Highresolution regional gravity field recovery from Poisson wavelets using heterogeneous observational techniques
 Yihao Wu^{1},
 Zhicai Luo^{1, 2}Email author,
 Wu Chen^{3} and
 Yongqi Chen^{3}
DOI: 10.1186/s4062301706182
© The Author(s) 2017
Received: 8 November 2016
Accepted: 13 February 2017
Published: 24 February 2017
Abstract
Keywords
Regional gravity field recovery Poisson wavelets Tikhonov regularization Quasigeoid/geoid GPS/leveling dataBackground
Highresolution regional gravity field recovery is of considerable importance not only for surveying and mapping, but also for research fields, such as oceanography (understanding ocean circulation and currents), geophysics (investigating the structure of seismic activities and the lithosphere), and geodynamics (Kuroishi 2009; Panet et al. 2011; Shih et al. 2015).
Typically, middle and shortwavelength gravity field signals down to a few kilometers are extracted from highresolution groundbased measurements, e.g., terrestrial and shipborne gravity data, which are only available in geographically limited regions (Wang et al. 2012; Odera and Fukuda 2014; Lieb et al. 2016). In contrast, longwavelength signals from tens of kilometers or larger are often recovered using global geopotential models (GGMs) derived from satellite observations. Over the last 10 years, launches of the Gravity Field and Climate Experiment (GRACE) (Tapley et al. 2004) and Gravity Field and SteadyState Ocean Circulation Explorer (GOCE) (Rummel et al. 2002) missions have greatly contributed to improving the spatial resolution and accuracy of GGMs. Moreover, developing various observational techniques, e.g., GPS, airborne gravimetric measurements, and satellite altimetry missions, can further improve regional gravity fields (Hwang et al. 2006; Jiang and Wang 2016; Wu and Luo 2016). Combined, these data sets form a solid basis for modeling highresolution and highquality regional gravity fields. However, these data have heterogeneous spatial coverage and resolutions, various error characteristics, and different spectral contents, which make their use an open issue. Thus, the aim of this study is to adopt an approach that combines heterogeneous data and extracts different spectral contents from various observational techniques for regional gravity field recovery.
The Stokes/Molodensky integral makes it difficult to combine heterogeneous data, while the least squares collocation (LSC) is numerically inefficient in managing cases that involve a large number of pointwise data (Wittwer 2009). The new gravity field described in this study is parameterized using Poisson wavelets. Poisson wavelets are radially symmetric basis functions that have localizing properties in both the spatial and frequency domains, which have been used extensively in regional gravity field modeling and potential field analysis (Tenzer and Klees 2008; Hayn et al. 2012; Bentel et al. 2013).
We also investigate several aspects that affect the solution quality derived from Poisson wavelets. To begin with, as heterogeneous data have different spatial coverage and resolutions, the derived least squares system from Poisson wavelets is typically illconditioned, where regularization is mandatory for deriving reliable results (Wittwer 2009). One of the key points that affect the quality of regularization is the choice of regularization matrices (Chambodut et al. 2005). Unlike using diagonal regularization matrices in global gravity field modeling from spherical harmonics (Kusche and Klees 2002), the regularization matrices derived from various constraints in regional scale are no longer entirely diagonal. The choice of regularization matrices may affect the solution quality, which is investigated in this study.
Moreover, mainly due to commission errors in the GGMs and uncorrected systematic errors in the data, the computed gravimetric quasigeoid/geoid usually deviates from local values observed from GPS/leveling data by a centimeter level or larger (Wu et al. 2016). Generally speaking, corrector surface (Featherstone 2000; Fotopoulos 2005; Nahavandchi and Soltanpour 2006) or more complicated algorithms, e.g., least squares collocation (Tscherning 1978) and boundaryvalue methodology (Klees and Prutkin 2008; Prutkin and Klees 2008), can be applied to reduce systematic errors and properly combine GPS/leveling data and gravimetric solutions. However, given the difficulty in choosing a proper corrector surface, as well as the associated algorithmic complexity for the leastsquares collocation and boundaryvalue approach, we propose a direct methodology for removing the inconsistency between these two data sets. The GPS/leveling data are treated as an independent observation group and added to the functional model for the gravity field computation. Using this method, the quasigeoid/geoid that fits the local leveling system is computed in one step, no systematic errors exist between the data, and postprocessing procedures, e.g., corrector surface or LSC approaches, for calibrating systematic errors are not required.
The rest of the paper is organized as follows: the main principle of regional gravity modeling from Poisson wavelets is first introduced, and heterogeneous gravityrelated observations are linked to the functional model parameterized by Poisson wavelets. The weights for different observation groups are determined through the variance component estimation (VCE) approach, and unknown coefficients are estimated through the least squares adjustment. Further, the Tikhonov regularization method is introduced to deal with the illconditioned system, where a rapid synthesis method for computing zero and firstorder regularization matrices is provided. In the following section, Hong Kong is selected as the study area and heterogeneous data are introduced. The numerical results are also shown in this part, where the new height reference surface, HKGEOID2016, is determined. In addition, HKGEOID2016 is compared with existing models, e.g., HKGEOID2000 (Luo et al. 2005) and recently published GGMs, such as EIGEN6C4 (European Improved Gravity Model of the Earth by New Techniques 6C4) and EIGEN6C3STAT (Förste et al. 2012, 2014), for crossvalidation. The last section contains the main summary and conclusions.
Methods
Functional model and parameters estimation
Choice of regularization matrix
The regularization matrix is the key element controlling the quality of the regularized solutions, which describes the signal energy decreasing from large to small scale (Chambodut et al. 2005). For selecting the optimal regularization matrix, the zero and firstorder regularization matrices are investigated.
Zeroorder Tikhonov regularization
Firstorder Tikhonov regularization
The zeroorder Tikhonov regularization derived above is suitable for functions restricted to a sphere σ _{U} belonging to L _{2}(σ _{U}). However, this function space is quite large and includes many unsmooth functions. To introduce a scalar product suitable for smoother functions, a target function is selected as the firstorder derivative of the Poisson wavelets, and the inner products of the target functions are used to derive the entries of the firstorder regularization matrix.
Results and discussion
Study area and data
Global geopotential model and digital terrain model
GPS/leveling data
Gravity data
Statistics of the difference between altimetry and marine gravity data (Units: mGal)
Max  Min  Mean  SD 

12.6  −30.0  −4.8  4.3 
Statistics of the residual gravity observations (Units: mGal)
Max  Min  Mean  SD  

Terrestrial Δg − Δg _{EGM2008}  57.2  −29.5  −3.0  11.9 
Terrestrial Δg − Δg _{EGM2008} − Δg _{RTM}  13.7  −18.6  −1.1  4.4 
Shipborne Δg − Δg _{EGM2008}  13.0  −30.4  −8.6  7.2 
Shipborne Δg − Δg _{EGM2008} − Δg _{RTM}  12.9  −20.6  −5.7  6.0 
Satellite altimetry Δg − Δg _{EGM2008}  18.9  −10.9  −0.8  2.3 
Satellite altimetry Δg − Δg _{EGM2008} − Δg _{RTM}  9.8  −8.8  −0.1  1.4 
Numerical results
Determination the regularization matrix and regularization parameter
Evaluation of different geoids with various regularization parameters when zeroorder regularization is used (Units: cm)
Regularization parameters  10^{−14}  10^{−13}  10^{−12}  10^{−11}  10^{−10.8}  10^{−10.7}  10^{−10.6}  10^{−10.4}  10^{−10.2}  10^{−10}  10^{−9}  10^{−8}  10^{−7} 

2.8  2.5  2.3  2.1  2.0  1.9  1.8  1.9  2.0  2.2  2.4  2.6  3.0 
Evaluation of different geoids with various regularization parameters when firstorder regularization is used (Units: cm)
Regularization parameters  10^{−14}  10^{−13}  10^{−12}  10^{−11.8}  10^{−11.6}  10^{−11.4}  10^{−11.3}  10^{−11.2}  10^{−11}  10^{−10}  10^{−9}  10^{−8}  10^{−7} 

Radial constraint  2.4  2.2  2.1  1.9  1.8  1.7  1.6  1.7  1.7  1.8  1.9  2.1  2.3 
Horizontal constraint  2.4  2.2  2.1  1.9  1.8  1.7  1.6  1.7  1.7  1.8  1.9  2.1  2.3 
Optimal network design for Poisson wavelets
A new height reference surface over Hong Kong
Statistics of the residuals of gravity data (Units: mGal)
Max  Min  Mean  SD  

Terrestrial  5.0  −5.7  0.0  1.2 
Marine  5.6  −6.3  0.0  1.1 
Satellite altimetry  15.4  −18.4  0.0  4.8 
Statistics of the residuals of geoidal heights at GPS/leveling points (Units: cm)
Max  Min  Mean  SD  

Group I  0.5  −0.6  0.0  0.2 
Group II  0.8  −1.1  0.0  0.3 
External accuracy of HKGEOID2016 (Units: cm)
Max  Min  Mean  SD 

1.8  −1.9  0.0  1.1 
Statistical differences between GPS/leveling data and geoidal heights based on various geoids (Units: cm)
Max  Min  Mean  SD  

HKGEOID2016  1.8  −1.9  0.0  0.6 
HKGEOID2000  6.9  −6.0  0.8  2.4 
Statistical differences between GPS/leveling data and GGMderived geoidal heights (Units: cm)
Mean  SD  

EGM2008  17.8  4.2 
EIGEN6C4  20.4  4.5 
EIGEN6C3STAT  23.3  4.2 
Conclusions
Poisson wavelets are used for modeling the regional gravity field using data from various observational techniques. The method combines data with different spatial coverage, various noise levels, and spectral contents. As a case study, terrestrial, shipborne, and satellite altimetry gravity data, as well as GPS/leveling measurements, are incorporated in the Poisson wavelet model for regional gravity field recovery over Hong Kong.
The Tikhonov regularization is introduced to manage the illconditioned least squares system; in particular, the performances of various regularization matrices are investigated. The numerical results show solutions with firstorder regularization provide better results, where the accuracy of the local solution increases by 0.2 cm compared to that obtained from zeroorder regularization. These results also indicate that firstorder regularization may be more preferable in regional gravity field recovery using Poisson wavelets. Moreover, a direct approach is proposed to properly combine the gravimetric quasigeoid/geoid and GPS/leveling data; a subset of the GPS/leveling data is treated as an independent observation group to formulate the new functional model, and the quasigeoid/geoid that fits the local leveling system can be modeled in a single step. The results show the SD is approximately 0.3 cm for the residuals on the first two GPS/leveling groups, regarded as observations. The zero mean value indicates the gravimetric model and GPS/leveling data can be properly combined through this direct approach. In addition, an external validation with 61 independent GPS/leveling points shows that the accuracy of the new geoid, HKGEOID2016, is approximately 1.1 cm. Compared with the original solution, HKGEOID2000, the SD of the differences between observed and computed geoidal heights at all GPS/leveling points is reduced from 2.4 to 0.6 cm when HKGEOID2016 is incorporated. This is a significant improvement. In addition, the performances of three recently published GGMs, EGM2008, EIGEN6C3STAT and EIGEN6C4, are investigated in Hong Kong. The corresponding results show the accuracies of these GGMs are all below 4 cm. The deviation from the local geoid at the decimeter level also indicates the GGM alone cannot recover a highquality gravity field in this region, and local refinement is necessary.
Several issues should be carefully considered to make further improvements to the local geoid. Usually, the geoid is poorly modeled in the coastal areas due to unfavorable data coverage (Hipkin et al. 2004; Filmer and Featherstone 2012). In the study region, the distribution of shipborne data is quite sparse, and the quality of the altimetry data degrades in the vicinity of coastal areas. Therefore, the densification of shipborne data would further improve the local geoid in future work. In addition, the developments of appropriate altimeter waveform retracking approaches may also contribute to improving the geoid over coastal areas (Hwang et al. 2006; Andersen and Knudsen 2009). Errors in the GGM inevitably propagate into regional solutions because we usually consider the GGM as errorfree data implemented in the removerestore framework. The magnitude of these commission errors in the GGMs reaches the centimeter scale or larger (Pavlis et al. 2012), which should be carefully considered when computing the quasigeoid/geoid at centimeter accuracy. These commission errors are actually calibrated with GPS/leveling data using the direct approach proposed in this study. However, the errors in the GGM could also be quantified if the full error variance–covariance matrix of the spherical coefficients is known, and a more realistic error variance–covariance matrix of the data could be estimated through error propagation. In this manner, the weights of different observation groups may be more properly determined, and the accuracy of the quasigeoid/geoid, especially for the purely gravimetric one, may be further improved.
Abbreviations
 GPS:

Global Positioning System
 GGM:

global geopotential model
 GRACE:

Gravity Field and Climate Experiment
 GOCE:

Gravity Field and SteadyState Ocean Circulation Explorer
 LSC:

least squares collocation
 VCE:

variance component estimation
 EIGEN:

European Improved Gravity Model of the Earth by New Techniques
 RCR:

removecomputerestore
 RTM:

residual terrain model
 3D:

three dimensional
 DTU:

Technical University of Denmark
 d/o:

degree and order
 DTM:

digital terrain model
 SRTM:

Shuttle Radar Topography Mission
 GEBCO:

General Bathymetric Chart of the Oceans
 MES:

mean elevation surface
Declarations
Authors’ contributions
YW and ZL initiated the study, designed the numerical experiments, and wrote the manuscript. WC and YC provided the data and supplied beneficial suggestions. YW finalized the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to give our sincerest thanks to the two anonymous reviewers for their beneficial suggestions and comments, which helped us improve the manuscript. We would also like to thank the anonymous editor and the editorial office for their work on language editing. We acknowledge the assistance of Prof. Roland Klees and Dr. Cornelis Slobbe from Delft University of Technology for kindly providing the original software. This research was primarily supported by the National Natural Science Foundation of China (41374023), China Postdoctoral Science Foundation (No. 2016M602301), Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, Wuhan University (150208), the State Scholarship Fund from Chinese Scholarship Council (201306270014) and project KZS0J, ‘Development of a Hong Kong Positioning Infrastructure based on GPS, Beidou, and Ground based Augmentation System.’ Generic Mapping Tools (GMT) was used to generate the figures.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis 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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