# A software package for computing a regional gravimetric geoid model by the KTH method

## Authors

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s12145-014-0149-3

- Cite this article as:
- Abbak, R.A. & Ustun, A. Earth Sci Inform (2015) 8: 255. doi:10.1007/s12145-014-0149-3

## Abstract

Nowadays, the geodetic community has aimed to determine 1-cm accuracy gravimetric geoid model, which satisfies the demands of most engineering applications. However, the gravimetric geoid determination is a difficult mission which needs an exclusive attention to obtain reliable results for this purpose. Today, Least-Squares Modification of Stokes (LSMS) formula which is so-called the KTH method (Swedish Royal Institute of Technology) has been performed in the regional geoid studies. Based upon the earlier investigations, the KTH method provides more reasonable results than the Remove Compute Restore technique, especially in roughly terrain with sparse terrestrial gravity data. Nevertheless, a compact and practical software package is now not available for users and researchers in geosciences. Thus, in this paper, a scientific software called “LSMSSOFT” is developed and presented by adding a new algorithm which speeds up the evaluation of Stokes’ integral. Afterwards, the LSMSSOFT is applied to a case study for the construction of a geoid model over the Auvergne test area in France. Consequently, the algorithm treated in the software and its results imply that the LSMSSOFT is an alternative software package for modelling the gravimetric geoid by the KTH method.

### Keywords

Evaluation of Stokes’ integralGravimetric geoidModification of Stokes’ formulaSoftware packageTerrestrial gravity data## Introduction

The geoid, which is a equipotential surface of the Earth’s gravity field, coincides with the mean sea level more or less by extending inside the continents. Due to irregular distribution of topographical masses and their densities, the geoidal heights vary from −105 m to +85 m with respect to the best fitting reference ellipsoid. Hence, the knowledge about the position of the geoid and/or its surface normals known as vertical directions highly contributes to the studies of the Earth’s interior, oceanographical processes, geophysical interpretations etc. Furthermore, the precise geoid information plays an important role in the geodetic infrastructure because the topographic elevations and depths of the sea are referred from it. In other words, a variety of implementations in geodesy, geophysics, oceanography and engineering requires such kind of heights which are related to the geoid. For this purpose, the spirit levelling technique presents an accurate height determination. Nevertheless, this technique is very time consuming, laborious and costly task for the geodesists to reach the reasonable results. During the two last decades, the Global Positioning System (GPS) provides us an alternative technique to obtain the accurate heights associated with the Earth’s gravity field. However, this alternative method requests a refined gravimetric geoid model in order to supply a comparable accuracy with the spirit levelling, i.e. a few centimetres.

The gravimetric geoid model can be precisely computed by Stokes’ integral with the help of terrestrial gravity anomalies distributed globally. For practical implementations, however, the integration domain is spatially restricted owing to the limited availability of terrestrial gravity measurements over the entire Earth. Molodensky et al. (1962) proposed that the truncation bias (due to ignorance of gravity data at remote zone) can be reduced if the terrestrial gravity data is combined with the long-wavelength information of the gravity field derived from Global Geopotential Model (GGM). This method is called modification of Stokes’ formula. Advancing in technology and geodetic theory, nowadays, this combination has been very effective method for reaching 1-cm accuracy gravimetric geoid model.

The modifications of Stokes’ formula can be broadly split into two categories; *deterministic* and *stochastic* approaches. While the deterministic approaches aim to minimize the truncation error caused by the lack of terrestrial gravity data in remote zone by assuming that GGM coefficients are errorless, stochastic methods are used to reduce global mean square error of the truncation error as well as random errors of GGM and terrestrial gravity data (Sjöberg 2003a). Some other researchers such as Wong and Gore (1969), Meissl (1971), Vincent and Marsch (1974), and Vaníček and Kleusberg (1987) applied different deterministic methods. Then, Sjöberg (1981, 1984, 1991, 2003a) suggested a stochastic method (the KTH method) for the modification. His proposals consider not only the truncation error but also errors of the GGM and terrestrial data, and GGM and local gravity dataset are combined in the spectral domain (i.e. frequency domain) to determine a precise regional geoid model. Sjöberg and Hunegnaw (2000) compared deterministic and stochastic methods in the geoid modelling, and they demonstrated that the reasonable way to obtain a more accurate model is to combine GGMs and local data via the stochastic methods. A numerical study comparing some deterministic and stochastic modification methods can also be found in Ellmann (2005b). Also, some comparisons show that the KTH method is the most precise method in the gravimetric geoid modelling (Ågren et al. 2009; Abbak et al. 2012a; Yildiz et al. 2012).

Although the KTH is known as a successful method in the gravimetric geoid determination, still a compact and practical software package is not available for users and researchers studying in geodesy and related disciplines, e.g. geophysics. Therefore, in this paper, a scientific software package which is called “LSMSSOFT” is developed in the C/C++ programming language by adding a new algorithm and a range of options. By this way, the LSMSSOFT is faster, simpler and more user-friendly than the KTH-GEOLAB developed in the FORTRAN programming language by Kiamehr and Sjöberg (2010).

The current paper starts with a brief review of the computational scheme of the KTH method with additive corrections. Then, we introduce the LSMSSOFT that uses an algorithm of the KTH method and its corrections. Afterwards, we explain the input data and procedure of gridding terrestrial gravity anomalies over the target area. We apply the LSMSSOFT for the construction of a gravimetric geoid model of the Auvergne region by comparing with GPS/levelling data in an absolute and relative sense. Finally, a short summary concludes this paper.

## The KTH Method with additive corrections

The KTH method based on the Least-Squares Modification of Stokes’ (LSMS) Formula has been reviewed in several geodetic literature (e.g. Ellmann 2004; Kiamehr 2006; Ulotu 2009; Abbak 2012b). In this section, we briefly explain the computation of geoidal height by the KTH method and its additive corrections.

*N̂*can be summarized by the following formula (Sjöberg 2003a):

*N*

_{DWC}is the downward continuation (DWC) of gravity anomaly, \(\delta N_{\mbox {\tiny comb}}^{\mbox {\tiny Atm}}\) is the combined atmospheric correction containing (direct and indirect) atmospherical effects on the geoid and δ

*N*

_{ell}is the ellipsoidal correction for the spherical approximation of the geoid in Stokes’ formula to the reference ellipsoidal surface.

*b*

_{n}is the set of spectral parameters that enables to use the observed (original) gravity anomalies instead of residual anomalies in Eq. 2.

*R*is the mean Earth’s radius,

*γ*is the normal gravity at computation point,

*S*

^{L}(

*ψ*) is the modified Stokes’ function,

*L*is maximum degree of the modification, Δ

*g*

_{n}is the gravity anomaly computed from the GGM,

*M*is the upper limit of the spherical harmonic expension.

*G*is the Newtonian gravitational constant,

*ρ*is the Earth’s crust density, and

*H*is the elevation of the topography at the computation point,

*P*.

*r*

_{P}=

*R*+

*H*

_{P}is the spherical radius of the point

*P*,

*H*

_{P}is the orthometric height of point

*P*,

*Q*represents the moving integration point.

*ρ*

^{a}is the density of the atmosphere at sea level. Generally, it is selected that

*ρ*

^{a}is equal to 1.23 kg/m

^{3}for the computations (Sjöberg 1999; Ecker and Mittermayer 1969).

## The LSMSSOFT for computing a gravimetric geoid model

This section comprehensively explains the general structure of the software encoded in the GNU C/C++ platform. It is named “The Least-Squares Modification of Stokes’ formula SOFTware (LSMSSOFT)”. Subsequently, with the aim of speeding up the integration procedures, a new approach designed by the authors is rigorously treated in this software. The next subsection introduces the new approach briefly.

### Functions in the software

MAIN takes the names of data files and user’s options from the command line. In this context, there must be totally three data files: GGM, anomaly and elevation. The GGM file is publicly available from the International Centre for Global Earth Models (ICGEM) website (http://icgem.gfz-potsdam.de/ICGEM). The software has the ability to read all available GGM files in the ICGEM website directly as an input file without any change. The anomaly file in which data is stored in the point format (

*φ*,*λ*, Δ*g*), must cover the target area and at least 2 degrees the nearest vicinity of the target area. The elevation file must coincide with the anomaly file in respect to the resolution and coverage as well. Some parameters (*M*,*ψ*,*C*0) and boundary of the target area with the resolution should be entered to program. After all, the computations according to the theory given in the previous section, are completed by considering all parameters and options, then, the gridded geoid heights (and all segments if desired) in the target area are printed to the stdout sequentially. The flowchart that shows the input files, their formats and outputs are illustrated in Fig. 1.- CHLS generates the inversion of the design matrix
**A**of unknown modification coefficients by using the method of Cholesky Decomposition. Cholesky decomposition for the least-square solution estimates the modification parameter vector**s**from the linear system of equations,where$$ \textbf{A}\textbf{s}=\textbf{h} $$(11)**A**and**h**contain the coefficients of the linear sytem and constant terms, respectively (Ellmann 2005a). Because the algorithm of Cholesky Decomposition is more suitable for a symmetric and positive-definite matrix, this algorithm is employed for the “biased” version of the KTH method. SVD performs the singular value decomposition of the design matrix

**A**. Since the design matrix becomes ill-conditioned in the “unbiased” and “optimum” version of the KTH method, the solution for**s**becomes numerically unstable. Thus, there is an alternative method which is the Singular Value Decomposition (SVD) that avoids the unstability of the inversion of a symmetric matrix. The algorithm in this software was adopted from Press et al. (2007).COEFFICIENT estimates the modification parameters

*s*_{n}and*b*_{n}by the least-squares method. The function considers three stochastic modifications, i.e. the biased (Sjöberg 1984), unbiased (Sjöberg 1991) and optimum (Sjöberg 2003a) version of the KTH method. In this function, the whole computations are realized according to Ellmann (2005a), and results converges his studies.LEVELELL generates the normal gravity field by concerning the reference ellipsoid. As a default, the normal geopotential coefficients are calculated based upon GRS80 reference ellipsoid. However, any geopotential model with the different

*GM*value can be used in the software as it rescales the coefficients only.LEGENDRE calculates the fully normalized associated legendre polynomials of the first and second kind by using the recursion formulae. The parameters of this function are the spherical latitude and maximum degree of the polynomial.

HELP supplies the general information about the usage of the software. In the case where “-H” or unexpected options are used, the HELP function becomes active. It shows how the LSMSSOFT works together with the options, and what parameter is needed for each option.

The software should be compiled in the Linux based server or personal computer, or in Windows operating system by installing free like-linux platforms, e.g. Cygwin. Cygwin can be downloaded from http://www.cygwin.com.

### The new algorithm

The main topic of this subsection is the efficient evaluation of the convolution integral on the sphere, i.e. the Stokesian integration, by means of the new approach. With the intention of expediting the computational procedures of the integration in the Stokes’ formula, a new algorithm designed by the authors was adopted to the LSMSSOFT properly. A similar philosophy for the algorithm was succesfully adapted by Huang et al. (2000). Our algorithm solely considers a compartment for looking a running point *Q*, which is varying according to the grid and integration cap sizes. This structure will make it faster than earlier approach.

The new algorithm can be briefly defined as follows: The curser starts the integration from the minimum latitude and longitude introduced by the user. The research area (hereafter compartment) is rectangular shaped according to the cap and grid sizes (Fig. 2). Considering the cap size *ψ*_{0} (specified by the user), the curser begins to look for the running point *Q*. If the radial distance between the computation point *P* and running point *Q* is less than or equal to the cap size *ψ*_{0}, the modification part of Stokes’ function *S*^{M}(*ψ*) is calculated, otherwise it is passed. This calculated modification value is encoded with respect to its position with *x* and *y* coordinate pairs in the compartment (i.e *x* and *y* are the numbers of grid from the lower left corner of the compartment). All integrations in the east-west direction are made with the help of previously computed Stokesian modification parts. When we move to the next latitude, the modification parts are recomputed at the minimum longitude. Similarly, all integrations can be repeated for every parallel where the geoidal heights are computed. The new algorithm is portrayed in Fig. 2.

The performance of the new strategy was tested against to the classical pointwise method for a gravity data set in central France (i.e current study area). For the area with a maximum latitude difference of 2 degrees, the new method can be up to 0.17 millimetres bias compared to the classical pointwise integration, which should be comfortably neglected in the practice. The mean and standard deviation of the differences are 0.00 mm and 0.04 mm, recpectively. Related with the time difference, the new method finishes the integration in 0.53 seconds on a computer with 2.0 GHz Intel processor, whereas the pointwise method is up to 29.00 seconds on the same processor. This means that the classical method is 55 times worse than the new method in point of view of CPU time.

## A numerical example in the Central France

In this section, we test the potential of the LSMSSOFT by using an example data from Auvergne test area. At first, we make an overview of the datasets, which will be utilized in the LSMSSOFT for geoid modelling procedures. Then, we explain our scheme of gridding the gravity anomalies in the target area. These gravity anomalies are used in the construction of a gravimetric geoid model by applying the KTH approach via the LSMSSOFT. Finally, we evaluate this new model by means of the GPS/levelling data in an absolute and relative sense.

### Input data

^{2}including inner waters. The lowest and the highest points in the target area are 0 m at the southeastern part of the area and 2800 m at the peak of Alps. The area has the average height about 500 m. Its geographical limits are from 45° to 47° northern latitude and from 2° to 4° eastern longitude (Fig. 3). This region was selected because it is one of the most complicated areas in France from the point of view of rough topography and geoidal height variation.

The gravity data used in the current research are obtained from Duquenne (2007). The gravity points are related to the International Gravity Standardization Net 1971 (IGSN71), and also its geographical datum is the World Geodetic System 1984 (WGS84). The geographical distribution of all gravity data points in the target area is shown in Fig. 3b.

The gravity data coverage is satisfactory. The number of gravity points within the target area is about 190 000, which produce a density of one point per 2 km ^{2}, approximately. As it is well known, the density of the gravity data directly affects the accuracy of any geoid model. Also the procedure of the LSMSSOFT needs the gravity data outside of the edges of study area. In this sense, the gravity data in the nearest vicinity of the target area is available.

In 2000, the National Aeronautics and Space Administration (NASA) organized the Shuttle Radar Topography Mission (SRTM) project to generate the most complete high-resolution digital topographic database of the Earth to date. Thus, a Digital Elevation Model (DEM) with 3 ×3 arc-second resolution was created and made publicly available on the Internet. The DEM will fulfill the elevation needs in the current study for the topographical, atmospherical and downward continuation corrections. And also the DEM will be utilized to predict the mean gravity anomalies in our gridding procedure.

GGMs are the representation of Earth’s gravitational potential in terms of spherical harmonic coefficients obtained from space-borne and terrestrial data sources. Every model is different from the other with respect to the resolution, data source, analysis method, quality and density of data. Thus the selection of any GGM in geoid determination directly affects the regional geoid solution (e.g. Abbak 2012b).

ITG-GRACE2010S will be used for the regional geoid modelling with the LSMSSOFT in this study. This model is a satellite-only model derived from 84 months of GRACE tracking data, which is complete to degree and order 180 (Mayer-Guerr et al. 2010). We choose this satellite-only model because this is an appropriate satellite-only GGM for long-wavelength components of gravity knowledge, comparing with the spectral tools (Ustun and Abbak 2010). Additionally, the GGM and ground gravity data are correlated especially when the combined GGM is used in Stokes’ formula. Satellite-only models may avoid this correlation when they are used as a reference GGM.

The EGM Development Team of the National Geospatial-Intelligence Agency (NGA) in USA, has been released the Earth Gravitational Model 2008 (EGM2008) on the internet freely (Pavlis et al. 2012). EGM2008 presents remarkable improvement compared to previous combined GGMs due to the release of new gravity data from more sources, such as terrestrial gravity, GRACE tracking and satellite altimetry data. Therefore, EGM2008 is the best combined model all over the world, comparing via external data (for details see all papers in Newton’s Bulletin Nr. 4 entitled “External Quality Evaluation Reports of EGM2008”). Thus, in the current study EGM2008 will be used for the filling the gravity gaps in both the water bodies and mountainous regions where terrestrial gravity data is unsurveyed.

In the target area, there are 56 GPS/levelling benchmarks which validate the new geoid model in an absolute and relative sense. The accuracy of the ellipsoidal heights determined by GPS is estimated to be ±3 cm, whereas the accuracy of orthometric heights can be estimated to be ±2 cm in the study area (Duquenne 2007).

### Gridding the gravity anomalies

The observed gravity data on physical surface of Earth can be generally represented as *Free-air* or *Bouguer* gravity anomalies in geodesy and geophysics. The free-air anomalies are used for geoid modelling, whereas the Bouguer anomalies are better fitted for interpolation due to their smoothness.

There are two types of Bouguer gravity anomalies: simple and refined/complete. Janak and Vaníček (2005) suggest that the complete Bouguer anomaly is inevitable in high mountains with the sparse gravity points while in the low-elevations the easier simple Bouguer approach could be sufficient in the interpolation. Since our target area is partially mountainous and is of overdetermined gravity data, in this research the simple Bouguer anomalies were regarded enough to predict the mean gravity anomalies. Nonetheless, the complete Bouguer anomalies should be preferably used in rough terrain for the gravity anomaly prediction.

### The KTH geoid model

This subsection presents the computation processes of the new geoid model which is named as Auvergne Geoid Model via the KTH method 2013 (AG13). The LSMSSOFT was used for the whole numerical evaluations. Firstly, the Stokesian integration with gridded free-air gravity anomalies was implied in the target area to obtain approximate geoid undulations. Subsequently, other corrections were separately added to these approximate undulations to get the accurate ones.

Due to the shortage of terrestrial gravity data out of the target area, the integration cap size (*ψ*_{0}) around the computation point is limited to only few degrees while the modification of Stokes’ formula is mainly trying to reduce the truncation error (Kiamehr 2006). Also the selection of upper bound of Stokes’ function *L* plays an essential role in geoid modelling procedure in terms of the computational efficiency. Whereas higher degree GGMs (*M*) may significantly balance the lack of gravity data, the errors of the potential coefficients of GGM increase gradually. Therefore, a reasonable compensation should be determined among these parameters. For example, this procedure can be done such that the value of a parameter is changing whilst the others are keeping constant. The value of the parameter which gives the best result compared to the GPS/levelling data is fixed up (e.g. Abbak 2012b).

*ψ*

_{0}= 1.0°,

*M*=

*L*= 120, \(\sigma ^{2}_{\Delta g}=25\) mGal

^{2}(Fig. 5). Additionally, the additive corrections over the territory of the Auvergne were calculated by using same parameters (Fig. 6).

### Validation of the AG13 model

There are two ways to evaluate the accuracy of any gravimetric geoid model via the GPS/leveling data: the absolute and relative techniques. The absolute technique is estimation of the RMS errors of the discrepancies between the gravimetric and geometric geoid models. A plenty of mathematical models can be used to absorb the long-wavelengths and systematic errors in the geoid model which are ranging from a simple linear regression to a 7-parameter similarity transformation (Kotsakis and Sideris 1999). In this research, the 7-parameter model was preffered since it gives the minimum standard deviation for gravimetric geoid model of the target area. While the one parameter model is utilized for the comparison, the standard deviation of AG13 and EGM08 models are 99.4 mm and 47.2 mm, respectively.

*N*

_{ij}, which are derived from geometric and gravimetric methods are simply calculated for the whole baselines. These differences can be represented in the relative form in parts per million (ppm):

*D*

_{ij}is the distance between

*i*

^{th}and

*j*

^{th}points (unit: km),

*N*

^{Gra}and

*N*

^{Geo}are the gravimetric and geometric undulations of these points (unit: mm).

Evaluating gravimetric geoid models by means of GPS/levelling data in the absolute and relative senses

Geoid model | Absolute statistics (mm) | Relative statistics (ppm) | |||||
---|---|---|---|---|---|---|---|

Min | Max | Mean | RMS | Min | Max | Mean | |

AG13 | −64.4 | 47.8 | 0.5 | 25.6 | 0.0 | 3.1 | 0.4 |

EGM08 | −64.5 | 88.5 | 3.6 | 34.3 | 0.0 | 5.7 | 0.5 |

## Conclusions

The present paper summarized the theory of the KTH method with the additive corrections to a precise regional geoid determination. Then, a scientific software package that is constructed according to the aforementioned theory of the KTH method was presented by adding a new algorithm with a wide number of the options. Finally, the paper was followed by a sample numerical application in partially mountainous part of France where the terrestrial data is overdetermined. The numerical analyses show that the software yields us more reasonable results in the target area, comparing with GPS/leveling data.

This study contributes to the important computational aspects of the KTH method with the additive corrections. In order to find the best solution in the computation of a gravimetric geoid model by the LSMSSOFT, the interested users are strongly advised to experiment with their own data and some alternative modification parameters, and then compare them, finally draw conclusions from the test results.

On the other hand, the software is intentionally designed to be composed of a range of various sub-functions. This structure lets users to replace their own sub-functions (e.g. an alternative singular value decomposition) without any negative influence on the remained part of the software.

Ultimately, the software is completely structural and user friendly, and can be simply applied for the academic purposes. We hope that the availability of the LSMSSOFT increases the eagerness of the geoscience researchers to conduct the KTH method for their own data in the different target areas.

## Acknowledgments

This study was financially supported by The Research Fund of Selcuk University of Turkey under grant 09-101-009. The first author developed some parts of the software while he was at the Royal Institute of Technology in Sweden under supervision of Prof. Sjöberg and at the Tallinn Technology University in Estonia under supervision of Prof. Ellmann. The authors cordially appreciate the constructive remarks made by two anonymous reviewers in the first version of this paper. The authors would like to give a special thanks to Prof. Ramin Kiamehr for his valuable comments on the original manuscript.