# Application of variance components estimation to calibrate geoid error models

- Dong-Mei Guo
^{1}Email author and - Hou-Ze Xu
^{1}

**Received: **17 April 2015

**Accepted: **3 August 2015

**Published: **20 August 2015

## Abstract

The method of using Global Positioning System-leveling data to obtain orthometric heights has been well studied. A simple formulation for the weighted least squares problem has been presented in an earlier work. This formulation allows one directly employing the errors-in-variables models which completely descript the covariance matrices of the observables. However, an important question that what accuracy level can be achieved has not yet to be satisfactorily solved by this traditional formulation. One of the main reasons for this is the incorrectness of the stochastic models in the adjustment, which in turn allows improving the stochastic models of measurement noises. Therefore the issue of determining the stochastic modeling of observables in the combined adjustment with heterogeneous height types will be a main focus point in this paper. Firstly, the well-known method of variance component estimation is employed to calibrate the errors of heterogeneous height data in a combined least square adjustment of ellipsoidal, orthometric and gravimetric geoid. Specifically, the iterative algorithms of minimum norm quadratic unbiased estimation are used to estimate the variance components for each of heterogeneous observations. Secondly, two different statistical models are presented to illustrate the theory. The first method directly uses the errors-in-variables as a priori covariance matrices and the second method analyzes the biases of variance components and then proposes bias-corrected variance component estimators. Several numerical test results show the capability and effectiveness of the variance components estimation procedure in combined adjustment for calibrating geoid error model.

## Keywords

## Background

*h*is ellipsoidal height obtained from GPS measurements,

*H*is orthometric height obtained from leveling survey,

*N*is gravimetric geoid data calculated from global geopotential models or regional gravimetric geoid. Based on this inherent appeal of the relationship between ellipsoidal, orthometric heights and gravimetric geoid data, if any two of the heights are given, the third can be derived with Eq. (1).

In practical terms, the performance of Eq. (1) is more complex due to the adaptation of the parametric models for the systematic errors (e.g. long-wavelength systematic errors in N, distortions in the orthometric height due to an over constrained adjustment of leveling network, etc.) and the correctness for the stochastic model of the observable noises due to the covariance (CV) matrices for the height types obtained from separate adjustments of the individual height types.

Up until today, the former problem has been a topic of interest and well studied by different kinds of parametric functions to absorb the systematic errors of ellipsoidal, orthometric heights and gravimetric geoidal undulations. Many models have been performed ranging from a simple linear regression to more complicated models, such as polynomial fit, multiquadric function fit, Spline fit method and so on (Sansò and Sideris 2013). Since it is evident that the suitability of the parameter model depends on the density, distribution and quality of height network data, there is no universal model applicable to any situation. Here the method of multi-surface function is used after evaluation and assessment of parametric models in this paper.

As for the latter issue, the problem of stochastic modeling for observables in the combined adjustment of heterogeneous height types has not yet to be satisfactorily solved by this traditional formulation. This problem will be the main focal point of this paper. In order to reach the best unbiased estimators of the unknown parameters, a proper CV matrix of the observables is required. Here the well-known method of variance component estimation (VCE) is employed to the combined least-squares (LS) adjustment in Eq. (1) (Teunissen and Amiri-Simkooei 2008; Amiri-Simkooei 2013; Wensch et al. 2013). There are many reasons for performing VCE. For example, VCE is an effective statistical tool to test the noise level. This method is very flexible and easily understood. It can be conducted to estimate variance and CV components for linear and nonlinear stochastic model. The main idea of this paper is to give a detailed analysis of the combination adjustment and to estimate all errors of the heterogeneous heights data and to develop a method for calibrating the geoid error models using the method of VCE. In practice, the iterative minimum norm quadratic unbiased estimation scheme is implemented via a combined adjustment using existing ellipsoidal, orthometric heights and gravimetric geoid data.

## Implement of MINQUE to the combined LS adjustment

### Combined LS adjustment of GPS-leveling heights and gravimetric geoid data

*A*is the

*m*×

*t*matrix of known coefficients depends on the parametric model (

*m*is the number of observational equations,

*t*is the number of unknown parameters),

*x*is

*t*× 1 vector of the unknown parameters depending on the parametric model, \(E\left\{ \bullet \right\}\) is the mathematical expectation operator,

*Q*

_{ v }is the

*m*×

*m*CV symmetric cofactor matrix of the observables,

*σ*is the variance or CV components, and the observations vector

*l*consists of the height ‘misclosure’ at the GPS-leveling benchmark as follows,

*A*corresponding to the parametric model the model of Multi-surface function fit is being used,

*φ*and

*λ*are latitude, longitude respectively, and

*a*

_{ i }are the coefficients.

*m*× 1

*)*vector of unobservable random error with zero mean, for each of heterogeneous height data types are given by,

*B*is the block-structured matrix \(B = \left[ {I{-}I{-}I} \right]\), such that

*I*is an

*m*×

*m*unit matrix.

*C*is a positive-definite symmetric matrix. Here we assume that there are no correlations between the heterogeneous height types.

*Q*(⋅) are known positive cofactor matrices for ellipsoidal, orthometric heights and gravimetric geoid data, and \(\sigma_{h}^{2}\), \(\sigma_{H}^{2}\), \(\sigma_{N}^{2}\) are the corresponding variance components.

*P*associated with the observations take the forms,

*P*(⋅) is assumed diagonal with elements which is the reciprocal of the error variances. According to LS adjustment, the unknown parameters and adjusted residuals of observations can be easily solved. The unknown parameters is given as follows,

According to this combined LS adjustment approach, the solution can be achieved depending on two issues, one is the appropriateness of the parametric models, which refers to the correction for the data inconsistencies and the systematic errors, and another is the residuals of the height data types which allow for the calibration of data covariance matrices. Since the former problem has been well studied, the main work of next section focus on a description of the implement of VCE schemes to the combined LS height adjustment.

### Application of MINQUE to the Combined LS Adjustment

There are many methods available to implement VCE within the LS adjustment (Helmert 1924; Rao 1970). The first solution was proposed by Helmert. And an independent solution was proposed by Rao (1970) who put forward a method known as minimum norm quadratic unbiased estimation (MINQUE). Assuming the observations are normally distributed the approach of Helmert’s and Rao’s are equivalent. In this paper the MINQUE procedure is employed since this procedure does not require distributional assumptions.

*S*is denoted by

*S*

_{ ij }in the matrix is derived by

*tr*(·) is the trace operator, \(i,j = h,H,N\), and

*Q*(·) is the cofactor matrix for observables. Considering the matrix

*S*may not be of full rank, the algorithm of pseudo-inverse should be used to solve Eq. (15).

*R*is a matrix denoted by

*A*is an appropriate matrix corresponding to the parametric model as in Eq. (4) and

*C*

_{1}is the CV matrix of the observables. A CV matrix model for the heterogeneous observation types can be written as the following linear model,

*q*can be expressed as

It is evident from the expression in Eq. (18) that an iterative process should be used because the unknown variance components \(\sigma_{h}^{2}\), \(\sigma_{H}^{2}\) and \(\sigma_{N}^{2}\) are embedded in *C*
_{1}. So, the estimation for the variance components must be conducted with a convergence criterion. Here the iterative performance of MINQUE is employed to determine the variance components. In practice, a convergence criterion should be specified to terminate the computation. In this paper, the computation should not be stopped until all the values are equal.

## Numerical results and discussions

### Description of data

Here a rugged area bounded by 39°N and 41°N, and 82°W and 85°W, is chosen for testing the theory. Several numerical studies are implemented with the two datasets composed of GPS-leveling heights, gravimetric geoid data obtained from gravimetrical data and the initial cofactor matrices for ellipsoidal, orthometric heights and gravimetric geoid.

*g*

_{ fa }is the FA anomaly, Δ

*g*

_{ ref }is the effect of the reference field of EGM2008 on the gravity anomalies (Pavlis et al. 2008), Δ

*g*

_{ rtm }is the terrain effect of RTM reduction with the smooth reference surface, e.g. of resolution 120 km, and Δ

*g*

_{ res }is the residual gravity anomalous obtained by subtracting the effect of a global Earth gravity field model Δ

*g*

_{ ref }and the RTM effect of the topography Δ

*g*

_{ rtm }from Δ

*g*

_{ fa }, \(\Delta g_{res} = \Delta g_{fa} - \Delta g_{\text{ref}} - \Delta g_{\text{rtm}}\). From Table 1, it is obvious that the reduced gravity anomaly is significantly smoother than the FA gravity anomalous.

Statistics of the gravity anomalous (unit: mGal)

Min | Max | Mean | STD | |
---|---|---|---|---|

Δ | −52.5 | 56.4 | 0.2 | 22.9 |

Δ | −47.5 | 17.4 | −1.8 | 15.6 |

Δ | −12.4 | 13.5 | −0.6 | 3.7 |

Δ | −37.1 | 39.1 | 2.5 | 15.0 |

Initial CV matrix characteristics (unit: cm)

GPS | Leveling | Geoid | |
---|---|---|---|

Average σ | 0.7 | 0.6 | 2.0 |

### Iterative MINQUE scheme

When estimating variance components, convergence depends on the given initial values of weight of the measurements. Here we will study an algorithm to improve the convergence speed and behavior of estimating variance components.

*j*th step \(\hat{\sigma }_{h}^{j}\), \(\hat{\sigma }_{H}^{j}\), \(\hat{\sigma }_{N}^{j}\). To continue the next iteration, we compute the new set of two variance components, from which, the weights for the next iteration can be further computed,

### Numerical examples for applying MINQUE to combined LS adjustment

A popular approach is to compute the difference between the gravimetrical geoidal undulation *N*, with ellipsoidal height *h* from GPS measurement and orthometric height *H* from leveling measurement \(\Delta N = h{-}H - N\). The discrepancies Δ*N* denote an evaluation for the accuracy of gravimetric geoid. In order to estimate the accuracy of gravimetric geoid, the different between gravimetric geoid solutions are compared with 165 GPS-leveling heights.

Since the main idea of this paper is to discuss the influence of a priori CV matrices on the final computed variance components, two numerical examples are conducted to assess the gravimetric geoid accuracy in this section. In the scheme of case study II, combined height adjustment is conducted by using the a priori CV matrices as given in Table 2. In the scheme of case study I, combined height adjustment is conducted by using variance-component estimated by MINQUE method. The widely used approach in case study I is the comparison of different geoid model computed by the scheme of case study II over the same region, which is an external method to evaluate the accuracy of gravimetric geoid.

Statistical results of the absolute and relative accuracies of gravimetric geoid (unit: cm)

Fit-model | Inside precision | Outside precision | ||||||
---|---|---|---|---|---|---|---|---|

Max | Min | Mean | SD | Max | Min | Mean | SD | |

Case study I | 2.24 | −1.97 | 0.52 | 1.25 | 8.98 | −7.42 | 2.41 | 4.70 |

CASE study II | 1.32 | −0.89 | 0.16 | 0.29 | 5.40 | −4.63 | 1.15 | 1.91 |

## Conclusions

The main focus point of this study is to give a procedure of geoid determined with available high precise gravimetric geoid and GPS and leveling heights and to improve the accuracy of geoid. In the context, a precise formula for the geoid computation is derived, and the relationship between ellipsoidal, orthometric heights and gravimetric geoid data is conducted. And a detailed algorithm uses the complete description of the CV matrices of the observation vector and of the coefficient matrix, possibly with unknown components of each. In the practice of this context, the iterative application of MINQUE is employed to test the variance components matrices for heterogeneous height data. Numerical case studies shows that the best results were obtained combined with the variance components estimation, with an outside precision of 1.91 cm and an inside precision of 0.29 cm, while the classical method with a-prior CV matrix is a bit worse, with an outside precision of 4.70 cm and an inside precision of 1.25 cm. The results of numerical examples show the capability and effectiveness of variance components estimation procedure in combined adjustment for calibrating geoid error model.

## Declarations

### Authors’ contributions

DG has made substantial contributions to conception and design, or acquisition of data, or analysis and interpretation of data and involved in drafting the manuscript. HX has given final approval of the version to be published; and agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. Both authors read and approved the final manuscript.

### Acknowledgements

This work has been support by the National Natural Science Foundation of China (Grant No: 41321063, 41274050, and 41374021), and Hubei province natural science foundation (Grant No: 2014CFB171).

### Compliance with ethical guidelines

**Competing interests** The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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