毕业论文附录英文翻译

附录

SLAC-PUB-3620

April 1985

(A)

APPLICATION OF GPS IN A HIGH PRECISION

ENGINEERING SURVEY NET WORK

ROBERT RULAND, ALFRED LEICK

ABSTRACT.A GPS satellite survey was carried out with the Macrometer to sup-port construction at the Stanford Linear Accelerator Center(SLAC).The networkconsists of 16 stations of which 9 stations were part of the Macrometer network.The horizontal and vertical accuracy of the GPS survey is estimated to be l-2 m m and2-3 m m respectively.The horizontal accuracy of the terrestrial survey,consisting of angles and distances,equals that of the GPS survey only in the“loop”portion ofthe network.All stations are part of a precise level network.The ellipsoidal heightsobtained from the GPS survey and the orthometric heights of the level network are used to compute geoid undulations.A geoid profile along the linac was computed by the National Geodetic Survey in 1963.This profile agreed with the observed geoid within the standard deviation of the GPS survey.Angles and distances were adjusted together(TERRA),and all terrestrial observations were combined with the GPS vector observations in a combination adjustment(COMB).A comparison of C O M B and TERRA revealed systematic errors in the terrestrial solution.A scale factor of 1.5 ppm f.8 ppm was estimated.This value is of the same magnitude as the over-all horizontal accuracy of both networks.

INTRODUCTION

At the Stanford Linear Accelerator Center a new project is under construction,the Stanford Linear Collider(SLC).The shape of the completed SLC will be like a tennis racket with the handle being the existing linac and the curved parts being the new North and South collider arcs.The diameter formed by the loop will be about 1 km.To

position the approximately 1000 magnets in the arc tunnels,a network of nearby reference marks is necessary(Pietryka 1985).An error analysis has shown that a tunnel traverse cannot supply reference points with the required accuracy.Therefore,a https://www.wendangku.net/doc/8b14594135.html,work with vertical-penetrations will support the tunnel traverses.-The required absolute positional accuracy of a control point is f 2 m m(Friedsam -1984).

This two-dimensional surface net must be oriented to the same datum as defined by the design coordinate system.This design coordinate system is used to express the theoretical positions of all beam guiding elements.Since this coordinate system defines the direction of the existing two mile long linear accelerator(linac)as its Z-axis,the SLC coordinate system must integrate points along the linac in order to pick up its direction.Therefore,three linac stations have been added to the SLC net.Figure 1 shows the resulting network configuration.

The disadvantageous configuration is obvious,especially since there is no intervisibility between linac stations 1,10 and 19 to stations other than to 42 and 20.To improve this configuration,one would have to add stations northerly and southerly of the linac.However,due to local topography,doing that would have tripled the survey costs.

This was the situation when it was decided to try GPS technology,although it was at that time not yet proven that the required 2 m m standard deviation positional accuracy could be obtained.

SURVEY DESIGN

The horizontal control network consists of 16 stations,12 in the…loop?,and 4 along the linac.Because of financial considerations,not all 16 stations have been included in the GPS survey.Only the 4 linac and 5…loop?stations were occupied by the GPS survey.The intent was to determine the coordinates of the loop stations,including

station 42,by conventional means,i.e.triangulation and trilateration,followed by an inner constraint adjustment.Then the GPS information would be used to orient the net to the direction of the linac(Ruland 1985).

Conventional Horizontal Net

All monuments are equipped with forced centering systems and built either as massive concretears or steel frame towers,both with independant observation platforms.The observation schedule consists of directions and distances with standard deviations of 0.3 mgon and 2 mm,respectively.

Conventional Vertical Net

All 16 stations are part of a high precision level network.To minimize errors and simplify repeated leveling,both benchmarks and turning points are permanently monumented.Doublerunning the entire net requires about 700 setups.The standard deviation for a 1 km double-run

line is 0.3 mm.

GPS Survey

The GPS survey,which utilized the five available satellites,was carried out in August 1984 by Geo-Hydro Inc.The whole observation window was used for each station.In general three Macrometers were put to use.

Linac Laser Alignment System

For the frequent realignment of the linear accelerator,the linac laser alignment system was designed and installed.This system is capable of determining positions perpendicular to the axis of the linac(X and Y)to better than f.l m m over the total length of 3050 m.To do so,a straight line is defined between a point source of light and a detector.At each of the 274 support points,a target is supported on a remotely actuated hinge.To check the alignment at a desired point,the target at that point is inserted into the lightbeam by actuating the hinge mechanism.The target is actually a rectangular Fresnel lens with the correct focal length so that an image of the light source is formed on the plane of the detector.This image is then scanned by the

detector in both the vertical and the horizontal directions to determine the displacement of the target from the predetermined line.The targets are mounted in a 60 cm diameter aluminum pipe which is the basic support girder for the accelerator.The support girder is evacuated to about 10/.Lof Hg to prevent air refraction effects from distorting or deflecting the alignment image(Hermannsfeldt 1965).

Using this system it was possible to determine the X-coordinates of the four linac stations,independant of terrestrial or GPS survey techniques,to better than ±0.l mm.

ANALYSIS OF LEVELING DATA

To check for blunders,the L-l norm adjustment technique was applied(FUCHS 1983).Several blunders have been identified and cleared.A L-2 norm adjustment was then carried out with CATGPS(Collins 1985)in a minimally constrained fashion by fixing the height of station 41 to its published value of 64.259m.The choice of this particular station as well as the specific numerical value is,of course arbitrary for the purpose of the adjustment.CATGPS is suitable for adjusting leveling data if the latitudes and longitudes of the stations are fixed.The results of the level adjustment are summarized in Table 1(Column Level).

ANALYSIS OF GPS DATA

All GPS vectors and their respective(3x3)covariance matrices as received from Geo-Hydro were subjected to an inner constraint least squares solution for the purpose of blunder detection and to get an unconstraint estimate of the obtained accuracy.

Table 1 Summary of Adjustment Results

Inner Constraint GPS Solution

Applying data snooping(Baarda 1976)on the residuals the vector observation(39-42)was suspected-of containing a blunder of about 1.3 cm.A recomputation was carried out at GeoHydro and,indeed,the time bias was not fixed in the original computation.Fixing the time biasin the case of short vectors is the standard procedure in Macrometer vector computation.The components of the recomputed vector agreed within 2 m m with the adjusted values of the original network solution.Upon implementing the corrected observations the residuals did not suggest the existence of other blunders.The inner constraint solution was carried out with MAC(Leick 1984);the results are documented in Table 1,Table 2,and Fig.2.The quality and homogeneity of the GPS network is well documented by the tables and the figure.The standard deviations for the horizontal positions are between 1 and 2 m m and for the vertical positions between 2 and 3 mm respectively.

If one computes the standard deviations and the adjusted length for all observed

vectors and their ratios,then the average ratio is 1:690000.This value yields another

characterization of the horizontal accuracy achieved in this GPS survey.

Minimum Constraint GPS Solution

This solution defines the reference datum.The most simple set of minimal constraints are i imposed by fixing one station to account for the translatory component of the GPS polyhedron. The rotation and the scale are inherent in the Macrometer vector measurement and processing technique.The published geodetic latitude and longitude(NAD 1927)are adopted for station 41.The ellipsodial height for this station is equated to its orthometric height given above.Thus_the defined ellipsoid differs only slightly from the classical definition of a local reference ellipsoid(At the initial point the geodetic latitude and longitude equal astronomical latitude and longitude respectively;one geodetic and one astronomical azimuth are equated,and the ellipsodial height is taken as zero.)This classical definition makes the ellipsoid tangent to the equipotential surface at the initial point.Since the choice of the numerical values for station 41 are totally immaterial as far as the adjustment of GPS vectors is concerned,the classical definition of the local reference ellipsoid could have been used as well.The deflections of the vertical happen to be known in his adequate for this project as long as the correction of the measured horizontal angles due to deflections of the vertical are negligible since no attempt is made to apply these corrections.

Table 2 Standard Deviations of GPS Solution

Figure 2 Error Ellipses from GPS Inner Constraint Solution

SHAPE OF THE GEOID

The shape of the geoid in the area of the survey follows readily from a comparison of the ellipsoidal and orthometric heights according to

H=h-N

Figure3 shows the geoidal profile along the linear accelerator.

The figure shows an unexpected dip of the-observed geoid at station 20.It so happens that this station required an observation tower of 20 m for the terrestrial measurements and that the height above the ground monument was measured trigonometrically.Assuming that the geoid follows the dashed line one can deduce an error in the height of the tower platform of about 8mm.In the context of an earlier survey for the construction of the linear accelerator the Coast and Geodetic Survey computed a geoid profile between stations 1 and 42.The report(Rice 1966)lists the components of the deflection of the vertical for stations 1 and 42,and for a non-existing station halfway between stations 10 and 19.From these values the Coast and Ge9detic Survey computed a function for the undulation.All linear values are in feet.The variable z is measured from station 1. It is stated in the report that this function gives undulations with an accuracy estimate of better than 0.001 ft.No procedure is given as to how this accuracy estimate was obtained.The undulation curve,derived from the following function,is shown in Fig.3.:

N =11.102*3142106)(10*0629.6)(10*4331.11)(10x x x ---+-

The.deviation between this curve and the observed geoid just barely exceeds,at station

10,the standard deviation for the Macrometer determined height difference from 1 to 10,and is within the standard deviation at stations 19 and 42.

Figure 3 Geoid Profile

Incidentally,the over-all slope of the observed geoid is a consequence of adopting geodetic rather than astronomic positions as minimal constraints at station 41.The east-west component of the deflection of the vertical at station 42 is 1.84 arcsec which accounts for 27 m m of the 22 mm geoidal slope between stations 1 and 42.

Figure 4 Geoid Undulation Contours

Figure 4 shows an attempt to draw contours of equal geoid heights.The small number of G P S stat&rs and their area1 distribution effects the accuracy of the contours.

ANALYSIS OF THE TERRES TRIAL OBSERV ATIONS

The triangulation and trilateration data were also checked for blunders applying the L-l norm technique(Fuchs 1980).The terrestial observations are then adjusted using the S-dimensional model of CATGPS.The reference ellipsoid is the one defined

above for the minimal constraint G P S vector solution,i.e.the same numerical values for station 41 are held fixed.The orientation in azimuth is achieved by holding the latitude of station 35 fixed to the numerical value computed for the minimal constraint GPS solution.The height of station 41 is constrained to the GPS solution as well.A consequence of this definition is that the terrestrial system(U)and the satellite system(S)coincide.Since the triangulation and trilateration observations do not contain much information in the third dimension,the ellipsoidal heights of the remaining stations are introduced as observed parameters.The heights are shown in Table 3.

Table 3 Orthometric Height H and Ellipsidal Height H

The elliposidal heights for the GPS stations follow immediately from the&iinrmal c&straint GPS vector adjustment,whereas the ellipsoidal heights of the remaining points are computed from the orthometric heights and the interpolated geoid undulations.The standard deviations for the latter set of heights are derived from a guess for the accuracy of the geoid interpolations.

In order to investigate the relative weighting of theles and the distances,two separate adjustments are ried out with CATGPS,each having only one type observation.The result is shown in Table 1.The le for the angle adjustment is provided by fixing the gitude of station 35.The stations 1,10,and 19 are luded from these adjustments because of the weak of that part of the network.In the next step angles

and distances are combined in a common ustment which excludes(TERRA A)and includes(TE RRA B)th e 1m?at stations 1,10,and 19 respectively

COMBINED ADJUSTMENT

CATGPS is finally used to adjust the terrestrial observations and the GPS vectors together.The minimal constraints are implemented by assigning to the latitude and longitude of station 41,to the latitude of station 35,and to the ellipsoidal heights of stations 1,33,and 39 the minimum constraint GPS results as constants.In this way the GPS vector observations will determine the heights of all stations,i.e.the leveled orthometric heights do not enter this adjustment at all.Table 1 shows that the estimated rotation parameters differ only insignificantly from zero.Their theoretical value is zero because of the specific choice of the numerical values of the coordinates held fixed.A different selection for the fixed coordinate values at station 41,e.g.astronomical positions,would have resulted in estimated rotation parameters significantly different from zero.The estimated scale factor is 1.5 ppm which is about twice its estimated standard deviation.

INTERPRETATION

Table 1 shows the a-posteriori variances of unit weight for all adjustments.It is seen that these values for the adjustments GPS,ANGLES,and DIST are all slightly above one,but are acceptable at a significance level of.05.Since the three variances of unit weight(1.13,1.11.1.17) are of nearly the same size,one could scale the variance of the GPS vectors,the angles,and the distances by a common scale.This would formally reduce the a-posteriori variances for TERRA(A),TERRA(B),and COMB,but would not change the outcome..of the adjustments.There appears to be no need to scale the variance for the GPS vector observation,the terrestrial angles and distances by separate(different)factors.

Table 4 Compilation of Adjustment Results

Table 4 shows the adjusted coordinates for the GPS vector adjustment,the combined angle and distance adjustment TERRA(B),and the combination solution COMB.The column“COMB-TERRA”shows for each coordinate the discrepancies in milhmeters between the cornbinedmsolution and the terrestrial solution.The comparison is permissable since solutions in the same terrestrial system(U)are compared.There is a large discrepancy in latitude at station1.However,this discrepancy can be readily explained by a weakness of the terrestrial solution TERRA.The lateral position(with respect to the linac)is only determined by the angles(33-20-1)and(20-N-l).Note that the separation of stations 20-l and 10-l is 3500m and 2500m respectively.The discrepancies COMB-TERRA(B)are shown in Fig.5.There appears to be a systematic effect along the linac in the ter-I I Irestrial observations.The deviation definitely exceeds what can be expected from the formal standard deviations of the terrestrial solution TERRA(B).Several partial solutions were carried out and the residuals were inspected in all cases.No evidence could be

found for the existance of blunders in the data.If one excludes the stations 1,10,and

19,then the combination solution and terrestrial solution agree within 1 mm.

A verification of whether either the GPS or the terrestrial observations along the linac are systematically debased could finally be obtained through utilizing the linac laser alignment system.A comparison of the X-coordinates of the linac stations from the TERRA and COM

B solution with those determined using the linac alignment system was done by means of a seven parameter transformation after the ellipsoidal coordinates had been converted into Cartesian coordinates.The results are shown in table 5.Looking at the(LINAC-COMB)CO~UIIUI,the values of the differences are insignificant with respect to the standard deviations of the COMB-solution.In other words,the COMB-solution reflects the correct geometry of the linac;whereas the significant differences in the(LINAC-TERRA)column indicate that the geometry of the stations in the systems is not congruent.

The column GPS-COMB shows only small discrepancies.The latitudinal differences are all smaller than 2 mm.The discrepancies in the east-west direction are somewhat larger.A proper interpretation of these discrepancies requires that one distinguish between the two coordinate systems involved.The combination solution C O M B(as well as TERRA)refers to the terrestrial coordinate system(U).B ecause of the specific choice of the coordinates of the fixed station 41 and the futed latitude of station 10,the terrestial coordinate system(U)and the satellite system(S)are

parallel.This is confirmed by the estimates of the rotation angles listed in Table

1.However,the same table lists a scale of±l.5 ppm.Going back to the definition of these transformation parameters it is seen that a positive scale estimate implies that the polyhedron determined by GPS observations(satellite system)is bigger than the one determined from the terrestrial observations.This is readily confirmed by comparing the longitudes of stations 1,41,and 35 for the GPS and the C O M B solutions in Table 4.The scale factor is,of course,also present in the latitudinal discrepancies,but to a lesser extent,because of the predominently east-west extension of the whole network.The longitudinal effect of the scale factor onaation 1 relative to station 41 is 1.5 ppm*3200 m=5.4 mm.This is the value by which the longitudinal separation of stations 1 and 41 should be increased in COMB.In fact,the effect of the

scale on the longitudes of all stations is computed as(-5,-3,-2,0, -1,0,1,2)in

millimeters.Differencing these values with those listed in Table 4 under column“GPS-COMB”yields the discrepancies in which the effect of the scale is eliminated.The values are(O,O,-l,O,-,-l,-l,O,-3) in millimeters.These values and those listed for the latitude are of the same size.They reflect the“non-scale”discrepancies between the GPS solution and the combination solution.Their smallness reflects the dominance of the GPS vector observations in the combination solution.

Table 5 Linac Comparison

CONCLUSIONS

The leveling data were used only to compute(interpolate)the geoid

undulations.The accuracy of these undulations depends directly on the accuracy of the leveling and the vertical components of the GPS survey.Processing the phase observations“line by line”yielded a co mpletely acceptable accuracy for this https://www.wendangku.net/doc/8b14594135.html,parison with the terrestrial observations demonstratesthat the_GPS accuracy statements(standard deviations)are,indeed,meaningful and not toooptimistic. Compared against the standard of the precise network and especially the linac laser alignment system measurements,it could be proven that the GPS technique in a close range application is capable of producing results with standard deviations in the range of l-3 m m and,therefore,can be applied for engineering networks.

The GPS survey has made it possible for the weak network of the linac(stations 1,10,19,42)to be tied accurately to the loop network.The terrestrial observations did not control the latitudinal position of station 1 accurately.To determine station 1 accurately with terrestrial observations would have required the design of a“classical”network which would have been difficult and expensive because of the visibility constraints due to topography and buildings(which did not exist during the first survey for the linac).

The GPS survey served as a standard of comparison for the terrestrial solution and revealed the existence of systematic errors in the latter solution even though a thorough analysis of the terrestrial observations did not reveal such errors.

Since the estimated scale factor of 1.5 ppm f.8 ppm is of the same magnitude as the over-all horizontal accuracy of both networks,no conclusion can be drawn as to internal scale problems of either the electronic distance measurement devices or the Macrometer.

REFERENCES

Baarda,W.(1976):Reliability and Precision of Networks,Presented Paper to the VIIth International Course for Engineering Survey of High Precision,Darmstadt.

Collins,J.,Leick A.(1985):Analysis of Macrometer Network with Emphesis on the Montgomery(PA)County Survey,Presented Paper to the First International

Symposium on Precise Positioning with the Global Positioning System,Rockville.

Fuchs,H(31980):Untersuchungen zur Ausgleichung durch Minimierender Absolutsummeder Verbesserungen,Dissertation,Technische Universitlt Graz.

Fuchs,H.,Hofmann-Wellenhof,B.,Schuh W.-D.(1983):Adjustment and Gross Error Detection of Leveling Networks,in:H.Pelzer and W.Niemeier(Editors):Precise Levelling,Diimmler Verlag,Bonn,pp.391-409.

Friedsam,H.,OrenW.,PietrykaM.,PitthanR.,Ruland R.(1984):SLC-Alignment Handbook,in:Stanford Linear Collider Design Handbook,Stanford,pp.8-3-8-85.

Hermannsfeldt,W.(1965):L?mat Alignment Techniques,Paper presented to the IEEE Particle Accelerator Conference,Washington D.C.

Leick A.(1984):M August 1984.acrometer Surveying,Journal of Surveying Engineering,V ol.110,No.2

Pietryka,M,Friedsam H.,Oren W.,Pitthan R.,Ruland R.(1985):The Alignment of Stanford?s new Electron-Positron Collider,Presented Paper to the 45th ASP-ASCM Convention,Washington D.C.

Rice,D.(1966):Vertical Alignment-Stanford Linear Accelerator-,in:Earth Movement Investigations and Geodetic Control for Stanford Linear Accelerator Center,Aetron-Blume-Atkinson,Report No.ABA 106.

Ruland,R.,Leick,A.(1985):Usability of GPS in Engineering Surveys,Presented Paper to the 45th ASP-ASCM Convention,Washington D.C.

附录

斯坦福直线加速器中心-3620

1985年4月

(A)

GPS在精密工程测量网中的应用

RobertRuland，AlfredLeiek

摘要：测距仪被用来进行GPS卫星测量，以支援斯坦福直线加速器中心(SLAC)

的建设。该测量网由16个测站组成，其中有9个是瓦lacrometer网的测站。GPS 测量的平面和高程精度，估计分别为1mm到2mm和2mm到3mm。由边角测量组成的地面测量的平面精度仅在该网的“环形”部分与GPS测量精度相同。所有测站都是精密水准网的一部分。由GPS测得的大地高和水准测量网的正高，可用来计算大地水准面差距。美国大地测量局于1963年对一条沿直线加速器方向的大地水准面剖面进行了计算。此剖面与上述水准面的吻合程度在GPS测量的标准差允许范围以内。之后将其角度和边长一起进行了平差，还将全部地面观测值与GPS 向量观测值一起，进行了一次联合平差。比较COMB和TERRA的结果，发现在地面网的解算中存在着系统误差。估计尺度因子为 1.5ppm±0.8ppm。此值与两网总的平面精度具有相同的量值。

引言

斯坦福直线加速器中心（SLAC）正在建设一项新的工程——斯坦福直线碰撞器(SLC)。它建成后的形状如同一把带把的网球拍。拍柄是已有的直线加速器，而弯曲部分是新碰撞器的北、南两条弧，其环形的直径约一公里。为了在弧形隧道内定出近千块磁铁的位置，有必要由附近的参考标志组成一个控制网(pietryka 1985)。误差分析表明，仅用一条隧道导线是不能以所需要的精度提供参考点的。因此，建立了一个(可从顶部)垂直贯通的控制网，以支持隧道导线。控制点所需要的绝对定位精度为±2mm(Friedsman 1984)。

这个二维地面网应根据设计坐标系时所规定的那个基准进行定向。所设计的这个坐标系，是用来表示所有的射束导向元件的理论位置的。该坐标系规定，将现有两英里长的直线加速器(linac)的方向作为其Z轴，SLC坐标系必须与沿直线加速器的那些点结合起来，以得到它的方向。因此，三个直线加速器测站也被纳入SLC网。图1表示了该网最后的形状。

该网的形状不佳是显而易见的，特别是由于直线加速器上测站1、10和19到其它测站之间不存在通视条件(除了40号测站和20号测站以外)。为了改善该网的构形，必须在直线加速器的北面和南面增设一些测站。但由于局部地形的限制，将使测量费用增加两倍。

以上就是当时决定试验GPS方法的背景，尽管当时GPS能否达到所要求的2mm标准差的定位精度尚未被证实。

测量方案

平面控制网由16个测站组成:环形部分12个测站，沿直线加速器4个测站。出自经济方面的考虑，并非全部16个点都被纳入了GPS测量网，只有直线加速器部分4个站和环形部分5个站进行了GPS测量。这样做的目的，是用常规方法——三角测量和三边测量方法定出包括42号测站在内的环形部分测站的坐标，随之再进行一次内约束平差，然后用GPS信息将网调整至直线加速器的方向(Ruland，1955)。

1、常规平面网

全部标石都装有强制对中系统，并建造了坚固为混凝土测墩或钢架结构的站标。测墩和标石都建立了独立的观测台。观测项目包括方向和距离，其标准分别为0.3mgon和0.2mm。

2、常规高程网

全部15个测站都是精密水准网的一个组成部分。为将误差减至最小程度，并简化重复水准测量作业，水准点和转点上都埋没了永久性标石。整个网的双程测量大约需要设站700个。双程每公里标准差为0.3mm。

3、GPS测量

1984年8月，Geo-Hydro公司利用5个可用的卫星进行了GPS测量。每个测站都利用了整个观测窗口，通常使用三台测距仪。

4、直线加速器的激光准直系统

为了对直线加速器进行反复的经常性的调整，我们设计并安装了直线加速器的激光准直系统。该系统可用来测定直线加速器轴线之垂线方向的数值(X和Y)，在全长305米的范围内精度可优于 0.lmm。这样，点光源与探测器之间的直线即可确定。在274个支持点的每个点上，均有一个由遥控驱动关节支持的站标。为了检查待测点是否在准直线上，只要驱动关节机械，使该点的站标移至光束中。站标实际上是一个矩形Fresnel透镜，它具有已调准的焦距，以使光源在探测器平面上成像，然后再由探测器在垂直和水平方向对该成像进行扫描，以确定站标自预定直线的偏移量。站标安置在一个60cm直径的铝管内，而铝管又是加速器的基本支承梁。支承梁抽空到大约1加水银柱的大气压，以防止空气折射效应对准直成像产生畸变和偏转（Hermannsfeldt 1965)。

利用这一系统可以不依赖于地面测量或GPS测量技术，独立地确定直线加速器部分的四个测点的X坐标，其精度均优于±0.1mm。

水准测量资料的分析

为了检核粗差，曾使用了L-1范数平差技术(Fuchs，1983)，并检出和剔除了一些粗差。之后，将41号测站的高程固定在已知值64.259M上，用CATGPS(Collins，1985)程字按最小约束条件形式进行了一次L-2范数平差。就平差目的而言，选择这一特定点以及这样的特定数值，当然是任意的。如果测站的经纬度被固定，那么对水准测量数据平差来说，CATGPS将是非常适用的，水准网平差结果汇总于表1中(见水准测量一栏)。

GPS资料分析

为了检验粗差并得到所获精度的非约束条件估值，曾对从Geo-hydro公司接收到的全部GPS向量及它们各自的协方差矩阵，进行了一次内约束条件的最小二乘解算。

1、内约束条件的GPS解算

根据对残差进行的数据探测，怀疑39到42的向量观测包含着大约1.3cm 的粗差。Geo-Hydro公司对此进行了二次重算。在初始计算中，时偏实际上未加固定。对短向量情况来说，固定时偏是计算测距仪向量的一种标准处理方法。重算向量的分量同初始网的解算的平差值符合程度在2mm以内。完成观测量改正后，其残差并不能使人联想到其它粗差的存在。内约束条件的解算可借助于MAC

程序来完成，计算结果列于表1、表2和图2中。以上图表充分表明了GFS网的质量和均匀性。平面位置和高程位置的标准差，分别为1~2mm和2~3mm。如果对所有观测向量计算标准差和平差后的长度，以及它们的比率的话，则其平均比率为1:690000。此值给出了这次GPS测量所达到的平面测量精度的另一特性。

2、极小约束条件的GPS解算

这一解算确定了参考基准。最简单的一组极小约束条件是强制固定一个测站，以计算GPS多面体的平移分量。旋转和尺度(因子)是Hacromoter向量测量和数据处理技术中固有的。41号测站采用已公布的大地纬度(北美1927年基准)，并令该点的大地高与上面所给的正高相等。因此这样定义的椭球与经典定义的某一局部参考椭球(在原点上大地纬度和经度分别等于其天文纬度和经度，大地方位角与天文方位角相等，并取大地高为零)将相差甚微。这一经典定义使得椭球在原点与等位面相切。因此就GPS向量平差而论，对41号测站选择什么数值根本无关紧要，故局部参考椭球的定义同样可以利用。在这种情况下，垂线偏差恰好是已知的(见下文)。只要是垂线偏差而引起的水平角观侧的改正是微不足道的，任何关于局部参考系的定义对于这一方案都是适用的，更何况并没有打算利用这些改正数。

大地水准面形状

比较大地高和正高，根据H=h-N可容易得到测区的大地水准面形状。图3表示了沿直线加速器方向的大地水准面剖面图。

上图表明，所测得的大地水准面在20号侧站上出现了意料不到的凹陷。为了进行地面侧量正巧需在该测站上建一座20米高的观测站标。该坐标相对于地面标石的高度是用三角法测量的。假定大地水准面随虚线延伸，从而可以推断站标平台的高度约含有8mm 的误差。由于这个原因，为了建造直线加速器早先曾进行过一次测量。在那次测量中，美国海岸大地测量局计算了1至42号测站之间的大地水准面剖面。Rioe 在1966年的报告中列举了1号侧站和42号测站，以及10至19号测站正中的一个不存在的点的垂线偏差分量。根据这些值，海岸大地测量局对大地水准面的差求得一个函数。所有线值均以英尺为单位。变量x 从一号测站开始度量。报告指出该函数给出的大地水准面差距具有优于0.001英尺的估计精度，但并未给出怎样得到这一精度估值的过程。图3表示了按函数

N=3142106)(10*0629.6)(10*4331.11)(10*102.11x x x ---+-

求得的大地水准面差距曲线。这一曲线与测得的大地水准面之间的偏差，在10