Surveying adjustment datum and relative deformation accuracy analysis

Abstract In the surveying adjustment, unknown parameters are usually not direct observations, but the elements related to these direct observations. In order to determine the unknown parameters adequate known data should be provided, and these necessarily required known data are used to form the adjustment datum. Under different datums, different results will be obtained even with the same direct observations. However, in the practical adjustment calculation, the datum and its effect on the results are always ignored. In this paper, the adjustment datum is firstly discussed and defined as datum equations. Then an adjustment method based on the datum equations and least squares is presented. This method is a generic one, not only suited for the case in an ordinary datum but also in the gravity centre datum or a quasi-datum, and can be easily used to analyse different deformations. Based on this method, the transformation between different reference frames is derived. It shows that the calculation results, deformation and positioning accuracy under one kind of datum are relative and generic. A case study is further introduced and used to test this new method. Based on the case study, the conclusions are reached. It is found that the relative positional root mean square error of each point becomes bigger as the distance between the point and the datum increases, and the relative deformation offsets under different kinds of datum are helpful for reliable deformation analysis.


Introduction
In the surveying adjustment, the unknown parameters are usually not the direct observations, but the elements related to these observations. For example, the observations are the values of direction and distance in a plane control network whilst the unknown parameters are the plane coordinates of the control points. In order to determine the unknown parameters, enough known data should be provided. This known data is necessary and called the adjustment datum. When the rank of the observation equation's coefficient matrix is less than the number of unknown parameters this is called a rank deficient problem in the adjustment (Lu et al., 2007).
The n-dimensional (n51-4) space has 1 scale datum, n position datums and n(n-1)/2 azimuth (direction) datums. That is to say the number of datums for onedimensional space (height network) is 2 (1 scale datum and 1 position datum); for two-dimensional space (plane network) the number is 4 (1 scale datum, 2 position datums and 1 direction datum); for three-dimensional space the datum number is 7; and for four-dimensional space it is 11 (Cui et al., 2009;Liu and Feng, 2003;Wu, 2006).
It should be pointed out that we can obtain the coordinates of the unknown points or deformations and their accuracy (variance matrix), which are coordinates and variance matrix under a given relative datum from the adjustment for a control network or a monitoring network, according to a set of direct observations (direction, distance, GPS baseline vector, etc.) (Chen, 2003;Tang, 2011). Under different datums, different results will be obtained with the same observations. However, in the practical adjustment calculation, the datum and its effect on the adjustment results are always omitted. The issue of measurement datum has been emphasised in many studies (Huang, 2001;Lu, 2007;Ye and Huang, 2000).
In this paper, the adjustment datum is firstly discussed and defined as datum equations. Then a new adjustment method based on the datum equations and least squares is presented. This is a generic method and makes it easy to analyse the relative deformations among the monitoring stations (Wang and Xu, 2011;Wang et al., 2010).
The adjustment datum has a clear analytical relationship with the relative positional accuracy. Liu (1984) and Xu and Liu (1990) have outlined the complete definition of relative positional accuracy, which explains that it is not correct to understand the relative positional error as a mean square error of the coordinate differences of two points. In this paper, the transformational formula between different datums can be deduced by the method based on datum equations. The significance of the relative solutions, deformation, and positional accuracy attached to a certain datum are later explained.

Equation (1) is the observation equation for surveying adjustment
LzV~f (X ) (1) The linearised error equation of equation (1) can be expressed as follows

V~AdX {l
where L and V are the observations and their corrections,X is t unknown parameters, P is a weight matrix of the observations, and dX is a vector containing corrections toX . For a control network in a geometric space, it contains all the coordinates of the unknown points and other unknown parameters to form the adjustment datum. A is the coefficient matrix. It is a rank deficient matrix and the rank can be estimated with Rank(A)~t{d. The rank deficiency d is equal to the number of the required datums, and the relationship among the parameters is expressed as followŝ In order to get the unknown parameters dX we must select a set of datums, which means the parameters should be solved under the selected datums. Because all of the datums have a functional relationship with the unknown parameters, we can build an equation for each datum. For example, for the plane control network we can establish the following equations if we select points P 1 , P 2 to form the adjustment datum We can also select P 1 as a fixed position, the length S 23 between points P 2 , P 3 as the scale datum, and the azimuth a 45 of P 4 , P 5 as the direction datum. Then the new datum formed is as follows The above nonlinear function is linearised and can be expressed in a matrix form as These three expressions can all be called datum equations. The coefficient matrix G T k of the datum equation (4) is a full rank matrix whose rank should be equal to the number of datums d : Rank(G T k )~d and Rank(A T ,G k )~t.
As can be seen from above derivations, each set of datums corresponds to a group of datum equations. If we select a group of equations, their coefficient matrix is fully rank, and their rank is equal to the required datum number d. In this case we can also regard them as the datum equations.
Assume that there is an equation The rank of G is Rank(G)~d and it meets the condition AG50. Select the equation as a datum equation, which we generally call a gravity centre datum equation. For a plane control network with coordinates as unknown parameters, generally we can set G as The solution given under the gravity centre datum is also the solution of a rank deficient adjustment. The geometric meaning of the gravity centre datum is that: The position datum is fixed to the average values of each point's coordinates (the gravity coordinate); the scale datum is fixed to the weighted average of the ratios of the adjusted distances for each point to the gravity centre over their initial values; and the directional datum is fixed to the weighted average azimuth value of each point to the gravity centre.
Let G T be divided into two parts and set Set the datum equation This datum is said to fix the gravity centre of some points as the overall datum, which is also called the quasi-stable datum.

Solution based on datum equations Normal equation and its solution
In order to obtain the least squares solution dX X under the selected datum, we can use datum data to eliminate d datum parameters in dX X . This can make the design matrix A a full rank matrix and then we can get the solution through the least squares adjustment. However it is easy to overlook the significance of the datum. Therefore, it is necessary to use the following method.
The datum equations show the functional relationship among the unknown parameters, and they form the conditional equations. We can get the normal equations under parameter adjustment with constraints through combining equations (2) and (4) (2) Chen et al. Surveying adjustment datum and relative deformation accuracy analysis

Survey Review
where K is the connection number vector of the datum equations.
Left multiply the first equation in equation (7) with G T to get With AG50 and the rank of G T We get K50 Left multiply the second equation in equation (7) with G k and add it to the first equation, we get Equation (8) is a full rank equation, it could obtain the least squares solution based on the datum equation (4) dX k~Qk (A T Pl{G k W k ) where Q k~( A T PAzG k G T k ) {1 , the subscript k represents used here is similar as that in equation (4), which can be defined by forming different datums. Equation (8) is also called the normal equations, corresponding to the datum in equation (4).

Co-efficient factor of unknown parameters
Applying the covariance propagation law to equation (9), the co-efficient factor of dX X could be estimated as follows Since The co-efficient factors of corrected observationsL L and their corrections V could also be calculated Taking the second expression of equation (13) into equation (14) and considering equation (11), we have Then the estimation formula of unit weight variance can be calculated as followŝ

Solutions with gravity centre datum and quasistable datum
When the gravity centre datum is used (refer to equation (5)), the least squares solution can be expressed as equation (16) dX Equation (16) is a rank deficient solution. It can be proven that the solution (rank deficient solution) based on the gravity centre datum has the following characteristics (i) dX T g dX g~m in (ii) Tr(QX X g )~min (iii) unbiased when G T dX~0. When the quasi-stable datum is used, as described with equation (6), its observation and normal equations could be expressed as follows From above discussion it can be found that once the datums are decided the unknown parameters could be estimated with equations (9) and (12). These above methods are not only suitable for the solutions under the general datums, but also fit for the solutions under both the gravity centre datum and the quasi-stable datum.

Relative deformation and its positional accuracy
Suppose the datum expressed by equation (4) has a solution dX k and its corresponding covariance factor is Chen et al. Surveying adjustment datum and relative deformation accuracy analysis Survey Review QX X k . If there is another datum expressed as equation (18) The solution of the datum equation (18) could also be calculated using a similar approach as introduced above Next, to derive the relationship between dX j ,QX X j and dX k ,QX X k . Transform the above expression to Considering equation (7) and the relationship of equation (11), then According to the covariance propagation law, then (20) and (21) represent the relationship between dX j ,QX X j and dX k ,QX X k , and they can be used to transform solutions obtained under different datums.
It is not difficult to understand that to an adjustment problem, different datums will produce different solutions with different accuracies. Those different datums and their corresponding covariance factors could be transformed using equations (20) and (21). In other words, the solutions and accuracies of an adjustment problem are only meaningful when attached to a certain datum. As for a plane control network or three-dimensional control network, using equations (20) and (21) the solution and its corresponding positional accuracy under the one datum could be expressed as the results in another datum. This kind of relative positional accuracy is a generalised one. Compared with the relative positional accuracy based on the coordinate difference of two points, this expression has a more comprehensive meaning.
Suppose dX k and QX X k are the solution and associated covariance factor attained based on one datum in a plane monitoring network, the relative deformation and relative positional accuracy based on another datum could be obtained according to equations (20) and (21). If there are three points P 1 ,P 2 ,P 3 in a monitoring network and the points P 1 ,P 2 form the datum points, then the deformation value and relative positional error of P 3 relative to P 1 ,P 2 could be determined with equations (20) and (21). The same process can be used for P 2 when points P 1 ,P 3 are used as the datum and so on. We call them a three-point relative deformation value and three-point relative positional accuracy.

Example
There is a GPS deformation network shown as Fig. 1. If the plane coordinates based on a datum in the previous period had been determined, we want to analyse the deformation according to the new GPS observations acquired. Take the plane coordinates from G06 and G08 as the datum, and we could obtain the deformation value and the accuracy of each point in this datum, and then using equations (20) and (21) we can attain the results of each point by using any two points as a datum. Table 1 lists the deformation values (dx, dy and dz) and their corresponding accuracies (Mx, My and Mz) of points G03, G06 and G05, relative to any two datums. We could conclude from Table 1: (i) normally, the further a point is away from the datum points, the higher the relative positional root mean square error. (ii) in each kind of datum, point G05 shows a larger deformation value than the other points. However when G05 is not used as a datum point, G03, G06 and other points show smaller deformations but when G05 is included all of these points exhibit larger deformations. This phenomenon illustrates that G05 has a deformation of more than 4 cm.

Conclusions
From the above study the following conclusions can be drawn. 1. Datum data could be represented by the datum equations, and a set of datum equations could represent a set of datums.
2. Finding the least squares adjustment solution using datum equations is a universal approach. It is not only suitable for general datums, but also suitable for the gravity centre datum and the quasi-stable datum.
3. We could derive the transform formula to solve the solution of different datums in this way, and it is easy to analyse the relevant problem.
4. Relative solution, relative deformation and relative positional accuracy are generalised and comprehensive.
5. Utilising the concept of relative deformation and relative positional accuracy could make analysis of deformation much easier and more reliable.