CNES/GRGS 10-day gravity field models (release 2) and their evaluation

GRACE, designed to monitor temporal variations in the fluid mass at the surface of the Earth, is still operating and providing invaluable data 7 years after launch. One hundred and ninety-nine satellite-only geopotential solutions to degree and order 50 were recomputed per 10-day interval for the period 29 July 2002–27 May 2008 using an improved data editing and solution regularization procedure. These release 2 solutions are significantly improved compared to release 1 solutions, the noise over deserts and oceans in the form of North–South striping being reduced by 20–40%. This is thanks to the tailored regularization of each individual Stokes coefficient applied in the solution procedure, and to a time-variable reference model containing mean annual, semiannual and secular variations for degrees 2–50 towards which the variations per 10-day solution are constrained. It may attenuate signals of the order of a few percent, whereas this always occurs when applying a Gaussian smoother even with a half-width smoothing radius as small as 300 km. The uncertainty of an individual point in the time series of a basin expressed in equivalent water height inferred from the 10-day solutions is approximately 20 mm. Comparison of these 10-day solutions to monthly GRACE project solutions (CSR, GFZ and JPL) shows substantial differences. Even for the largest basin, the Amazon, a 15% difference in annual amplitude is found between CNES release 2 and CSR versus GFZ and JPL. The mass-loss estimates for East and West Greenland vary by 100%. Sometimes clear outliers are detected in the GFZ and JPL solutions when a particular basin is studied, which have to be eliminated. In view of the large differences detected between the time series for specific basins, it is hazardous to draw conclusions based on a single solution.     

Influence of time variable geopotential models on precise orbits of altimetry satellites,global and regional mean sea level trends

During the last decade a significant progress has been reached in the investigation of the gravity field of the Earth. Besides static, also time variable geopotential models have been recently created. In this paper we investigate the impact of the recent time variable geopotential models on altimetry satellite orbits and such altimetry products based on these orbits, as global and regional mean sea level trends. We show that the modeling of time variable gravity improves the orbit solutions, at least for the GRACE period where time variable gravity is sufficiently accurately observed by this mission. Our analysis includes six geopotential models jointly developed by GFZ German Research Centre for Geosciences and Space Geodesy Research Group (CNES/GRGS) Toulouse: the stationary model EIGEN-GL04S, a stationary version of EIGEN-6S (EIGEN-6S_stat), a corrected version of EIGEN-6S and three enhanced versions of EIGEN-6S called EIGEN-6S2, EIGEN-6S2A and EIGEN-6S2B. By “stationary” we mean “containing periodic parameters such as annual and semi-annual variations, but no secular (drift) terms”. We computed precise orbits for the radar altimetry satellites ERS-1, ERS-2, TOPEX/Poseidon, and Envisat over 20 years between 1991 and 2011. The orbit, single-mission and multi-mission altimetry crossover analyses show that the time variable models EIGEN-6S_corrected, EIGEN-6S2 and its two precursors EIGEN-6S2A/B perform notably better than the stationary models for the GRACE period from 2003 onwards. Thus, using EIGEN-6S2 and EIGEN-6S2A/B we have got 3.6% smaller root mean square fits of satellite laser ranging observations for Envisat, as when using EIGEN-GL04S. However, for the pre-GRACE period 1991–2003, the stationary geopotential models EIGEN-GL04S and EIGEN-6S_stat as well as EIGEN-6S2 having no drift terms for degree 3–50 at this time interval perform superior compared to EIGEN-6S_correct and EIGEN-6S2A/B which contain drifts for this period. We found, that the time variable geopotential models have a low (0.1–0.2 mm/yr) impact on our results for the global mean sea level trend. However, we found strong East/West differences up to 3 mm/yr in the regional mean sea level trends when using orbits of all four satellites based on time variable and stationary geopotential models. We show that these differences are related to the relative drifts of the centers-of-origin between the orbit solutions based on the time variable and stationary geopotential models. From the results of our detailed study, we conclude that the final version of the time variable gravity field model EIGEN-6S2 performs best for the four satellites tested. This model provides the most reliable and mission-consistent sea level estimates for the whole time period from 1992 to 2010. This model is of maximum spherical harmonic degree and order 260 and contains time series for drifts as well as annual and semiannual variations of the spherical harmonic coefficients for degree 2–50.     

GOCE orbit analysis: Long-wavelength gravity field determination using the acceleration approach

The restricted sensitivity of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) gradiometer instrument requires satellite gravity gradiometry to be supplemented by orbit analysis in order to resolve long-wavelength features of the geopotential. For the hitherto published releases of the GOCE time-wise (TIM) and GOCE space-wise gravity field series—two of the official ESA products—the energy conservation method has been adopted to exploit GPS-based satellite-to-satellite tracking information. On the other hand, gravity field recovery from data collected by the CHAllenging Mini-satellite Payload (CHAMP) satellite showed the energy conservation principle to be a sub-optimal choice. For this reason, we propose to estimate the low-frequency part of the gravity field by the point-wise solution of Newton’s equation of motion, also known as the acceleration approach. This approach balances the gravitational vector with satellite accelerations, and hence is characterized by (second-order) numerical differentiation of the kinematic orbit. In order to apply the method to GOCE, we present tailored processing strategies with regard to low-pass filtering, variance–covariance information handling, and robust parameter estimation. By comparison of our GIWF solutions (initials GI for “Geodätisches Institut” and IWF for “Institut für WeltraumForschung”) and the GOCE-TIM estimates with a state-of-the-art gravity field solution derived from GRACE (Gravity Recovery And Climate Experiment), we conclude that the acceleration approach is better suited for GOCE-only gravity field determination as opposed to the energy conservation method.     

Contribution of satellite laser ranging to combined gravity field models

In the framework of satellite-only gravity field modeling, satellite laser ranging (SLR) data is typically exploited to recover long-wavelength features. This contribution provides a detailed discussion of the SLR component of GOCO02S, the latest release of combined models within the GOCO series. Over a period of five years (January 2006 to December 2010), observations to LAGEOS-1, LAGEOS-2, Ajisai, Stella, and Starlette were analyzed. We conducted a series of closed-loop simulations and found that estimating monthly sets of spherical harmonic coefficients beyond degree five leads to exceedingly ill-posed normal equation systems. Therefore, we adopted degree five as the spectral resolution for real data analysis. We compared our monthly coefficient estimates of degree two with SLR and Gravity Recovery and Climate Experiment (GRACE) time series provided by the Center for Space Research (CSR) at Austin, Texas. Significant deviations in C₂₀ were noted between SLR and GRACE; the agreement is better for the non-zonal coefficients. Fitting sinusoids together with a linear trend to our C₂₀ time series yielded a rate of (−1.75 ± 0.6) × 10⁻¹¹/yr; this drift is equivalent to a geoid change from pole to equator of 0.35 ± 0.12 mm/yr or an apparent Greenland mass loss of 178.5 ± 61.2 km³/yr. The mean of all monthly solutions, averaged over the five-year period, served as input for the satellite-only model GOCO02S. The contribution of SLR to the combined gravity field model is highest for C₂₀, and hence is essential for the determination of the Earth’s oblateness.     

An alternative computation of a gravity field model from GOCE

GOCE is the first satellite with a gravitational gradiometer (SGG). This allows to determine a gravity field model with high spatial resolution and high accuracy. Four of the six independent components of the gravitational gradient tensors (GGT) are measured with high accuracy in the so-called measurement band (MB) from 5 to 100 mHz by the GOCE gradiometer. Based on more than 1 year of GOCE measurements, two gravity field models have been derived. Here, we introduce a strategy for spherical harmonic analysis (SHA) from GOCE measurements, with a bandpass filter applied to the SGG data, combined with orbit analysis based on the integral equation approach, and additional constraints (or stabilization) in the polar areas where no observation is available due to the orbit geometry. In addition, we combined the GOCE SGG part with a set of GRACE normal equations. This improves the accuracy of the gravity field in the long-wavelength parts, due to the complementarity of GOCE and GRACE. Comparison with other models and with external data shows that our results are rather close to the GPS-levelling data in well-selected test regions, with an uncertainty of 4–7 cm, for truncation at degree 200.     

DORIS-based point mascons for the long term stability of precise orbit solutions

In recent years non-tidal Time Varying Gravity (TVG) has emerged as the most important contributor in the error budget of Precision Orbit Determination (POD) solutions for altimeter satellites’ orbits. The Gravity Recovery And Climate Experiment (GRACE) mission has provided POD analysts with static and time-varying gravity models that are very accurate over the 2002–2012 time interval, but whose linear rates cannot be safely extrapolated before and after the GRACE lifespan. One such model based on a combination of data from GRACE and Lageos from 2002–2010, is used in the dynamic POD solutions developed for the Geophysical Data Records (GDRs) of the Jason series of altimeter missions and the equivalent products from lower altitude missions such as Envisat, Cryosat-2, and HY-2A. In order to accommodate long-term time-variable gravity variations not included in the background geopotential model, we assess the feasibility of using DORIS data to observe local mass variations using point mascons. In particular, we show that the point-mascon approach can stabilize the geographically correlated orbit errors which are of fundamental interest for the analysis of regional Mean Sea Level trends based on altimeter data, and can therefore provide an interim solution in the event of GRACE data loss. The time series of point-mass solutions for Greenland and Antarctica show good agreement with independent series derived from GRACE data, indicating a mass loss at rate of 210 Gt/year and 110 Gt/year respectively.     

GPS-only gravity field recovery with GOCE,CHAMP, and GRACE

Gravity missions such as the Gravity field and steady-state Ocean Circulation Explorer (GOCE) are equipped with onboard Global Positioning System (GPS) receivers for precise orbit determination (POD), instrument time-tagging, and the extraction of the long wavelength part of the Earth’s gravity field. The very low orbital altitude of the GOCE satellite and the availability of dense 1 s GPS tracking data are ideal characteristics to exploit the contribution of GPS high-low Satellite-to-Satellite Tracking (hl-SST) to gravity field determination. We present gravity field solutions based on about 8 months of GOCE GPS hl-SST data from 2009 and compare the results with those obtained from the CHAllenging Minisatellite Payload (CHAMP) and Gravity Recovery And Climate Experiment (GRACE) missions. The very low orbital altitude of GOCE significantly improves gravity field recovery from GPS hl-SST data above degree 20, but not for the degrees below 20, where the quality of the spherical harmonic coefficients remains essentially unchanged. Despite the limited time span of GOCE data used, the gravity field of the Earth can be resolved up to about degree 115 using GPS data only. Empirically determined phase center variations (PCVs) of the GOCE onboard GPS helix antenna are, however, mandatory to achieve this performance.     

New improved orbit solutions for the ERS-1 and ERS-2 satellites

Improved orbit solutions of the European Remote Sensing Satellites ERS-1 and ERS-2 have been computed in the ITRF2005 terrestrial reference frame using the recent models based mainly on IERS Conventions 2003. These solutions cover the periods 3 August 1991 to 8 July 1996 for ERS-1, and 3 May 1995 to 4 July 2003 for ERS-2. For each satellite, the final orbit solution is based on a combination of three separate orbit solutions independently computed at the Delft Institute of Earth Observation and Space Systems (DEOS) of the Delft University of Technology (The Netherlands), the Navigation Support Office of the European Space Operations Centre (ESOC, Germany) and the Helmholtz Centre Potsdam GFZ German Research Centre for Geosciences (Germany) using three different software packages for precise orbit determination, but using the same models in the same terrestrial reference frame within the European Space Agency (ESA) project ‘Reprocessing of Altimeter Products for ERS (REAPER)’. Validation using radar altimeter data indicates that the new combined orbits of ERS-1 and ERS-2 computed by us are significantly more accurate, approaching the 2–3 cm level in radial direction, than previously available orbit solutions.     

Recovery of the Earth’s gravity field from formation-flying satellites: Temporal aliasing issues

Temporal and mean gravity field models derived from the twin-satellite, leader–follower mission GRACE have provided a new type of information for Earth sciences. In this contribution, we study the potential of various alternative satellite formations for gravity field determination in the post-GRACE era in a simulation environment. In particular, the effects of spherical harmonic truncation and of temporal aliasing in the processing of gravity products from such future formations are investigated.     

The impact of error covariance matrix structure of GRACE’s gravity solution on the mass flux estimates of Greenland ice sheet

The error variance-covariance matrices of the monthly GRACE gravity field models are usually well-structured (e.g., order-leading) to contain the error information of the monthly gravity field models, and they are important information to further improve the accuracy of the estimated mass transportation by a post-processing scheme. Intensive studies have been performed to understand the impact of different approximations of the full error variance-covariance matrix on the mass estimates obtained with the conventional GRACE-post processing methods (e.g., the de-striping method). In this study, based on the variants of the mascon approach which treat monthly GRACE solutions as input, we consider four different structures to the error variance-covariance matrix, (i) full matrix, (ii) block diagonal matrix, (iii) diagonal matrix, and (iv) identity matrix, and examine their impact on the mass transport estimates in Greenland. Using both synthetic and real data, we analyze the results at four temporal scales: (i) the long-term decadal scale, (ii) the inter-annual scale, (iii) the seasonal scale, and (iv) the monthly scale. Based on the synthetic study, we find that for the recovery of the long-term trend, the application of the diagonal structure obtains the best estimates. This is caused by the amplification of the model error in the mascon approach when considering the full and block diagonal structures of error variance-covariance matrix, not due to any imperfection of them. Therefore, we emphasize that one should be aware of mascon model deficiency in the mascon approach. Furthermore, the best inter-annual, seasonal and monthly mass estimates are derived by considering the block diagonal and full structures. This is caused by the behavior of the bias and the unique parameterization error in the case of different structures. A similar finding is also presented in the real data case. Finally, our analysis denotes the necessity of releasing the block diagonal structure together with the official monthly gravity field model for the GRACE Follow-On mission, instead of releasing only the diagonal structure as done for the GRACE mission.     

基于自适应滤波的编队卫星实时相对定轨

提出基于自适应滤波的编队卫星实时相对定轨算法,利用2005-12-09—10两颗GRACE（Gravity Recovery and Climate Experiment）卫星的GPS（Global Positioning System）实测数据进行实时相对定轨试验计算,采用JPL（Jet Propulsion Laboratory）轨道对试验结果外部检核,结果表明:①自适应滤波相对定轨通过自适应因子,可以较好地平衡编队卫星的观测信息和相对动力学信息,其相对定轨结果精度优于Kalman滤波相对定轨结果;②自适应滤波相对定轨结果随着星间基线缩短而精度提高;③两颗GRACE卫星采用单频伪距和广播星历进行自适应滤波相对定轨,可以得到精度优于6cm的星间基线。     

Estimating low resolution gravity fields at short time intervals to reduce temporal aliasing errors

The Gravity Recovery and Climate Experiment (GRACE) satellite mission has been estimating temporal changes in the Earth’s gravitational field since its launch in 2002. While it is not yet fully resolved what the limiting source of error is for GRACE, studies on future missions have shown that temporal aliasing errors due to undersampling signals of interest (such as hydrological variations) and errors in atmospheric, ocean, and tide models will be a limiting source of error for missions taking advantage of improved technologies (flying drag-free with a laser interferometer). This paper explores the option of reducing the effects of temporal aliasing errors by directly estimating low degree and order gravity fields at short time intervals, ultimately resulting in data products with improved spatial resolution. Three potential architectures are considered: a single pair of polar orbiting satellites, two pairs of polar orbiting satellites, and a polar orbiting pair of satellites coupled with a lower inclined pair of satellites. Results show that improvements in spatial resolution are obtained when one estimates a low resolution gravity field every two days for the case of a single pair of satellites, and every day for the case of two polar pairs of satellites. However, the spatial resolution for these cases is still lower than that provided by simply destriping and smoothing the solutions via standard GRACE post-processing techniques. Alternately, estimating daily gravity fields for the case of a polar pair of satellites coupled with a lower inclined pair results in solutions with superior spatial resolution than that offered by simply destriping and smoothing the solutions.     

Improvement of the GRACE star camera data based on the revision of the combination method

The new release of the sensor and instrument data (Level-1B release 02) of the Gravity Recovery and Climate Experiment (GRACE) had a substantial impact on the improvement of the overall accuracy of the gravity field models. This has implied that improvements on the sensor data level can still significantly contribute to arriving closer to the GRACE baseline accuracy. The recent analysis of the GRACE star camera data (SCA1B RL02) revealed their unexpectedly higher noise. As the star camera (SCA) data are essential for the processing of the K-band ranging data and the accelerometer data, thorough investigation of the data set was needed. We fully reexamined the SCA data processing from Level-1A to Level-1B with focus on the combination method of the data delivered by the two SCA heads. In the first step, we produced and compared our own combined attitude solution by applying two different combination methods on the SCA Level-1A data. The first method introduces the information about the anisotropic accuracy of the star camera measurement in terms of a weighing matrix. This method was applied in the official processing as well. The alternative method merges only the well determined SCA boresight directions. This method was implemented on the GRACE SCA data for the first time. Both methods were expected to provide optimal solution characteristic by the full accuracy about all three axes, which was confirmed. In the second step, we analyzed the differences between the official SCA1B RL02 data generated by the Jet Propulsion Laboratory (JPL) and our solution. SCA1B RL02 contains systematically higher noise of about a factor 3–4. The data analysis revealed that the reason is the incorrect implementation of algorithms in the JPL processing routines. After correct implementation of the combination method, significant improvement within the whole spectrum was achieved. Based on these results, the official reprocessing of the SCA data is suggested, as the SCA attitude data are one of the key observations needed for the gravity field recovery.     

Gravity field models from kinematic orbits of CHAMP,GRACE and GOCE satellites

The aim of our work is to generate Earth’s gravity field models from GPS positions of low Earth orbiters. Our inversion method is based on Newton’s second law, which relates the observed acceleration of the satellite with forces acting on it. The observed acceleration is obtained as numerical second derivative of kinematic positions. Observation equations are formulated using the gradient of the spherical harmonic expansion of the geopotential. Other forces are either modelled (lunisolar perturbations, tides) or provided by onboard measurements (nongravitational perturbations). From this linear regression model the geopotential harmonic coefficients are obtained.     

Regional gravity field model in Pakistan area from the combination of CHAMP,GRACE and ground data using least squares collocation: A case study

This study describes a methodology of recovery of the Earth’s gravity field from CHAMP and GRACE satellites data in Pakistan using least squares collocation (LSC) based downward continuation technique. The CHAMP height anomalies and GRACE gravity disturbances derived from the observed satellite data have been used in combination solution using LSC with observed gravity values at the Earth surface. The combined covariance functions of height anomalies and/or gravity disturbances at satellite altitudes and observed gravity anomalies at Earth surface have been used as the basis for combination and downward continuation solution. The variance of predicted gravity anomalies from GRACE gravity disturbances is relatively lower than the corresponding results of gravity anomalies from CHAMP height anomalies. This fact may be attributed partly to the amplification of noise and partly to the unstable inverse transformation process of height anomalies to gravity anomalies. The impact of data error variance has been studied in the context of smoothing and noise reduction in the final solution of downward continuation using least squares collocation. The raising of data error suppresses the noise and as a result a smooth final solution is obtained. The prediction results appear to be dependent on the quality of data and goodness of combined covariance function, which are fairly comparable for the CHAMP and GRACE data. The recovered gravity field from satellite data appears to contribute mainly to medium and long wavelength parts of total gravity field spectrum. Due to flexibility of data handling in least squares collocation, this procedure is applicable to any observable of gravity field being at different altitudes and with different data spacing.     

Lunar gravity field modeling critical analysis and challenges

This paper summarizes and provides a critical analysis of the historical developments of lunar gravitational models from the earliest use of ground based tracking systems of the Lunar Orbiter to the Lunar Prospector mission. This encompasses a comprehensive and critical analysis of the various methods used in the estimation of the gravity coefficients and the processing of large batches of diverse measurements and data types. It has been shown that weakness exists in the current models of the lunar gravity field, which is primarily due to the lack of far side lunar tracking data information, which makes the lunar potential modeling difficult but expected to be overcome as data from SELENE satellite-to-satellite tracking becomes available. Comparisons of various lunar models reveal an agreement in the low order coefficients of the spherical harmonics. However, substantial differences in the models exist in the higher-order harmonics. A numerical comparison has been presented showing the performance of all the contemporary lunar gravitational models used within the astrodynamics community and available in public domain. Improvements to the current models are part of a continuing process and the recent model improvements and future possibilities in lunar gravity modeling are discussed. A brief review of the recent missions has been presented. It is hoped that this critical review will benefit the researchers by presenting the historical as well as state of the art in this field.     

Characteristics and accuracies of the GRACE inter-satellite pointing

For almost 10 years, the Gravity Recovery and Climate Experiment (GRACE) has provided information about the Earth gravity field with unprecedented accuracy. Efforts are ongoing to approach the GRACE baseline accuracy as there still remains an order of magnitude between the present error level of the gravity field solutions and the GRACE baseline. At the current level of accuracy, thorough investigation of sensor related effects is necessary as they are one of the potential contributors to the error budget. In the science mode operations, the twin satellites are kept precisely pointed with their KBR antennas towards each other. It is the task of the onboard attitude and orbit control system (AOCS) to keep the satellites in the required formation. We analyzed long time series of the inter-satellite pointing variations as they reflect the AOCS performance and characteristics. We present significant systematic effects in the inter-satellite pointing and discuss their possible sources. Prominent features are especially related to the magnetic torquer characteristics, star cameras’ performance and KBR antenna calibration parameters. The relation between the magnetic torquer attitude control and the Earth magnetic field, impact of the different performance of the two star camera heads on the attitude control and the features due to uncertainties in the calibration parameters relating the star camera frame to K-frame are discussed in detail. Proper understanding of these effects will help to reduce their impact on the science data and subsequently increase the accuracy of the gravity field solutions. Moreover, understanding the complexity of the onboard system is essential not only for increasing the accuracy of the GRACE data but also for the development of the future gravity field satellite missions.     

A simulation of Martian gravity field recovery by using a near equatorial orbiter

An improvement to the Martian gravity field may be achieved by means of future orbiting spacecraft with small eccentricity and low altitude exemplified through a newly proposed mission design that may be tested in upcoming reconnaissance of Mars. Here, the near equatorial orbital character (with an inclination approximating 10°, eccentricity as 0.01 and semi-major axis as 4000 km) is considered, and its contribution to Martian gravity field solution is analyzed by comparing it with a hypothetical polar circular orbiter. The solution models are evaluated in terms of the following viewpoints: power spectra of gravity field coefficients, correlations of low degree zonal coefficients, precise orbit determination, and error distribution of both Mars free air gravity anomaly and areoid. At the same time, the contributions of the near equatorial orbiters in low degree zonal coefficients time variations are also considered. The present results show that the near equatorial orbiter allows us to improve the accuracy of the Martian gravity field solution, decrease correlation of low degree zonal coefficients, retrieve much better time variable information of low degree zonal coefficients, improve precise orbit determination, and provide more accurate Mars free air gravity anomaly and areoid around the equatorial region.     

On the usefulness of atmospheric and oceanic angular momentum in recovering polar motion and gravity field variations in a unified process

Earth rotation parameters (ERPs) are excited by variations in the mass distribution on the Earth’s surface and the exchange of angular momentum between the atmosphere and oceans and the solid Earth. The same mass redistribution causes temporal changes in the gravity field coefficients with the second degree harmonics related to the rotational deformation and hence to changes in the Earth’s inertial tensor. If precise models of the atmospheric and oceanic angular momentum (AM) are available solution for polar motion and degree 2 Stokes harmonics can be unified. In this study we utilize SLR tracking of LAGEOS to compare (i) degree 2 harmonics from ERPs and gravitation, and (ii) LAGEOS excitation functions and geophysical data (mass + motion). Finally, we investigate to what extent a unified approach is possible with current models for AM data and gravity mass change estimated from ERP within orbit determinations.     

Generation of vertical–horizontal and horizontal–horizontal gravity gradients using stochastically modified integral estimators

The Earth’s gravity field modelling is an ill-posed problem having a sensitive solution to the error of data. Satellite gravity gradiometry (SGG) is a space technique to measure the second-order derivatives of geopotential for modelling this field, but the measurements should be validated prior to use. The existing terrestrial gravity anomalies and Earth gravity models can be used for this purpose. In this paper, the second-order vertical–horizontal (VH) and horizontal–horizontal (HH) derivatives of the extended Stokes formula in the local north-oriented frame are modified using biased, unbiased and optimum types of least-squares modification. These modified integral estimators are used to generate the VH and HH gradients at 250 km level for validation purpose of the SGG data. It is shown that, unlike the integral estimator for generating the second-order radial derivative of geopotential, the system of equations from which the modification parameters are obtained is unstable for all types of modification, with large cap size and high degree, and regularization is strongly required for solving the system. Numerical studies in Fennoscandia show that the SGG data can be estimated with an accuracy of 1 mE using an integral estimator modified by a biased type least-squares modification. In this case an integration cap size of 2.5° and a degree of modification of 100 for integrating 30′ × 30′ gravity anomalies are required.