測量譯文翻譯(共26頁)

1、重慶大學(xué)本科學(xué)生畢業(yè)設(shè)計(jì)（論文）附件 附件C：譯文C1附件(fjin)C：相對(duì)精度的嚴(yán)格(yng)評(píng)估作者(zuzh)：Toms Soler, M.ASCE1; and Dru Smith2文摘:這一技術(shù)論文引入了一個(gè)“相對(duì)精度”的一種嚴(yán)格計(jì)算?！跋鄬?duì)精度”一詞指的是相對(duì)誤差。在相對(duì)平面三維參考系統(tǒng)（e，n，u）中，任意兩點(diǎn)之間（兩點(diǎn)相對(duì)獨(dú)立）都可以用任意一種大地測量技術(shù)連接光學(xué)程序、全球定位系統(tǒng)等等。它利用全局笛卡爾函數(shù)，通過點(diǎn)的（x，y，z）坐標(biāo)組成的方差協(xié)方差矩陣來定義空間網(wǎng)絡(luò)。數(shù)字對(duì)象標(biāo)識(shí):10.1061/ ASCE SU.1943-5428.0000023CE數(shù)據(jù)庫主題詞:調(diào)查、數(shù)據(jù)

2、分析的準(zhǔn)確性。作者關(guān)鍵詞:土地測量;調(diào)整數(shù)據(jù);相對(duì)的精度。摘要C2相對(duì)精度的概念，盡管比較直觀，但仍沒有在數(shù)學(xué)界得到它應(yīng)有的關(guān)注。最近，Burkholder做了第一次嘗試來精確的定義網(wǎng)絡(luò)與相對(duì)精度之間的不同。網(wǎng)絡(luò)精確度測量的是坐標(biāo)與無誤差數(shù)據(jù)的接近程度。美國聯(lián)邦大地測量管制委員會(huì)和聯(lián)邦地理數(shù)據(jù)委員會(huì)1998年。在美國，參考數(shù)據(jù)是由連續(xù)運(yùn)轉(zhuǎn)參考站（即CORS）網(wǎng)絡(luò)中的坐標(biāo)定義的框架(Snay and Soler 2008)。網(wǎng)絡(luò)精度可以直接從一種三維網(wǎng)絡(luò)方案（例如CORS）中的全球標(biāo)準(zhǔn)化方差協(xié)方差矩陣（x，y，z）中獲得。這個(gè)矩陣可以從任意一種平差3D網(wǎng)絡(luò)的軟件的輸出中得到。它是由每個(gè)點(diǎn)的方差協(xié)

3、方差組成，并且不包含兩個(gè)點(diǎn)的協(xié)方差的一個(gè)3*3的滿秩對(duì)角矩陣。Burkholder（2008）得出的方法在概念上似乎是正確的，但這一理論還需要被廣泛化。這一技術(shù)摘要擴(kuò)充了先前的成果，提出了在計(jì)算中突出的方差-協(xié)方差矩陣和非對(duì)稱互協(xié)方差矩陣。盡管在笛卡爾坐標(biāo)系中當(dāng)?shù)鼐龋鳛闃?biāo)準(zhǔn)誤差向量的基本組件，可以用一些GPS商用軟件包計(jì)算得到，但這些精度通常不會(huì)轉(zhuǎn)換成當(dāng)?shù)仄矫娲蟮刈鴺?biāo)框架。如果用的話，也是用來描述不嚴(yán)密的理論。相對(duì)精度假設(shè)(jish)我們想測定（x1，y1，z1）和（x1，y1，z1）兩點(diǎn)之間的相對(duì)精度。我們知道（e.g., Soler 1976; Soler and Hothem 198

4、8; Leick 2004）在全球(qunqi)笛卡爾坐標(biāo)系統(tǒng)（e.g., geocentric）中的位置誤差可以用下面(xi mian)的公式轉(zhuǎn)換成當(dāng)?shù)卮蟮仄矫嬷担╡ast, north, up=vertical）?？臻g參考系統(tǒng)，國家大地測量，國家海洋和大氣行政機(jī)構(gòu)的首席科學(xué)家，出生于1315 East-West Hwy, Silver Spring, MD 20910，郵箱：Tom.S國家大地測量，國家海洋和大氣行政機(jī)構(gòu)的首席大地測量家，出生于1315 East-West Hwy, Silver Spring，MD 20910. 郵箱：Dru.Smithnoaa.g

5、ov這篇原稿2009年5月26日完成，2009年11月17日通過核審，2009年12月29日發(fā)表。一直到2011年1月1日都被討論，不同的討論必須屈服于這個(gè)報(bào)紙，這個(gè)技術(shù)摘要出自2010年10月1日的Journal of Surveying Engineering 第3頁136卷，編碼 ISSN 0733-9453/2010/3-120125/$25.00。其中，表示笛卡爾參考框架（xi , yi , zi） 和相對(duì)平面框架（ei ,ni ,ui）之間轉(zhuǎn)換的旋轉(zhuǎn)矩陣，具體表示為其中，表示點(diǎn)i的大地經(jīng)緯度，上面的公式與指向正東方向的大地經(jīng)線一致。我們也假定兩個(gè)點(diǎn)的全球方差協(xié)方差矩陣是已知對(duì)陣矩陣

6、其中的非對(duì)稱塊，是點(diǎn)之間的交叉矩陣，不是對(duì)稱矩陣重慶大學(xué)本科學(xué)生畢業(yè)設(shè)計(jì)（論文）附件 附件C：譯文C3方程（4）里的矩陣可以從一個(gè)相當(dāng)(xingdng)大的矩陣（幾十或幾百個(gè)點(diǎn)）中得到。兩點(diǎn)的相對(duì)精度可以用下面簡單的數(shù)學(xué)模型通過點(diǎn)2參考點(diǎn)1的相對(duì)位置的傳播誤差獲得注意的是，方程6是三維空間的普遍(pbin)普遍定義。然而，“相對(duì)(xingdu)”意味著，實(shí)際上它只能應(yīng)用在相對(duì)距離比較小的點(diǎn)之間，例如小于100千米。對(duì)于距離很大的點(diǎn)，這個(gè)定義在概念上沒有意義。這個(gè)問題轉(zhuǎn)化成了根據(jù)由方程（4）給出的已知方差協(xié)方差矩陣函數(shù)，計(jì)算方差協(xié)方差矩陣的問題。的方差協(xié)方差矩陣將已知的誤差傳播律應(yīng)用到方程（6）

7、，可以寫成其中導(dǎo)數(shù)矩陣這樣表示C4為了得到方程（7）的解，我們首先應(yīng)得到由已知的全球方差協(xié)方差矩陣得到的函數(shù)的值。詳細(xì)的計(jì)算在下一部分。的方差協(xié)方差矩陣將誤差傳播定律同時(shí)應(yīng)用到方程（1）和方程（2），得到其中(qzhng)代入上面方程（9）中的行列式，得到(d do)下列結(jié)果由上，可以證明任何一個(gè)(y )大型的方差協(xié)方差矩陣轉(zhuǎn)化成相對(duì)應(yīng)的相對(duì)平面方差協(xié)方差矩陣，都可以通過下面擴(kuò)展的一般模式得以實(shí)現(xiàn)。 這是一個(gè)全球笛卡爾方差協(xié)方差矩陣轉(zhuǎn)換到一個(gè)相對(duì)大地平面方差協(xié)方差矩陣的塊到塊的公式。據(jù)我所知，這是第一次明確提出這一轉(zhuǎn)換矩陣。最終方程式如果將方程（11）的方差協(xié)方差矩陣代入方程（7），我們最終得

8、到乘以上面的一個(gè)矩陣塊后，我們得到從上面的方程，我們可以直接得到，點(diǎn)1和點(diǎn)2之間的相對(duì)精度與點(diǎn)2和點(diǎn)1之間的相對(duì)精度計(jì)算方程是一樣的，因此精度也是一樣的。盡管相對(duì)精度看起來好像與兩點(diǎn)間的距離無關(guān)，但這不是完全正確的。顯然，距離遠(yuǎn)的點(diǎn)，由于受較大的系統(tǒng)誤差影響（例如大氣折射，軌道誤差等），因此相應(yīng)的方差(fn ch)協(xié)方差矩陣塊也有較大的誤差。但是，這些系統(tǒng)誤差對(duì)距離較近C5的點(diǎn)影響較小。方程（14）的另一個(gè)(y )優(yōu)勢，就是它很容易記憶。上面所給出的所有(suyu)方差協(xié)方差矩陣的值都是以線性單元表示。如果需要，大地?cái)?shù)據(jù)的方差協(xié)方差矩陣也可用曲線角單元表示（例如經(jīng)緯度的弧秒）。轉(zhuǎn)換成角度單元的

9、嚴(yán)密公式會(huì)在下面給出。用角度單元(dnyun)表示的方差協(xié)方差矩陣線性單元和角度單元在正交曲線大地坐標(biāo)系中的轉(zhuǎn)換，已經(jīng)通過被稱為十進(jìn)制或者Lam的矩陣實(shí)現(xiàn)，并且用符號(hào)H表示（詳見Soler 1976; Soler and Johnson 1987）。線性與曲線大地坐標(biāo)系間的微分流形轉(zhuǎn)換具體表示為其中 N和M分別表示卯酉圈和子午圈的主曲率半徑a =橢圓體的長半軸；e =第一偏心率C6由上，由角度單元（弧度）表示的大地坐標(biāo)系的方差協(xié)方差矩陣，用方程（15）擴(kuò)展到兩個(gè)點(diǎn)，用上面(shng min)的方法計(jì)算傳播誤差，可以得到最終，用角度(jiod)單位表示的相對(duì)精度可以寫成例證(lzhng)下面是一

10、個(gè)簡單的應(yīng)用方程（14）的實(shí)際案例。假定點(diǎn)1的坐標(biāo)是，點(diǎn)2的坐標(biāo)是。這個(gè)例子在這里強(qiáng)調(diào)了它的數(shù)學(xué)特性，因?yàn)閮烧局g的夾角是90，那相對(duì)精度的概念可能意義不完整。也就是說，圖一表述了相對(duì)平面框架和參考全球大地坐標(biāo)系的相對(duì)位置和相對(duì)方向。然后，我們可以寫成C7作為預(yù)期值，旋轉(zhuǎn)(xunzhun)矩陣的行是點(diǎn)1和點(diǎn)2之間沿著相對(duì)笛卡爾軸線的單位向量的組成部分。最終，將上面的矩陣代入方程（14），得到上面的矩陣表示即使對(duì)于比相對(duì)精度更簡單的例子，兩點(diǎn)之間的方差協(xié)方差矩陣也是很復(fù)雜(fz)，不能直觀計(jì)算。它們非常依賴于相對(duì)平面大地框架的方向，而這個(gè)方向是隨著點(diǎn)不斷地變化的。然而，點(diǎn)1和點(diǎn)2的相對(duì)誤差的北向

11、分量僅僅與Z軸的誤差有關(guān)。這就證明兩點(diǎn)間的相對(duì)平面中n軸和z軸是互相(h xing)平行的。而它的e軸和u軸卻不是平行的。在這種情況下，因?yàn)檫@些軸都在赤道平面上，所以他們不受Z坐標(biāo)誤差的影響。注意，這里的方程（21）（29）只適用于假想的特殊案例。C8另外(ln wi)，將方程（21）（24）代入方程(fngchng)（11），參考點(diǎn)1和點(diǎn)2的相對(duì)平面框架（見圖1）的方差協(xié)方差矩陣，作為全球(qunqi)笛卡爾框架的方差協(xié)方差矩陣的函數(shù)，或者形象的寫成具體形式是 在這一假定案例中，運(yùn)算結(jié)果的幾何影響更明顯。盡管是在一般情況下進(jìn)行的分析證明，但是讀者在這個(gè)例子中也可以根據(jù)在圖1中的坐標(biāo)軸的指向，

12、很容易地確認(rèn)全球和大地框架的方差、協(xié)方差和交叉協(xié)方差的一致性。運(yùn)算結(jié)果非常直觀。方程（25）、（26）的運(yùn)算結(jié)果可以用參考點(diǎn)1和點(diǎn)2所在的相對(duì)平面框架的方差協(xié)方差的函數(shù)來表示，如下：我們注意到，如果將誤差傳播到方程（6），可以直接獲得上面的方差協(xié)方差矩陣。然而，我們的前提條件是將作為所給的方差協(xié)方差矩陣的函數(shù)，在這個(gè)實(shí)際案例中，最后的結(jié)果由方程（25）得到。這一理論是由GPS的出現(xiàn)，使用指定的笛卡爾框架（例如ITRF00）調(diào)整過的、由一些歷元（例如1,997.00）的衛(wèi)星軌道定義的的3D地心網(wǎng)絡(luò)支持的。交換方程（27）中的分指數(shù)矩陣1和2，證實(shí)了關(guān)于方程（14）中點(diǎn)1和點(diǎn)2的相對(duì)精度相同的論證

13、。然而，按相同的邏輯由點(diǎn)2到點(diǎn)1來證明點(diǎn)1到點(diǎn)2和點(diǎn)2到點(diǎn)1的相對(duì)精度的一致性，對(duì)方程（25）卻是行不通的。換句話說，在方程（25）中我們不能用交換點(diǎn)1和點(diǎn)2來證明兩點(diǎn)間的相對(duì)精度具有相反性。顯然，這一結(jié)論可以用相對(duì)大地框架的指向隨點(diǎn)的變化而變化這一事實(shí)來證明。因此，這些框架有時(shí)被稱作“移動(dòng)框架”（Soler 1976）?，F(xiàn)在我們(w men)舉一些假想的例子。圖2中，我們描述了相對(duì)全球框架和大地框架，分別用、，。經(jīng)定義(dngy)，相對(duì)全球框架（圖2中的和C9）在每個(gè)點(diǎn)上總是平行(pngxng)于地心框架，像上面提到的，隨點(diǎn)的變化而變化。圖2中，我們將兩種框架分別疊加在點(diǎn)1和點(diǎn)2上。從圖像中

14、（在特定簡化后的案例中），我們可以看到點(diǎn)1的u1軸取代了點(diǎn)2的u2軸的位置，所以，當(dāng)相對(duì)大地(dd)框架向點(diǎn)2移動(dòng)后，x1軸與y2軸方向一致（見圖2）。這同樣可以應(yīng)用到其它相對(duì)坐標(biāo)軸。因此，我們可以根據(jù)點(diǎn)1和點(diǎn)2軸的一致性象征性地寫成相似(xin s)的，倒過來可以寫成如果我們與上面對(duì)應(yīng)，改變方程（25）的笛卡爾指數(shù)，會(huì)發(fā)現(xiàn)矩陣中的元素保持不變，這就證明了點(diǎn)1到點(diǎn)2與點(diǎn)2到點(diǎn)1的相對(duì)精度是相同的。綜上所述，相對(duì)精度可以由應(yīng)用方程（14）或方程（27），然后將參數(shù)值代入方程（11）中的右手矩陣獲得。方程（27）不僅適用于這個(gè)假想的案例，而且當(dāng)方差協(xié)方差矩陣參考已知的點(diǎn)1、點(diǎn)2所在的相對(duì)平面框架時(shí)

15、，方程（27）適用于任C10何案例(n l)中相對(duì)精度的計(jì)算。結(jié)論(jiln)據(jù)我所知，描述相對(duì)精度計(jì)算的嚴(yán)密理論至今還沒有人發(fā)布。這一技術(shù)說明是第一次接近這一空缺領(lǐng)域的嘗試。這一理論提出了如何計(jì)算兩點(diǎn)間的相對(duì)精度的準(zhǔn)確方程。順便，也引進(jìn)了全球笛卡爾與相對(duì)平面的方差協(xié)方差矩陣的矩陣轉(zhuǎn)換方程。最后，我們假設(shè)了一個(gè)兩點(diǎn)位于以x 軸、y軸為原點(diǎn)的赤道橢球面上，其結(jié)果(ji gu)很直觀地證明這一數(shù)學(xué)公式是正確的 。作者也很看重這一理論的教育性，它對(duì)于讀者來說是直觀的公式，至今比較復(fù)雜的數(shù)學(xué)理論也可以很容易被理解。致謝作者想感謝Nishanthi Wijekoon畫的圖。參考Burkholder, E

16、. F. （2008）. The 3-D spatial data model. Foundation of the spatial data infrastructure, CRC, Boca Raton, Fla.Federal Geodetic Control Subcommittee (FGCS) and Federal Geographic Data Committee （FGDC）.（1998）. “Geoespatial positioning accuracystandards. Part 2: Standards for geodetic networks.” Rep. No

17、. FGCD-STD-007.2-1998, National Geodetic Survey, Silver Spring, Md.Leick, A. （2004）. GPS satellite surveying, 3rd Ed., Wiley, Hoboken, N.J.Snay, R. A., and Soler, T. （2008）. “Continuously operating reference station （CORS）: History, applications, and future enhancements.” J. Surv. Eng., 134（4）, 9510

18、4.Soler, T. （1976）. “On differential transformations between Cartesian and curvilinear _geodetic_ coordinates.” Rep. No. 236, Dept. of Geodetic Science, The Ohio State Univ., Columbus, Ohio.Soler, T., and Hothem, L. D. （1988）. “Coordinate systems used in geodesy: Basic definitions and concepts.” J.

19、Surv. Eng., 114（2）, 8497.Soler, T., and Johnson, S. D. （1987）. “Alternative geometric determination of altazimuthal-distance covariance matrices.” J. Surv. Eng., 113（2）, 5769.譯文原文出處：Journal of Surveying Engineering. Aug2010, Vol. 136 Issue 3, p120-125. 6p. 2 Diagrams.重慶大學(xué)本科學(xué)生畢業(yè)設(shè)計(jì)（論文）附件 附件D：譯文原文D1附件(

20、fjin)D：Rigorous Estimation of Local AccuraciesToms Soler, M.ASCE1; and Dru Smith2Abstract: This technical note introduces a rigorous calculation of “l(fā)ocal accuracies.” The term local accuracies refers to the relative error between two arbitrary points（uncertainty of one point with respect to the oth

21、er）, expressed in a local horizon three-dimensional reference frame（e, n, u）, that were connected using any geodetic-surveying methodology optical procedures, global positioning system (GPS),etc. It is calculated as a function of the full global Cartesian variance-covariance matrix of the (x , y , z

22、 )coordinates of the points defining the spatial network.DOI: 10.1061/ ASCE SU.1943-5428.0000023CE: Database subject headings:Surveys; Data analysis; Accuracy .Author keywords: Land surveying; Adjustment of data; Local accuracies.IntroductionThe concept of local accuracies, although intuitive, has n

23、ot re-ceived the mathematical attention that it deserves. Recently, Burkholder(2008)made a first attempt to define and express mathematically the differences between network and local accuracies. Network accuracy measures how well coordinates approach an ideal error-free datum Federal Geodetic Contr

24、ol Subcommittee( FGCS) and Federal Geographic Data Committee(FGDC) 1998. In the case of the United States, the datum of reference is the frame defined by the coordinates of the continuously operating reference station (CORS) network( Snay and Soler 2008) . Network accuracies are obtained directly fr

25、om the standard global variance-covariance matrix of a particular three-dimensional (3D) network solution, e.g., tied to the CORS.This matrix is available from the output of any software adjusting 3D networks, and it comprises a full matrix containing 3*3 block diagonals matrices, with the variances

26、 and covariances of each point, and nonblock diagonals containing the covariances between points.The treatment given in Burkholder (2008) may be assumed conceptually correct, but it needs to be generalized. This technical note extends the previous work to present the final equa-tion where all varian

27、ce-covariance matrices, D2as well as the nondi-agonal cross-covariance matrices, are explicitly taken into account.Although, the equivalence of local accuracies in Carte-sian coordinates are computed by some GPS commercial software packages as standard errors of vector baseline components, these acc

28、uracies are generally not transformed to the local horizon geo-detic frame, and if they are, perhaps not in a rigorous manner that follows the theory described below.Local AccuraciesAssume we would like to determine the local accuracies between two points and . It is well known (e.g., Soler 1976; So

29、ler and Hothem 1988; Leick 2004) that the positional errors in global Cartesian coordinates (e.g., geocentric) could be transformed to geodetic local horizon values (east, north, up= vertical) using the equations1Chief Technical Officer, Spatial Reference System Div., National Geodetic Survey, NOS,

30、National Oceanic and Atmospheric Administration, 1315 East-West Hwy, Silver Spring, MD 20910 (corresponding author) . E-mail: Tom.S.2Chief Geodesist, National Geodetic Survey, NOS, National Oceanic and Atmospheric Administration, 1315 East-West Hwy, Silver Spring, MD 20910. E-mail: Dru.S

31、.Note. This manuscript was submitted on May 26, 2009; approved on November 17, 2009; published online on December 29, 2009. Discussion period open until January 1, 2011; separate discussions must be submitted for individual papers. This technical note is part of the Journal of Sur-veying

32、 Engineering, Vol. 136, No. 3, August 1, 2010. ASCE, ISSN 0733-9453/2010/3-120125/$25.00.Where rotation matrices of the transformation be-tween the local Cartesian reference frame and the local horizon frame given D3explicitly byWhere = geodetic longitude and latitude, respectively, at each point i

33、. The above formulation is consistent with the value of the geodetic longitude taken positive toward the east.We are also assuming that the global variance-covariance matrix of the two points is known and given by the symmetric matrixwhere the nondiagonal blocks, whose elements are the cross covaria

34、nces between points, are not symmetric andThe matrix in Eq.(4) can be extracted from a much larger matrix (for tens or hundreds of points) . The local accuracies between the two points could be obtained by propagating errors to the relative local position of point 2 with respect to point 1 using the

35、 simple mathematical modelA note of caution, Eq.(6) is the general definition in 3D space. However, the word “l(fā)ocal” implies that, in practice, it should be applied only to points not too distant from each other, e.g.,100 km. For points separated by large distances, the definitiondoes not make sense

36、 conceptually. Thus, the problem translates to calculate the variance-covariance matrix as a function of the known variance-covariance matrix given by Eq.(4) .Variance-Covariance Matrix of Applying the well-known law of error propagation to Eq.(6) ,we can writeD4where the Jacobian matrix J is given

37、by the expressionIn order to solve the matrix Eq. (7) , we need first to determine the value of as a function of the given global variance-covariance matrix . The detailed calculation fol-lows in the next section.Variance-Covariance Matrix of Applying the error propagation law simultaneously to Eqs.

38、 (1) and (2) , we can writeWhereSubstituting the above Jacobian in Eq. (9) , we arrive to the following resultAs a consequence of the above, it can be proved that the generaltransformation of any large (x,y,z) variance-covariance matrix into its local horizon (e,n,u) counterpart D5follows the extend

39、ed general pattern:which is a block by block transformation, from a global Cartesian variance-covariance matrix to a local geodetic horizon variance-covariance matrix. To the writers knowledge, this transformation matrix is provided here explicitly for the first time.Final EquationIf the variance-co

40、variance matrix of Eq. (11) is introduced into Eq. (7) , we finally arrive atand after multiplying the above block matrices, we getFrom the above equation it immediately follows that the local accuracy between points 1 and 2 is identical to the local accuracy between points 2 and 1, thus connected p

41、oints have equal local accuracies. Although the local accuracies appear to be independent of the distance between the points, this is not exactly the case. Obviously, the blocks of variance-covariance matrices of distant points have larger errors caused by the greater effects of systematic errors (e

42、.g., atmospheric refraction, orbit error, etc.) which affect less to points that are closer together. An unexpected advantageous result of Eq. (14) is that it is simple to memorize.All values in the variance-covariance matrices shown above are given in linear units. If desired, the variance-covarian

43、ce matrix of the geodetic quantities can also be given in curvilinear angular units (e.g., seconds of arc for longitude and latitude). Therigorous transformation to angular units is given below.Variance-Covariance Matrix of Given in Angular UnitsD6The conversion between linear and angular units when

44、 using orthogonal curvilinear geodetic coordinates is achieved through the so-called metric or Lam matrix, denoted by the symbol H (e.g.,see Soler 1976; Soler and Johnson 1987) .Specifically, the transformation on a differential manifold between linear and curvilinear geodetic coordinates is written

45、 byWhereand N and M = principal radius of curvature along the prime vertical and the meridian, respectivelya = semimajor axis of the ellipsoid; e = first eccentricityBased on the above, the variance-covariance matrix of the geodetic coordinates in angular units (radians ), after extending Eq.(15) to

46、 two points and propagating errors following the same logic used before, will be D7and finally, the local accuracies expressed in angular units takes the formHypothetical ExampleA simple practical example of the application of Eq. (14) will now be presented. Assume that point 1 is located at the fol

47、lowing coordinates and, similarly, the coordinates of point 2 are .It should be emphasizedthat this example is used here to illustrate the mathematics involved because the concept of local accuracies may not have full meaning for two stations separated by an angle of 90. This said, Fig. 1 depicts th

48、e relative location and orientation of the local horizon frames and with respect to the global coordinate system .Then, we can write D8As expected (Soler 1976), the rows of the rotation matrices are the components of the unit vectors along the local Cartesian axes (e,n,u) located at points 1 and 2.

49、Then finally, substituting the above matrices into Eq. (14) we arrive atThe matrix above shows even for this simplified example than the elements of the local accuracies variance-covariance matrix between two points are complicated to guess and not very intuitive to rationalize. They are very much d

50、ependent of the orientation of the local horizon geodetic frames which are constantly changing from point to point.However, notice that the relative error between points 1 and 2 along the north component is only dependent on the errors of the z coordinate. This can be explained by the fact that the

51、local horizon n -axes are parallel to z -axis at both points. The same situation is not repeated with the e - and u -axes. In this case, because these axes are on the equatorial plane, their errors are independent of the z -coordinate errors. Note that Eqs. (21) (29) herein are valid only for the sp

52、ecial case shown in the hypotheti-cal example.Incidentally, substituting Eqs. (21) (24) into Eq. (11) , the variance-covariance matrix referred to the local horizon frames at points 1 and 2 (see Fig. 1 ), as a function of the variance-covariance matrix of the global Cartesian frame , or written symb

53、olicallytakes the explicit formD9The geometric dependence of the results with the assumptions of this simple exercise is here more evident. Although the general case was proved analytically, the reader will have no difficulty in immediately identifying on this example the correspondence between the

54、variances, covariances, and cross covariances of the global and local frames according to the orientation of coordinate axes shown in Fig.1. The interpretation of results is remarkably self-explanatory.Using the results from Eq. (26) , Eq. (25) can be expressed as a function of the variances and cov

55、ariances referred to the local horizon frames (e,n,u) at points 1 and 2, as follows:Notice that the above variance-covariance matrix could have been obtained directly if we propagate errors to Eq.(6) . However, we started from the premise of determining as a function of the given covariance matrix ,

56、 and in this practical example the results were given by Eq.(25) . This requirement is supported by the fact that with the advent of GPS, 3D geocentric networks are adjusted using a prespecified Cartesian (x,y,z) frame (e.g., ITRF00) defined by the satellite orbits at some selected epoch (e.g., 1,99

57、7.00). Switching the subindexes 1 and 2 in the matrix of Eq. (27) corroborates the statement made in connection with Eq. (14) that the local accuracies for points 1 and 2 are identical. However, the same logic of exchanging the 1 by 2 toprove that the local accuracies from point 1 to 2 are identical

58、 to point 2 to 1 does not work for Eq. (25) . In another words, we should not change 1 to 2 and 2 to 1 in Eq. (25) to prove reciprocal property of local accuracies between two points. This is explained by the fact that, by definition, the orientation of the local geodetic frames (e,n,u) changes from

59、 point to point. For this particular reason, these frames are sometimes referred as “moving frames”(Soler 1976). Let us concentrate now on the hypothetical example at hand. In Fig. 2, we have D10depicted the local global and geodetic frames, respectively, and , i=1,2 at points 1 and 2. By definition

60、, the local global frames, and in Fig. 2 are always parallel to the geocentric frame at each point. However, the orientation of the local geodetic frame, as mentioned above, changes from point to point. In Fig. 2 we have superimposed both frames (local global and geodetic) at points 1 and 2. From th