Affine distortion of small GPS networks with large height differences

Abstract

GPS is a promising tool for real-time monitoring of deformations of slopes or large structures. However, remaining systematic effects in GPS phase observations after double differencing and application of a priori models affect the resulting coordinates. They complicate the proper separation of the actual deformations from pseudo-deformations induced by the systematic effects. This paper shows that for small monitoring networks (baseline lengths <5 km) only affine distortions of the network geometry are generated by the remaining distance dependent systematic effects, e.g. unmodelled tropospheric and ionospheric propagation effects, or satellite orbit errors. Hence, a generic correction model is given by a three-dimensional affine transformation involving a maximum of 12 transformation parameters. For the determination of these parameters, four high quality GPS stations are necessary which are not affected by the actual deformations to be monitored. Based on the analysis of network geometries of synthetic GPS networks with large height differences and considering the physics of the GPS observations it is shown, however, that less than 12 parameters are sufficient for the computation of the corrections. The proposed 8 parameter model was applied to the GPS monitoring network of the Gradenbach landslide. For this small network with large height differences, it was shown that the distortions can be reduced by about 75%.

Motivation

In engineering geodesy, GPS is used to monitor the deformations of large structures like towers (e.g. Loves et al. 1995), bridges (e.g. Brunner 2005), landslides (e.g. Brunner et al. 2003) or open pit mines (e.g. Kim et al. 2003). The special conditions in these applications are rather detrimental for the use of GPS due to multipathing or large height differences (up to1,000 m) and resulting large variations of the meteorological conditions. In addition, the use of only L1 receivers as well as online processing exclude the application of standard methods for the reduction of systematic effects, e.g. like the ionosphere-free linear combination. Hence, systematic effects remain in the phase data after the double differencing (DD) and application of a priori correction models. These effects create network distortions and complicate the proper determination of the actual deformations. Therefore, appropriate correction approaches should be implemented prior to the deformation analysis in order to eliminate or at least reduce the main impact of the systematic effects.

The principal features of systematic effects have been described by e.g. Beutler et al. (1988) and Santerre (1991). In the context of network RTK various methods have been proposed for a reduction of systematic effects like collocation techniques, low order surface models, or the concept of virtual reference stations. An overview is given in, e.g. Lachapelle and Alves (2002), see also the references therein. All these studies have been carried out for almost horizontal networks. However, large height differences (up to 1,000 m) occur, e.g. for GPS monitoring networks of landslides. Since these networks are often small (baseline lengths <5 km), the ratios of vertical to horizontal network extension can easily reach 30%. Therefore, it is worth studying such three-dimensional networks.

In this paper, we are interested in a general formulation of the influence of distance dependent systematic effects (such as unmodelled tropospheric delays, ionospheric phase advances or orbit errors) in small GPS networks with large height differences. All other systematic effects like, e.g. multipath are not considered. Our investigations will start with a short review of the mathematical concepts.

In the next part, the generic structure of network distortions induced by distance dependent systematic effects is investigated. It will be algebraically shown that for small GPS networks these distortions are affine, i.e. they can be described by a superposition of rotation, translation, shear and strain. As a direct consequence, the formalism of a three-dimensional affine transformation represents a simple but very efficient, generic class of correction models. This was first pointed out by Geiger (1990) and Brunner (1994).

In the third part, it will be shown that less than 12 parameters are sufficient to correct the major part of the systematic distortions for small GPS networks with large height differences. As an example, the proposed 8 parameter model is applied to the GPS monitoring network of the landslide Gradenbach to correct the network distortion.

Mathematical concepts

Least-squares parameter estimation

We briefly summarize the least-squares procedure which is often used to estimate the point coordinates based on GPS double differenced phase observations (DD) (Leick 2004). Let us assume that the ambiguities were already resolved in a previous step. Then, the corresponding model for the network adjustment reads

$$ \begin{aligned}& E{\left\{{\mathbf{y}} \right\}}{\mathbf{= A}} {\mathbf{d}}{\varvec{\upxi}},\quad \\& D{\left\{{\mathbf{y}} \right\}} = \,{\mathbf{M}}_{{\rm DD}} {\mathbf{\Sigma}}_{{\rm ZD}} {\mathbf{M}}^{T}_{{\rm DD}} = {\mathbf{\Sigma}}, \\ \end{aligned} $$

(1)

where y denotes the n ×  1 vector of observed-minus-computed (O − C) values for the DD, ${\mathbf{A}}$ is the n ×  u design matrix for the DD, ${\mathbf{d}}{\varvec{\upxi}}$ the vector of the coordinate correction. The n ×  n regular variance–covariance matrix (VCM) of the DD is given by $D{\left\{{\mathbf{l}} \right\}},$ where ${\mathbf{M}}_{{\rm DD}}$ denotes the matrix operator for double differencing, and ${\mathbf{\Sigma}}_{{\rm ZD}}$ the VCM of the undifferenced GPS phase observations.

Basically, the datum of the GPS network is defined by the introduction of satellites orbits (e.g. from IGS). However, for small networks the information about the origin is only very weakly represented in the design matrix ${\mathbf{A}}$ that introduces a column rank defect of three. In practice, the rank defect can be solved as follows: in a first step, one network point located in stable terrain (bedrock) is chosen as a reference point for the local network. Its precise coordinates are determined with respect to IGS or ITRF stations at a chosen reference epoch. In a second step, these coordinates are held fixed during all consecutive adjustments of the coordinates of the monitoring stations. Assuming an accuracy better than 10 cm for the obtained ITRF coordinates of the reference point, the network distortions induced by this procedure (strain of 3 ppb and rotation of 1.6·10⁻⁹ rad, Beutler et al. 1988) are very small and can therefore be neglected. Larger errors can occur if only an approximate position for the reference station is used.

Assuming the fixed reference station to be the first station in the vector of coordinates, the least-squares estimated coordinate corrections ${\mathbf{d}} \hat {\mathbf{\xi}}$ read

$$ {\mathbf{d}} \hat{\mathbf{\xi}} = {\left({\begin{array}{*{20}c} {{{\mathbf{d}} \hat{\varvec{\upxi}}_{{\rm ref}}}} \\ {{{\mathbf{d}} \hat{\varvec{\upxi}}_{{\rm mon}}}} \\ \end{array}} \right)} = {\left({\begin{array}{*{20}c} {{\mathbf{0}}} \\ {{{\left({{\tilde{\mathbf{{A}}}}^{T} {\mathbf{\Sigma}}^{{- 1}} {\tilde{\mathbf{{A}}}}} \right)}^{{- 1}} {\tilde{\mathbf{{A}}}}^{T} {\mathbf{\Sigma}}^{{- 1}}}} \\ \end{array}} \right)}{\mathbf{y = Uy}}, $$

(2)

where ${{{\mathbf{d}} \hat{\varvec{\upxi}}_{{\rm mon}}}}$ denotes the (u−3) ×  1 vector of coordinate corrections of the monitoring stations and ${\tilde{\mathbf{{A}}}}$ the n ×  (u−3) submatrix of A where the columns associated with the fixed reference station are cancelled.

The vector y can be split into the vector y _S of the residual distance dependent systematic effects and a vector y _R containing the remaining parts, like the random observation errors or the deformation signal

$$ {\mathbf{y = y}}_{{\mathbf{S}}} + {\mathbf{y}}_{{\mathbf{R}}}. $$

(3)

Similarly, the vector of coordinate corrections is decomposed in

$$ {\mathbf{d}} \hat{\mathbf{\xi}}= \mathbf{Uy} = \mathbf{Uy}_{{\mathbf{S}}} + {\mathbf{Uy}}_{{\mathbf{R}}} = {\mathbf{d}} \hat{\varvec{\upxi}}_{{\mathbf{S}}} + {\mathbf{d}} \hat{\varvec{\upxi}}_{{\mathbf{R}}}. $$

(4)

Affine network transformation

Let ${\varvec{\upxi}}^{0}$ denote the point coordinates of the initial, distortion-free network that were used for the linearization in the adjustment and ${\mathbf{d}} \hat{\varvec{\upxi}}_{{\mathbf{S}}} = {\varvec{\upxi}}^{0} + {\mathbf{d}} \hat{\varvec{\upxi}}_{{\mathbf{S}}}$ the coordinates of the network distorted by remaining systematic effects. If we assume a differential affine transformation between ${\varvec{\upxi}}^{0}$ and ${\hat{\varvec{\upxi}}}_{{\mathbf{S}}}$ then for each station i the transformation reads

$$ {\hat{\varvec{\upxi}}}_{{{\mathbf{S}} i}} = {\varvec{\upxi}}^{0} _{i} + {{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} i}} = {\mathbf{F}} {\varvec{\upxi}}^{0} _{i} + {\mathbf{b}}. $$

(5)

The regular 3 ×  3 transformation matrix F can be decomposed in an infinitesimal rotation and an infinitesimal homogeneous strain relation (Jaeger 1969)

$$ {\mathbf{F}} = {\mathbf{dR}} + {\mathbf{E}}, $$

(6)

with the differential rotation matrix ${\mathbf{dR}},$ and the symmetric differential strain matrix ${\mathbf{E}}$

$$ {\mathbf{dR}} = {\left({\begin{array}{*{20}c} {1}& {{- r_{z}}}& {{r_{y}}} \\ {{r_{z}}}& {1}& {{- r_{x}}} \\ {{- r_{y}}}& {{r_{x}}}& {1} \\ \end{array}} \right)} \quad\hbox{and}\quad {\mathbf{E}} = {\left({\begin{array}{*{20}c} {{e_{{xx}}}}& {{e_{{xy}}}}& {{e_{{xz}}}} \\ {{e_{{xy}}}}& {{e_{{yy}}}}& {{e_{{yz}}}} \\ {{e_{{xz}}}}& {{e_{{yz}}}}& {{e_{{zz}}}} \\ \end{array}} \right)}, $$

(7)

where e _xx , e _yy , e _zz denote the three scale parameters, e _xy , e _xz , e _yz the three shear strains, and r _x , r _y , r _z the three differential rotation angles (Jaeger 1969). The three translations (t _x , t _y , t _z ) compose the 3 ×  1 vector ${\mathbf{b}}.$ These 12 parameters are identical for all network points.

For GPS networks the vector of translations is directly determined by

$$ {\mathbf{b =}} - ({\mathbf{F}} - {\mathbf{I}})\,{\varvec{\upxi}}^{0}_{{\rm ref}}, $$

(8)

where ${\varvec{\upxi}}^{0}_{{\rm ref}}$ denotes the coordinates of reference station, that were held fixed in the adjustment (i.e. ${{\mathbf{d}} \hat{\varvec{\upxi}}}_{{\rm ref}} = {\mathbf{0}}).$ Substituting Eq. (8) in Eq. (5) and stacking the coordinate corrections for all p network points in one vector yields

$$ {{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} }} = {\left({\begin{array}{*{20}c} {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 1}}}} \\ {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 2}}}} \\ {\vdots} \\ {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} p}}}} \\ \end{array}} \right)} = {\left({\begin{array}{*{20}c} {{{\mathbf{F - I}}}}& {{\mathbf{0}}}& {\cdots}& {{\mathbf{0}}} \\ {{\mathbf{0}}}& {{{\mathbf{F - I}}}}& {{}}& {\vdots} \\ {\vdots}& {{}}& {\ddots}& {{\mathbf{0}}} \\ {{\mathbf{0}}}& {\cdots}& {{\mathbf{0}}}& {{{\mathbf{F - I}}}} \\ \end{array}} \right)}{\left({\begin{array}{*{20}c} {{{\varvec{\upxi}}^{0}_{1} - {\varvec{\upxi}}^{0}_{{\rm ref}}}} \\ {{{\varvec{\upxi}}^{0}_{2} - {\varvec{\upxi}}^{0}_{{\rm ref}}}} \\ {\vdots} \\ {{{\varvec{\upxi}}^{0}_{p} - {\varvec{\upxi}}^{0}_{{\rm ref}}}} \\ \end{array}} \right)} = {\mathbf{H}} {\mathbf{\Delta}}{\varvec{\upxi}}^{0}, $$

(9)

where H is the 3p ×  3p block diagonal transformation matrix and ${\mathbf{\Delta}}{\varvec{\upxi}}^{0}$ the 3p ×  1 vector of coordinate differences associated with the distortion-free geometry.

Affine network distortions

Structure of distance dependent systematic effects in GPS DD

The magnitude of systematic effects in GPS DD that are caused by orbit errors and unmodelled atmospheric propagation effects depends linearly on the baseline length (Beutler et al. 1988). For small GPS networks (baseline lengths smaller than 5 km and height differences smaller than 1,000 m) we can therefore assume a linear dependence of the elements of y _S on the coordinate difference of the baseline endpoints

$$ {\mathbf{y}}_{{{\mathbf{S}} A,B}} {\mathbf{= T}}{\left({{\varvec{\upxi}}_{B} {\varvec{- \upxi}}_{A}} \right)}, $$

(10)

where ${\mathbf{y}}_{{{\mathbf{S}} A,B}}$ denotes the n × 1 vector of distance dependent systematic effects for DD O − C of the baseline from A to B, ${\mathbf{T}}$ the n ×  3 matrix of the resulting coefficients of the linear relationship, and ${\varvec{\upxi}}_{A}, {\varvec{\upxi}}_{B}$ the coordinates of the baseline endpoints.

Analysis of the transfer of systematic effects

In this section, we will analyse the structure of the matrix U (Eq. (2)) that linearly transfers the systematic effects of the DD O − C y to the coordinates. Considering a GPS network of p point, p − 1 linear independent baselines can be computed. Furthermore, if only the $(\bar{n} + 1)$ satellites that are visible at all stations are considered, Eq. (2) reads explicitly

$$ {{\mathbf{d}} \hat{\varvec{\upxi}}}_{{\mathbf{S}}} = {\left({\begin{array}{*{20}c} {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 1}}}} \\ {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 2}}}} \\ {\vdots} \\ {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} p}}}} \\ \end{array}} \right)} = {\left({\begin{array}{*{20}c} {{{\mathbf{U}}_{{1,1}}}}& {{{\mathbf{U}}_{{1,2}}}}& {\cdots}& {{{\mathbf{U}}_{{1,{\text{p}} - 1}}}} \\ {{{\mathbf{U}}_{{2,1}}}}& {{{\mathbf{U}}_{{2,2}}}}& {{}}& {\vdots} \\ {\vdots}& {{}}& {\ddots}& {{}} \\ {{{\mathbf{U}}_{{{\text{p}},1}}}}& {\cdots}& {{}}& {{{\mathbf{U}}_{{{\text{p,p}} - 1}}}} \\ \end{array}} \right)}{\left({\begin{array}{*{20}c} {{{\mathbf{y}}_{{{\mathbf{S}} 1}}}} \\ {{{\mathbf{y}}_{{{\mathbf{S}} 2}}}} \\ {\vdots} \\ {{{\mathbf{y}}_{{{\mathbf{S}} {\text{p - 1}}}}}} \\ \end{array}} \right)}, $$

(11)

where ${\mathbf{y}}_{{{\mathbf{S}} j}}$ denotes the $(\bar{n} \times 1)$ vector of systematic effects in the DD O − C associated with the baseline j. The $3 \times \bar{n}$ submatrix ${\mathbf{U}}_{{{\text{i,j}}}}$ of U describes the impact of the systematic effects on the coordinates of station i.

If only variance models are considered that are based exclusively on the satellite geometry, e.g. the elevation-dependent model, the matrix ${\mathbf{U}}$ depends only on the network geometry and the satellite distribution. Therefore its elements can be computed without actually carrying out measurements. Various simulation studies for networks with different number of common satellites, varying baseline lengths and different number of stations involved were carried out in MATLAB to study the structure of the matrix U. Two cases can be distinguished for the magnitude of the elements in the submatrices ${\mathbf{U}}_{{{\text{i,j}}}}:$

• Case (i): All elements of ${\mathbf{U}}_{{{\text{i,j}}}}$ are close to zero.

• Case (ii): Almost all elements of ${\mathbf{U}}_{{{\text{i,j}}}}$ are significantly different from zero.

Whether a submatrix ${\mathbf{U}}_{{{\text{i,j}}}}$ is of case (i) or case (ii) depends on the differencing schemes applied and the strategy used to solve the rank defect of the design matrix.

For small networks the structure of the matrix U can be further simplified: first, the submatrices ${\mathbf{U}}_{\rm i,j}$ of case (i) can be considered as matrices of zeros ${\mathbf{0}}_{{3} \times {\bar{n}}}.$ Second, the submatrices ${\mathbf{U}}_{{{\text{i,j}}}}$ of case (ii) can be considered as identical (besides an eventual change of sign of all elements of the submatrix). Both simplifications are allowed since they introduce only small additional network distortions. Using matrix and vector norms, the analysis of the results of the simulation studies showed that the changes in the norm of the coordinate vector of each station are smaller than 0.5 mm if the baseline lengths are smaller than 5 km and maximum systematic effects of 5 cm are considered for DD phase observations.

Figure 1 illustrates the structures of the matrix U and the magnitudes of its elements for different baseline scenarios. The grey shaded elements are significantly different from 0, white elements close to or equal 0. It can be stated that the magnitude of the submatrices of case (i) is about 10^{− 4} ... 10^{− 6} and the ratio in magnitude between the submatrices of both cases is of the order of 10⁵.

Fig. 1

Qualitative grey value plot for the matrix U for different small networks for one epoch. Each column corresponds to one double-differenced phase observation and each line to a coordinate component for the four baseline endpoints. The black triangle indicates the fixed reference station. The third column of the figure depicts the magnitude of the elements of the matrix |U| in log₁₀

The obtained structures of the matrix U are best comprehensible regarding the generic structure of the$3p \times p(\bar{n} + 1)$ transfer matrix V obtained by a free-network adjustment.

$$ {\mathbf{V =}}{\left({{\mathbf{A}}^{T} {\mathbf{\Sigma}}^{{- 1}} {\mathbf{A}}} \right)}^{+} {\mathbf{A}}^{T} {\mathbf{\Sigma}}^{{- 1}} {\mathbf{M}}_{{\rm DD}} {\mathbf{=}}{\left({\begin{array}{*{20}c} {{{\mathbf{V}}_{1}}}& {{\frac{{ - 1}} {{\text{p} - 1}} {\mathbf{V}}_{1}}}& {\cdots}& {{\frac{{ - 1}} {{\text{p} - 1}}{\mathbf{V}}_{1}}} \\ {{\frac{{ - 1}} {{\text{p} - 1}}{\mathbf{V}}_{1}}}& {{{\mathbf{V}}_{1}}}& {\ddots}& {\vdots} \\ {\vdots}& {\ddots}& {\ddots}& {{\frac{{ - 1}} {{\text{p} - 1}}{\mathbf{V}}_{1}}} \\ {{\frac{{ - 1}} {{\text{p} - 1}}{\mathbf{V}}_{1}}}& {\cdots}& {{\frac{{ - 1}} {{\text{p} - 1}}{\mathbf{V}}_{1}}}& {{{\mathbf{V}}_{1}}} \\ \end{array}} \right)}, $$

(12)

where ${\left({{\mathbf{A}}^{T} {\mathbf{\Sigma}}^{{- 1}} {\mathbf{A}}} \right)}^{+}$ is the pseudo-inverse of the normal equation, and V ₁ is the $3 \times (\bar{n} + 1)$ matrix that maps the undifferenced GPS phase observations to the coordinates. The specific choice of the matrix operator ${\mathbf{M}}_{{\rm DD}}$ of the double differencing strategy and the procedure to solve the rank defect of A leads to the special structure of the matrix U, depicted in Fig. 1. Even if the structure of U is quite different for all four network graphs, the obtained network distortions are affine.

Mathematical formulation of affine network distortions

In this section we will show that Eq. (12) can be represented as an affine transformation given by Eq. (9). We start rewriting the matrix U for the star network (all baselines have a common endpoint) given in Fig. 1a

$$ {{\mathbf{d}} \hat{\varvec{\upxi}}}_{{\mathbf{S}}} = {\left({\begin{array}{*{20}c} {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 1}}}} \\ {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 2}}}} \\ {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 3}}}} \\ {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 4}}}} \\ \end{array}} \right)} = {\left({\begin{array}{*{20}c} {{\mathbf{0}}}& {{\mathbf{0}}}& {{\mathbf{0}}} \\ {{\bar{\mathbf{{U}}}}}& {{\mathbf{0}}}& {{\mathbf{0}}} \\ {{\mathbf{0}}}& {{\bar{\mathbf{{U}}}}}& {{\mathbf{0}}} \\ {{\mathbf{0}}}& {{\mathbf{0}}}& {{\bar{\mathbf{{U}}}}} \\ \end{array}} \right)}{\left({\begin{array}{*{20}c} {{{\mathbf{y}}_{{{\mathbf{S}} 12}}}} \\ {{{\mathbf{y}}_{{{\mathbf{S}} 13}}}} \\ {{{\mathbf{y}}_{{{\mathbf{S}} 14}}}} \\ \end{array}} \right)} = {\left({\begin{array}{*{20}c} {{\mathbf{0}}}& {{\mathbf{0}}}& {{\mathbf{0}}} \\ {\bar{\mathbf{U}}}\mathbf{T}& {{\mathbf{0}}}& {{\mathbf{0}}} \\ {{\mathbf{0}}}& {\bar{\mathbf{U}}}\mathbf{T}& {{\mathbf{0}}} \\ {{\mathbf{0}}}& {{\mathbf{0}}}& {\bar{\mathbf{U}}}\mathbf{T} \\ \end{array}} \right)}{\left({\begin{array}{*{20}c} {{{\varvec{\upxi}}^{0}_{2} - {\varvec{\upxi}}^{0}_{1}}} \\ {{{\varvec{\upxi}}^{0}_{3} - {\varvec{\upxi}}^{0}_{1}}} \\ {{{\varvec{\upxi}}^{0}_{4} - {\varvec{\upxi}}^{0}_{1}}} \\ \end{array}} \right)}, $$

(13)

where ${\bar{\mathbf{{U}}}}$ denotes the $3 \times \bar{n}$ identical submatrices of case (ii). Equation (13) can be enlarged by the column for the coordinate difference of station 1 (the fixed reference station)

$$ {{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} }} = {\left({\begin{array}{*{20}c} {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 1}}}} \\ {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 2}}}} \\ {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 3}}}} \\ {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 4}}}} \\ \end{array}} \right)} = {\left({\begin{array}{*{20}c} {{{{\bar{\mathbf{U}}}\mathbf{T}}}}& {{\mathbf{0}}}& {{\mathbf{0}}}& {{\mathbf{0}}} \\ {{\mathbf{0}}}& {{{{\bar{\mathbf{U}}}\mathbf{T}}}}& {{\mathbf{0}}}& {{\mathbf{0}}} \\ {{\mathbf{0}}}& {{\mathbf{0}}}& {{{{\bar{\mathbf{U}}}\mathbf{T}}}}& {{\mathbf{0}}} \\ {{\mathbf{0}}}& {{\mathbf{0}}}& {{\mathbf{0}}}& {\bar{\mathbf{U}}}\mathbf{T} \\ \end{array}} \right)}{\left({\begin{array}{*{20}c} {{{\varvec{\upxi}}^{0}_{1} - {\varvec{\upxi}}^{0}_{1}}} \\ {{{\varvec{\upxi}}^{0}_{2} - {\varvec{\upxi}}^{0}_{1}}} \\ {{{\varvec{\upxi}}^{0}_{3} - {\varvec{\upxi}}^{0}_{1}}} \\ {{{\varvec{\upxi}}^{0}_{4} - {\varvec{\upxi}}^{0}_{1}}} \\ \end{array}} \right)}. $$

(14)

The structure of Eq. (14) is identical with that of an affine transformation given by Eq. (9). Therefore, the network distortions of the network depicted in Fig. 1a that are caused by errors which are a linear function of the station separation are affine. Since the diagonal structure expressed in Eq. (14) holds for star networks with an arbitrary number of stations, network distortions of any small star network are affine. Furthermore, since the different strategies of double differencing lead to identical results (Lindlohr and Wells 1985), all arbitrary baseline formations can be transformed into a star network. Consequently, we have algebraically shown that the network distortions of any small GPS network caused by errors which are a linear function of the station separation are affine.

In order to underline that the differencing scheme has no impact on these results, the small network depicted in Fig. 1c is considered. For this network Eq. (2) reads

$$ {{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} }} = {\left({\begin{array}{*{20}c} {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 1}}}} \\ {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 2}}}} \\ {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 3}}}} \\ {{{{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 4}}}} \\ \end{array}} \right)} = {\left({\begin{array}{*{20}c} {{\mathbf{0}}}& {{\mathbf{0}}}& {{\mathbf{0}}} \\ {\bar{\mathbf{{U}}}}& {{\mathbf{0}}}& {{\mathbf{0}}} \\ {\bar{\mathbf{{U}}}}& {\bar{\mathbf{{U}}}}& {{\mathbf{0}}} \\ {\bar{\mathbf{{U}}}}& {\bar{\mathbf{{U}}}}& {\bar{\mathbf{{U}}}} \\ \end{array}} \right)}{\left({\begin{array}{*{20}c} {{{\mathbf{y}}_{{{\mathbf{S}} 12}}}} \\ {{{\mathbf{y}}_{{{\mathbf{S}} 23}}}} \\ {{{\mathbf{y}}_{{{\mathbf{S}} 34}}}} \\ \end{array}} \right)} = {\left({\begin{array}{*{20}c} {{\mathbf{0}}}& {{\mathbf{0}}}& {{\mathbf{0}}} \\ {\bar{\mathbf{U}}}\mathbf{T}& {{\mathbf{0}}}& {{\mathbf{0}}} \\ {\bar{\mathbf{U}}}\mathbf{T}& {\bar{\mathbf{U}}}\mathbf{T}& {{\mathbf{0}}} \\ {\bar{\mathbf{U}}}\mathbf{T}& {\bar{\mathbf{U}}}\mathbf{T}& {\bar{\mathbf{U}}}\mathbf{T} \\ \end{array}} \right)}{\left({\begin{array}{*{20}c} {{{\varvec{\upxi}}^{0}_{2} - {\varvec{\upxi}}^{0}_{1}}} \\ {{{\varvec{\upxi}}^{0}_{3} - {\varvec{\upxi}}^{0}_{2}}} \\ {{{\varvec{\upxi}}^{0}_{4} - {\varvec{\upxi}}^{0}_{3}}} \\ \end{array}} \right)}. $$

(15)

Evaluating Eq. (15) line by line, it results that the distortion of each point depends only linearly on the coordinate differences between the considered station and the reference station ${\varvec{\upxi}}^{0}_{1}.$ For station 4 the corresponding line of Eq. (15) reads

$$ {{\mathbf{d}} \hat{\varvec{\upxi}}}_{{{\mathbf{S}} 4}} = {{\bar{\mathbf{U}}}\mathbf{T}}{\left({{\varvec{\upxi}}^{0}_{2} - {\varvec{\upxi}}^{0}_{1} + {\varvec{\upxi}}^{0}_{3} - {\varvec{\upxi}}^{0}_{2} + {\varvec{\upxi}}^{0}_{4} - {\varvec{\upxi}}^{0}_{3}} \right)} = {{\bar{\mathbf{U}}}\mathbf{T}}{\left({{\varvec{\upxi}}^{0}_{4} - {\varvec{\upxi}}^{0}_{1}} \right)}. $$

(16)

Therefore, Eq. (16) can be equivalently represented by Eq. (14).

In conclusion, for small GPS networks, we have shown that both, the special structure of the transfer matrix U and the linear dependence of the systematic effects on the station separation are sufficient conditions that the network distortions are affine. For our investigations, the following assumptions were performed:

• (i) Only satellites that are visible at all stations are considered.

• (ii) The baseline lengths are restricted to 5 km and the height differences to 1,000 m.

• (iii) The systematic effects considered depend linearly on the station separation, and the coefficients of this relationship are identical for all baselines.

• (iv) The variance model depends only on the satellite-station geometry.

Correction models for small networks with large height differences

General correction strategy

In the previous sections it was shown that the distance dependent systematic effects that remain after application of a priori correction models lead to affine distortions of small GPS networks. Therefore they can be corrected by a three-dimensional 12 parameters affine transformation. The general correction strategy consists of two steps. In the first step, using the linear relation in Eq. (5), the 12 transformation parameters p are determined from stations (the reference station and further calibration stations) that are not affected by the actual deformation to be monitored

$$ {{\mathbf{d}} \hat{\varvec{\upxi}}}_{{\rm cal}} = {\mathbf{G}}_{{\rm cal}} {\mathbf{p}}, $$

(17)

where ${{\mathbf{d}} \hat{\mathbf{\xi}}}_{{\rm cal}}$ denotes the $u_{{\rm cal}} \times 1$ vector of coordinate estimates (Eq. 2) of the c calibration stations. The general transformation parameters are the three translations and the nine elements of the matrix F: ${\mathbf{p}} = {\left({f_{{11}}, f_{{12}}, f_{{13}}, t_{x}, f_{{21}}, f_{{22}}, f_{{23}}, t_{y}, f_{{31}}, f_{{32}}, f_{{33}}, t_{z}} \right)}^{T}.$ The u _cal ×  12 matrix ${\mathbf{G}}_{{\rm cal}}$ reads

$$ {\mathbf{G}}_{{\rm cal}} = {\left({\begin{array}{*{20}c} {{{\mathbf{G}}_{1}}}& {{\mathbf{0}}}& {{\mathbf{0}}} \\ {{\mathbf{0}}}& {{{\mathbf{G}}_{1}}}& {{\mathbf{0}}} \\ {{\mathbf{0}}}& {{\mathbf{0}}}& {{{\mathbf{G}}_{1}}} \\ \end{array}} \right)},\quad {\text{with}}\quad {\mathbf{G}}_{1} = {\left({\begin{array}{*{20}c} {{x_{1}}}& {{y_{{1}}}}& {{z_{1}}}& {1} \\ {{x_{2}}}& {{y_{2}}}& {{z_{2}}}& {1} \\ {\vdots}& {\vdots}& {\vdots}& {\vdots} \\ {{x_{c}}}& {{y_{c}}}& {{z_{c}}}& {1} \\ \end{array}} \right)}. $$

(18)

In the second step, all stations are corrected by

$$ {{\mathbf{d}} \hat{\varvec{\upxi}}}^{{\rm corr}} = {{\mathbf{d}} \hat{\varvec{\upxi}}} - {\mathbf{G}} {\mathbf{p}}, $$

(19)

where ${{\mathbf{d}} \hat{\varvec{\upxi}}}^{{\rm corr}}$ is the u ×  1 vector of corrected coordinate estimates, and p are the determined parameters. The matrix G has the same structure as ${\mathbf{G}}_{{\rm cal}}$ in Eq. (18), but ${\mathbf{G}}_{1}$ contains now the coordinates of all network points. The vector ${{\mathbf{d}} \hat{\varvec{\upxi}}}^{{\rm corr}}$ is free of pseudo-deformations induced by remaining distant dependent systematic effects in the GPS DD O − C. It represents mainly the actual deformation signal, the noise of the coordinate estimation, and eventually other systematic effects, like, e.g. multipath.

This correction strategy however requires at least four “good” GPS stations (the reference station and three calibration stations) that are not affected by the actual deformation to be monitored. From a practical point of view this is very costly and even impossible to realize in some applications due to the specific topographic situation. Hence, it is a worthy investigation to determine if less than 12 parameters are sufficient for the correction. In this context, two aspects play a key role which we will investigate next.

• (i) How many transformation parameters can be precisely determined in local GPS monitoring networks?

• (ii) Which are the dominant distortion patterns created by the systematic effects?

Geometrical aspects

In this section we address the first question, i.e. we analyse how many transformation parameters can be precisely determined in small GPS networks with large height differences. For this task, the distribution of the network points in space has to be considered which has an important influence on the estimability of the transformation parameters.

For the determination of the parameters ${\mathbf{p}}$ the inverse of ${\mathbf{G}}_{{\rm cal}}$ must be computed. Due to the block diagonal structure, this inverse exists only if the submatrix ${\mathbf{G}}_{1}$ can be inverted, i.e. if $\det {\left({{\mathbf{G}}_{1}} \right)} \ne 0.$ From Eq. (18) we can see that a singularity occurs if all points are located close to a common plane in space. Geometrically, the problem (17) is then over-parameterized. This is algebraically expressed by $\det {\left({{\mathbf{G}}_{1}} \right)} \approx 0,$ i.e. ${\mathbf{G}}_{1}$ is ill-conditioned. Consequently, the solution of Eq. (17) is numerically not stable, and wrong parameters p can be obtained. To avoid these problems of over-parameterization in cases of a point distribution close to a plane in space, the following eight parameters can be used: ${\mathbf{p}} = {\left({f_{{11}}, f_{{12}}, t_{x}, f_{{21}}, f_{{22}}, t_{y}, f_{{33}}, t_{z}} \right)}^{T}$ for a parameterisation in a local level system. Alternatively they can be expressed by the six parameters of the two-dimensional affine transformation (e _xx , e _yy , e _xy , r _z , t _x , t _y ) and two additional parameters for the height (e _zz , t _z ). We will call this parameter set the eight parameter model.

In conclusion, from the geometric and algebraic point of view, the determination of 12 transformation parameters requires well-distributed points in space. Considering, e.g. GPS networks for the monitoring of landslides, the point distribution will probably be closer to a plane in space (the slope) than well distributed. In those cases, the earlier indicated eight parameters ought to be used.

Physical aspects

In this section we analyse which distortion patterns for small monitoring networks can be generated by distance dependent systematic effects in GPS DD. For horizontal networks the main distortion patterns have already been described by, e.g. Beutler et al. (1988), Brunner (1994), Geiger (1990), or Santerre (1991). Here, the results obtained by simulation studies in MATLAB are summarized that were carried out for various satellite distributions with the synthetic network depicted in Fig. 2.

Fig. 2

Representation of the synthetic network: the baselines are computed with respect to the central point (Ref), whose coordinates were held fixed during the adjustment

A total of 125 network points located in 5 superimposed layers are generated. The vertical separation between the layers is 500 m. In each layer 25 points are equidistantly located at the intersection of a grid of 1,000 m side. The coordinates of the central point located at the height of 1,500 m were held fixed. Without loss of generality a star network was chosen where all baselines start at the central point. So, independent baselines from 1 to 3 km are computed in all spatial directions. The elements of the vector y _S of systematic effects are simulated and neither noise nor signals are considered, cf. Eq. (4). All receivers and antennas are assumed to be identical and all visible satellites are tracked by all receivers. After each simulation run (one epoch), 12 transformation parameters are estimated from all 125 points of this 3D network in order to extract the main distortion patterns.

It is worth noting that due to the linearity of both, the adjustment formulation (Eq. 2) and the affine transformation (Eq. 5) the following holds: Each effect can be analysed separately. The whole network distortion is reconstituted by superposition of the individual network distortions associated with each effect. Each individual and the superimposed network distortions are described by affine transformations.

Orbits

In this investigation we assume that remaining systematic effects of precise IGS orbits can be neglected. Then, only for real-time monitoring, satellite errors may play a role if the used receivers cannot honour the broadcast health flag. In those cases, possible variations of a satellite from its nominal position are undetectable. To study this effect we simulate for one satellite only (in maximum elevation) a large offset of 100 m in each coordinate component. The maximum magnitudes in the resulting point coordinates are 0.01...0.03 m for different satellite constellations. The mechanisms that lead to the specific distortion patterns are explained in Beutler et al. (1988) and Brunner (1994). Here, we found mainly shear strain between the horizontal and vertical coordinate component.

Ionosphere

If two frequencies (L1, L2) are measured, the ionosphere-free linear combination can be computed to reduce the first-order part of the ionospheric effect. To minimize the hardware cost often only L1 receivers are used in local GPS networks. Therefore the ionospheric phase advance can be only partially reduced by approximate models and remains in the data. The Vertical Total Electron Content (VTEC) values from the CODE IONEX map of October 1, 2004 (Schaer 1999) are used to study exemplarily the ionospheric effects for the synthetic 3D network. As stated in e.g. Beutler et al. (1988), the ionospheric phase advance leads to a baseline shortening. For our synthetic 3D network, a network contraction in all three components is obtained. Here, maximum magnitudes of e _zz = −5 ppm occurred around noon in the vertical component. In addition in some simulated coordinate sets very small rotations about the local north axis can be found that are induced by horizontal VTEC gradients.

Troposphere

The wet tropospheric part is difficult to model. Therefore a large variability exists between the actual atmospheric conditions and those assumed by the a priori tropospheric model. For our simulation studies, the wet delay part given by the Saastamoinen (1973) model and the Niell (1996) wet mapping function is used to describe the remaining systematic effect. As stated in Beutler et al. (1988) relative tropospheric errors cause biases in the height component which magnitudes depend on the baseline inclination. For our 3D synthetic network, these results are reflected in a shortening and a translation of the network in the height component (e _zz =  − 100 ppm, t _z =  30 mm). This pattern is not affine. This is due to the asymmetry of the model values with respect to the reference point located at 1,500 m, i.e. the nonlinear height dependence of the model values over a range of 500 m...2,500 m in ellipsoidal height. For this range, the quadratic terms have to be added (Beutler et al., 1995, p 120ff) and Eq. (10) is not valid any more. However, if only points above the reference station are considered (height differences of 1,000 m), the tropospheric effect is linear and a scale parameter of e _zz = −60 ppm was obtained.

In conclusion, for small GPS networks with height differences smaller than 1,000 m not all 12 parameters are necessary to describe the dominant structure of systematic effects. In fact, mainly network distortions in the height component occur.

Application to the GPS monitoring network of the landslide Gradenbach

GPS monitoring network Gradenbach and data processing

Figure 3 shows a digital terrain model of the Gradenbach area (5 × 5 km²) and the location of the 6 GPS stations of the monitoring network of the landslide: three stations (R1, R2, R4) in located stable terrain (bedrock), and three monitoring stations MA, MC, and MD on the slope, Brunner at al. (2003). All six GPS stations are equipped with Ashtech receivers, and Ashtech choke-ring antennas with SCIS-type radome.

Fig. 3

Landslide Gradenbach area with GPS stations

The GPS data (3 s data rate) was processed by a Kalman-filter based Matlab toolbox developed by A. Wieser at EGMS TUGraz. For the processing of a representative 8-h data set (Oct 2, 2004, 20:00–Oct 3, 2004, 4:00) a cut-off angle of 10°, a spectral noise density of q =  10^{− 9} m² Hz, Klobuchar-style ionospheric correction, and the SIGMA-ɛ variance model (Hartinger and Brunner 1999) were applied. The coordinates from a 48-h static Bernese 5.0 solution (with estimated ZWDs every 2 h) were used as a priori values. The baselines were computed with respect to station R2, and the coordinates of R2 were held fixed during the adjustment. Table 1 indicates the baseline lengths and height differences.

Table 1 Overview on the baseline lengths and height differences in the Gradenbach network

Figure 4 shows the obtained coordinate variations for the North, East and Up components with respect to the static Bernese solution, respectively. The height component (Fig. 4c) has a typical height dependent pattern of the tropospherically induced bias: the larger the height differences with respect to the reference station, the larger the bias, cf. Gurtner et al. (1989) or Rührnößl et al. (1998). Here, a mean scale factor e _zz ≈−52 ppm is obtained.

Fig. 4

Variations of the coordinate time series with respect to the BENESE 48-h solution: a north component, b east component, c height, d mean horizontal variation

The variations of the horizontal components (Fig. 4a, b) show—at a first glance—no well pronounced systematic patterns. However, taking the median over the 8-h time series, a small bias in baseline direction remains (Fig. 4d). This bias may be explained by absolute tropospheric or remaining ionospheric effects (Beutler et al. 1988). Like for many GPS monitoring applications, the critical direction to be monitored by the Gradenbach network, i.e. the direction in which the landslide is most probably to move, coincides with the baseline direction. Therefore, for high-precision applications it is important to correct both, the vertical and the horizontal coordinate components.

Correction

For the correction in the coordinate domain, the eight parameter model (six parameters of a two-dimensional affine transformation and a translation and strain for the vertical component) was used. The translation parameters are determined by holding the coordinates of the reference station (R2) fixed. The remaining five parameters can be determined by at least two stable calibration stations, here R1 and R4. For further details refer to Schön (2006).

Figure 5 shows exemplarily the corrected height time series. In Table 2 the numerical results in terms of the median and rms of the time series are given. For the height component the bias is reduced about 75%, i.e. for the stations MC (MD) from 21 (24) mm to 5 mm. The coordinates of the station MA remains rather unchanged since it is located in the same height as the reference station R2. For the horizontal components, the improvement of the median of the time series is about 50% in North for MA and MD, and 28% for MC. For the east component the bias is reduced by 53% for MA, 36% for MC, and 23% for MD.

Fig. 5

Corrected coordinate time series for the height component

Table 2 Comparison of the numerical results before correction (a priori) and after correction (a posteriori) by the eight parameter correction model

Discussion

For the application of the correction model “good” GPS stations (the reference and calibration stations) are needed. Besides their stability (they should be not affected by the movement to be monitored), their position with respect to the vertical network extension is important. In fact, it is preferable that the heights of the calibration station and the reference station are upper and lower bounds to the heights of all monitoring stations in order to cover the extreme meteorological conditions and to avoid extrapolation in the correction formulas, cf. Schön (2006).

If only the dominant vertical distortions should be corrected, the 2-parameter model (Rührnößl et al 1998; Schön et al. 2005) is an interesting alternative, since only two instead of three stable stations have to be explored, installed, and maintained for the use of such a model.

Finally it should be noted that the proposed correction models (8 or 2 parameters) are software independent and directly applicable to the resulting coordinates processed by any scientific or commercial GPS software.

Conclusions

In this article it was shown that algebraically distance dependent systematic effects in GPS observations lead to an affine distortion of small networks (baselines <5 km, height differences <1,000 m). The basic conditions for affine distortions are (i) the linear dependence of the systematic effects on the station separation, (ii) the use of identical satellites at all stations, and (iii) the use of a geometry-based variance model of the DD. The generic class of proper correction models is therefore given by the 12 parameter differential affine transformation.

However, for many applications the use of all 12 parameters is not optimal, since (i) the determination of all 12 parameters is numerically instable, if—like for landslide applications—the points are close to a plane in space, (ii) at least four stable stations are necessary for the determination of all 12 parameters, which is often difficult to achieve due to the specific topographic or geologic situation, (iii) the dominant distortion patterns for networks with large height differences can be described by less than 12 parameters. Consequently, an eight parameter model was proposed that seems to be suitable for the correction of the affine network distortion. Using the example of the GPS monitoring network for the landslide Gradenbach, it was shown that the eight parameter model can reduce up to 78% of the network distortion, e.g. it reduces the height bias from 25 to 5 mm.