||
const days = Math.floor(diff / 86400000);
|
||
const hours = Math.floor((diff / 3600000) % 24);
|
||
const minutes = Math.floor((diff / 60000) % 60);
|
||
const seconds = Math.floor((diff / 1000) % 60);
|
||
runtime.textContent = `${days} 天 ${pad(hours)} 时 ${pad(minutes)} 分 ${pad(seconds)} 秒`;
|
||
}
|
||
|
||
updateRuntime();
|
||
window.setInterval(updateRuntime, 1000);
|
||
})();
|
||
</script></body></html><script>(function(){const postSlug = "e67166f3";
|
||
|
||
(() => {
|
||
const article = document.querySelector('.post-article');
|
||
const progressBar = document.querySelector('[data-reading-progress]');
|
||
const resume = document.querySelector('[data-reading-resume]');
|
||
const resumePercent = document.querySelector('[data-reading-resume-percent]');
|
||
const continueButton = document.querySelector('[data-reading-continue]');
|
||
const restartButton = document.querySelector('[data-reading-restart]');
|
||
if (!article || !progressBar || !resume) return;
|
||
|
||
const storageKey = `reading-progress:${postSlug}`;
|
||
let saved = null;
|
||
let canPersist = true;
|
||
let frame = 0;
|
||
|
||
try {
|
||
saved = JSON.parse(localStorage.getItem(storageKey) || 'null');
|
||
} catch {
|
||
saved = null;
|
||
}
|
||
|
||
const articleMetrics = () => {
|
||
const top = article.getBoundingClientRect().top + window.scrollY;
|
||
const distance = Math.max(1, article.offsetHeight - window.innerHeight);
|
||
return { top, distance };
|
||
};
|
||
|
||
const getProgress = () => {
|
||
const { top, distance } = articleMetrics();
|
||
return Math.min(1, Math.max(0, (window.scrollY - top) / distance));
|
||
};
|
||
|
||
const persist = (progress) => {
|
||
if (!canPersist) return;
|
||
try {
|
||
if (progress >= 0.97) localStorage.removeItem(storageKey);
|
||
else if (progress >= 0.02) localStorage.setItem(storageKey, JSON.stringify({ progress, savedAt: Date.now() }));
|
||
} catch {
|
||
// Storage can be unavailable in strict privacy modes.
|
||
}
|
||
};
|
||
|
||
const update = () => {
|
||
frame = 0;
|
||
const progress = getProgress();
|
||
progressBar.style.transform = `scaleX(${progress})`;
|
||
persist(progress);
|
||
};
|
||
|
||
const requestUpdate = () => {
|
||
if (!frame) frame = window.requestAnimationFrame(update);
|
||
};
|
||
|
||
const closeResume = () => {
|
||
resume.hidden = true;
|
||
canPersist = true;
|
||
};
|
||
|
||
const savedIsFresh = saved && Date.now() - Number(saved.savedAt) < 90 * 86400000;
|
||
const savedProgress = Number(saved?.progress);
|
||
if (savedIsFresh && savedProgress >= 0.05 && savedProgress < 0.97) {
|
||
canPersist = false;
|
||
if (resumePercent) resumePercent.textContent = `${Math.round(savedProgress * 100)}%`;
|
||
window.setTimeout(() => { resume.hidden = false; }, 350);
|
||
}
|
||
|
||
continueButton?.addEventListener('click', () => {
|
||
closeResume();
|
||
const { top, distance } = articleMetrics();
|
||
window.scrollTo({ top: top + distance * savedProgress, behavior: 'smooth' });
|
||
});
|
||
|
||
restartButton?.addEventListener('click', () => {
|
||
try { localStorage.removeItem(storageKey); } catch {}
|
||
closeResume();
|
||
window.scrollTo({ top: Math.max(0, articleMetrics().top - 90), behavior: 'smooth' });
|
||
});
|
||
|
||
update();
|
||
window.addEventListener('scroll', requestUpdate, { passive: true });
|
||
window.addEventListener('resize', requestUpdate);
|
||
window.addEventListener('pagehide', () => persist(getProgress()));
|
||
})();
|
||
})();</script><script>
|
||
(() => {
|
||
document.querySelectorAll('.article-outdated-note[data-freshness-at]').forEach((note) => {
|
||
const freshnessAt = new Date(note.dataset.freshnessAt || '');
|
||
const hasNewVersion = note.dataset.hasNewVersion === 'true';
|
||
if (Number.isNaN(freshnessAt.valueOf())) return;
|
||
|
||
const threshold = new Date();
|
||
threshold.setFullYear(threshold.getFullYear() - 1);
|
||
note.hidden = !hasNewVersion && freshnessAt >= threshold;
|
||
});
|
||
})();
|
||
</script><script>
|
||
(() => {
|
||
const prose = document.querySelector('.prose[data-toc]');
|
||
if (!prose) return;
|
||
|
||
let toc = [];
|
||
try {
|
||
toc = JSON.parse(prose.dataset.toc || '[]');
|
||
} catch {
|
||
toc = [];
|
||
}
|
||
|
||
prose.querySelectorAll('h1, h2, h3').forEach((heading, index) => {
|
||
const item = toc[index];
|
||
if (item?.id && !heading.id) heading.id = item.id;
|
||
});
|
||
|
||
const tocTree = document.querySelector('[data-toc-tree]');
|
||
if (tocTree && toc.length > 0) {
|
||
const links = [...tocTree.querySelectorAll('[data-toc-link]')];
|
||
const groups = [...tocTree.querySelectorAll('[data-toc-group]')];
|
||
const headings = toc
|
||
.map((item) => document.getElementById(item.id))
|
||
.filter(Boolean);
|
||
|
||
const setActiveToc = (id) => {
|
||
links.forEach((link) => {
|
||
const isActive = link.getAttribute('href') === `#${id}`;
|
||
link.classList.toggle('is-active', isActive);
|
||
if (isActive) link.setAttribute('aria-current', 'location');
|
||
else link.removeAttribute('aria-current');
|
||
});
|
||
|
||
const activeLink = tocTree.querySelector(`[data-toc-link][href="#${CSS.escape(id)}"]`);
|
||
const activeGroup = activeLink?.closest('[data-toc-group]');
|
||
groups.forEach((group) => group.classList.toggle('is-open', group === activeGroup));
|
||
};
|
||
|
||
const updateToc = () => {
|
||
const anchorOffset = 132;
|
||
let activeId = headings[0]?.id;
|
||
|
||
headings.forEach((heading) => {
|
||
if (heading.getBoundingClientRect().top <= anchorOffset) activeId = heading.id;
|
||
});
|
||
|
||
if (activeId) setActiveToc(activeId);
|
||
};
|
||
|
||
tocTree.addEventListener('click', (event) => {
|
||
const link = event.target.closest?.('[data-toc-link]');
|
||
if (!link) return;
|
||
|
||
const group = link.closest('[data-toc-group]');
|
||
groups.forEach((item) => item.classList.toggle('is-open', item === group));
|
||
});
|
||
|
||
updateToc();
|
||
window.addEventListener('scroll', updateToc, { passive: true });
|
||
window.addEventListener('resize', updateToc);
|
||
}
|
||
|
||
})();
|
||
</script><script>
|
||
(() => {
|
||
const prose = document.querySelector('.post-article .prose');
|
||
if (!prose) return;
|
||
|
||
const collapsedLineCount = 8;
|
||
const copyLabels = {
|
||
idle: '复制',
|
||
done: '已复制',
|
||
error: '复制失败',
|
||
};
|
||
|
||
const languageNames = {
|
||
bash: 'Bash',
|
||
c: 'C',
|
||
cpp: 'C++',
|
||
cs: 'C#',
|
||
css: 'CSS',
|
||
html: 'HTML',
|
||
js: 'JavaScript',
|
||
json: 'JSON',
|
||
jsx: 'JSX',
|
||
md: 'Markdown',
|
||
powershell: 'PowerShell',
|
||
py: 'Python',
|
||
shell: 'Shell',
|
||
sh: 'Shell',
|
||
ts: 'TypeScript',
|
||
tsx: 'TSX',
|
||
txt: 'Text',
|
||
yaml: 'YAML',
|
||
yml: 'YAML',
|
||
};
|
||
|
||
function getCodeLanguage(pre, code) {
|
||
const rawLanguage =
|
||
pre.dataset.language ||
|
||
code.dataset.language ||
|
||
[...pre.classList, ...code.classList]
|
||
.find((className) => className.startsWith('language-'))
|
||
?.replace(/^language-/, '') ||
|
||
'';
|
||
|
||
const normalized = rawLanguage.trim().toLowerCase();
|
||
return languageNames[normalized] || normalized.toUpperCase() || 'Code';
|
||
}
|
||
|
||
function getCodeText(code, lines) {
|
||
if (lines.length > 0) return lines.map((line) => line.textContent || '').join('\n');
|
||
return code.textContent || '';
|
||
}
|
||
|
||
prose.querySelectorAll('img').forEach((image) => {
|
||
if (image.closest('a, .article-header-cover')) return;
|
||
|
||
const src = image.currentSrc || image.getAttribute('src');
|
||
if (!src) return;
|
||
|
||
const link = document.createElement('a');
|
||
link.className = 'fancybox';
|
||
link.href = src;
|
||
link.dataset.fancybox = 'article-gallery';
|
||
link.dataset.caption = image.getAttribute('alt') || '';
|
||
image.parentNode?.insertBefore(link, image);
|
||
link.appendChild(image);
|
||
});
|
||
|
||
prose.querySelectorAll('table').forEach((table) => {
|
||
if (table.closest('.table-scroll')) return;
|
||
|
||
const wrapper = document.createElement('div');
|
||
wrapper.className = 'table-scroll';
|
||
table.parentNode?.insertBefore(wrapper, table);
|
||
wrapper.appendChild(table);
|
||
});
|
||
|
||
prose.querySelectorAll('.butterfly-hide-inline').forEach((item) => {
|
||
const button = item.querySelector('.butterfly-hide-toggle');
|
||
if (!button) return;
|
||
|
||
button.addEventListener('click', () => {
|
||
item.classList.toggle('is-open');
|
||
});
|
||
});
|
||
|
||
if (window.Fancybox) {
|
||
window.Fancybox.bind('[data-fancybox="article-gallery"]', {
|
||
compact: false,
|
||
dragToClose: true,
|
||
Images: {
|
||
zoom: true,
|
||
},
|
||
});
|
||
}
|
||
|
||
async function copyText(text) {
|
||
if (navigator.clipboard?.writeText) {
|
||
await navigator.clipboard.writeText(text);
|
||
return;
|
||
}
|
||
|
||
const textarea = document.createElement('textarea');
|
||
textarea.value = text;
|
||
textarea.setAttribute('readonly', '');
|
||
textarea.style.position = 'fixed';
|
||
textarea.style.top = '-9999px';
|
||
textarea.style.opacity = '0';
|
||
document.body.appendChild(textarea);
|
||
textarea.select();
|
||
document.execCommand('copy');
|
||
textarea.remove();
|
||
}
|
||
|
||
prose.querySelectorAll('pre').forEach((pre) => {
|
||
if (pre.closest('.code-block')) return;
|
||
|
||
const code = pre.querySelector('code');
|
||
if (!code) return;
|
||
|
||
const wrapper = document.createElement('div');
|
||
wrapper.className = 'code-block';
|
||
pre.parentNode?.insertBefore(wrapper, pre);
|
||
wrapper.appendChild(pre);
|
||
|
||
const header = document.createElement('div');
|
||
header.className = 'code-block-header';
|
||
|
||
const language = document.createElement('span');
|
||
language.className = 'code-block-language';
|
||
language.textContent = getCodeLanguage(pre, code);
|
||
|
||
const toolbar = document.createElement('div');
|
||
toolbar.className = 'code-block-toolbar';
|
||
|
||
const copyButton = document.createElement('button');
|
||
copyButton.className = 'code-block-copy';
|
||
copyButton.type = 'button';
|
||
copyButton.textContent = copyLabels.idle;
|
||
copyButton.setAttribute('aria-label', '复制代码');
|
||
|
||
const toggleButton = document.createElement('button');
|
||
toggleButton.className = 'code-block-toggle';
|
||
toggleButton.type = 'button';
|
||
toggleButton.textContent = '展开';
|
||
toggleButton.setAttribute('aria-expanded', 'false');
|
||
|
||
const expandCue = document.createElement('button');
|
||
expandCue.className = 'code-block-expand-cue';
|
||
expandCue.type = 'button';
|
||
expandCue.setAttribute('aria-label', 'Expand code block');
|
||
expandCue.setAttribute('aria-expanded', 'false');
|
||
expandCue.innerHTML = '<span aria-hidden="true"></span>';
|
||
|
||
toolbar.append(copyButton, toggleButton);
|
||
header.append(language, toolbar);
|
||
wrapper.appendChild(header);
|
||
wrapper.appendChild(expandCue);
|
||
|
||
const codeLines = [...code.querySelectorAll(':scope > .line')];
|
||
if (codeLines.length > 0) {
|
||
wrapper.classList.add('has-line-numbers');
|
||
wrapper.style.setProperty('--line-number-width', `${String(codeLines.length).length + 1}ch`);
|
||
codeLines.forEach((line, index) => {
|
||
line.dataset.line = String(index + 1);
|
||
});
|
||
}
|
||
|
||
const lineCount = codeLines.length || (code.textContent || '').split(/\r?\n/).length;
|
||
if (lineCount > collapsedLineCount) wrapper.classList.add('is-collapsible');
|
||
|
||
copyButton.addEventListener('click', async () => {
|
||
try {
|
||
await copyText(getCodeText(code, codeLines));
|
||
copyButton.textContent = copyLabels.done;
|
||
window.setTimeout(() => {
|
||
copyButton.textContent = copyLabels.idle;
|
||
}, 1400);
|
||
} catch {
|
||
copyButton.textContent = copyLabels.error;
|
||
window.setTimeout(() => {
|
||
copyButton.textContent = copyLabels.idle;
|
||
}, 1600);
|
||
}
|
||
});
|
||
|
||
function toggleExpanded() {
|
||
const isExpanded = wrapper.classList.toggle('is-expanded');
|
||
toggleButton.textContent = isExpanded ? '收起' : '展开';
|
||
toggleButton.setAttribute('aria-expanded', String(isExpanded));
|
||
expandCue.setAttribute('aria-expanded', String(isExpanded));
|
||
}
|
||
|
||
toggleButton.addEventListener('click', toggleExpanded);
|
||
expandCue.addEventListener('click', toggleExpanded);
|
||
});
|
||
})();
|
||
</script> |