On the influence of the ground track on the gravity field recovery from high–low satellite-to-satellite tracking missions: CHAMP monthly gravity field recovery using the energy balance approach revisited

In this paper, the influence of the ground track coverage on the quality of a monthly gravity field solution is investigated for the scenario of a high–low satellite- to-satellite tracking mission. Data from the CHAllenging Minisatellite Payload (champ) mission collected in the period April 2002 to February 2004 has been used to recover the gravity field to degree and order 70 on a monthly basis. The quality is primarily restricted by the accuracy of the instruments. Besides, champ passed through a 31/2 repeat mode three times during the period of interest resulting in an insufficient spatial sampling and a degraded solution. Contrary to the rule of thumb by Colombo (The global mapping of gravity with two satellites, Publications on Geodesy, vol 7(3), Netherlands Geodetic Commission, The Netherlands, 263 pp, 1984), see also Wagner (J Geod 80(2): 94–103, 2006), we found that the monthly solutions themselves could be recovered to about degree 30, not 15. In order to improve the monthly gravity solutions, two strategies have been developed: the restriction to a low degree, and the densification of the sampling by the introduction of additional sensitive measurements from contemporaneous satellite missions. The latter method is tested by combining the champ measurements with data from the Gravity Recovery And Climate Experiment (grace). Note that the two grace satellites are considered independent here, i.e. no use is made of the K-band ranging data. This way, we are able to almost entirely remove the influence of the ground track leaving the accuracy of the instruments as the primary restriction on the quality of a monthly solution. These findings are especially interesting for the upcoming swarm-mission since it will consist of a similar configuration as the combined champ and (grace) missions.     

Geographic information system–based morphometric characterization of sub-watersheds of Meenachil river basin,Kottayam district,Kerala, India

The study areas Tikovil and Payppara sub-watersheds of Meenachil river cover 158.9 and 111.9 km², respectively. These watersheds are parts of Western Ghats, which is an ecologically sensitive region. The drainage network of the sub-watersheds was delineated using SOI topographical maps on 1:50,000 scale using the Arc GIS software. The stream orders were calculated using the method proposed by Strahler's (1964 Strahler, A. N. 1964. “Quantitative geomorphology of drainage basins and channel networks”. In Hand book of applied hydrology. Vol. 4, Edited by: Chow, V. T. Vol. 4, 39–44.  [Google Scholar]). The drainage network shows that the terrain exhibits dendritic to sub-dendritic drainage pattern. Stream order ranges from the fifth to the sixth order. Drainage density varies between 1.69 and 2.62 km/km². The drainage texture of the drainage basins are 2.3 km^–1 and 6.98 km^–1 and categorized as coarse to very fine texture. Stream frequency is low in the case of Payappara sub-watershed (1.78 km^–2). Payappara sub-watershed has the highest constant of channel maintenance value of 0.59 indicating much fewer structural disturbances and fewer runoff conditions. The form factor value varies in between 0.42 and 0.55 suggesting elongated shape formed for Payappara sub-watershed and a rather more circular shape for Tikovil sub-watershed. The mean bifurcation ratio (3.5) indicates that both the sub-watersheds are within the natural stream system. Hence from the study it can be concluded that GIS techniques prove to be a competent tool in morphometric analysis.     

A model comparison in least squares collocation

The well known least squares collocation model (I) $$\ell = Ax + \left[ {\begin{array}{*{20}c} O \\ I \\ \end{array} } \right]^T \left[ {\begin{array}{*{20}c} s \\ {s' + n} \\ \end{array} } \right]$$ is compared with the model (II) $$\ell = Ax + \left[ {\begin{array}{*{20}c} R \\ I \\ \end{array} } \right]^T \left[ {\begin{array}{*{20}c} s \\ n \\ \end{array} } \right]$$ The basic differences of these two models in the framework of physical geodesy are pointed out by analyzing the validity of the equation $$s' = Rs$$ that transforms one model into the other, for different cases. For clarification purposes least squares filtering, prediction and collocation are discussed separately. In filtering problems the coefficient matrix R becomes the unit matrix and by this the two models become identical. For prediction and collocation problems the relation s′=Rs is only fulfilled in the global limit where s becomes either a continuous function on the earth or an intinite set of spherical harmonic coefficients. Applying Model (II), we see that for any finite dimension of s the operator equations of physical geodesy are approximated by a finite matrix relation whereas in Model (I) the operator equations are applied in their correct form on a continuous, approximate function \(\tilde s\) .     

The unexpected signal in GRACE estimates of $$C_{20}$$

For science applications of the gravity recovery and climate experiment (GRACE) monthly solutions, the GRACE estimates of \(C_{20}\) (or \(J_{2}\)) are typically replaced by the value determined from satellite laser ranging (SLR) due to an unexpectedly strong, clearly non-geophysical, variation at a period of \(\sim \)160 days. This signal has sometimes been referred to as a tide-like variation since the period is close to the perturbation period on the GRACE orbits due to the spherical harmonic coefficient pair \(C_{22}/S_{22}\) of S2 ocean tide. Errors in the S2 tide model used in GRACE data processing could produce a significant perturbation to the GRACE orbits, but it cannot contribute to the \(\sim \)160-day signal appearing in \(C_{20}\). Since the dominant contribution to the GRACE estimate of \(C_{20}\) is from the global positioning system tracking data, a time series of 138 monthly solutions up to degree and order 10 (\(10\times 10\)) were derived along with estimates of ocean tide parameters up to degree 6 for eight major tides. The results show that the \(\sim \)160-day signal remains in the \(C_{20}\) time series. Consequently, the anomalous signal in GRACE \(C_{20}\) cannot be attributed to aliasing from the errors in the S2 tide. A preliminary analysis of the cross-track forces acting on GRACE and the cross-track component of the accelerometer data suggests that a temperature-dependent systematic error in the accelerometer data could be a cause. Because a wide variety of science applications relies on the replacement values for \(C_{20}\), it is essential that the SLR estimates are as reliable as possible. An ongoing concern has been the influence of higher degree even zonal terms on the SLR estimates of \(C_{20}\), since only \(C_{20}\) and \(C_{40}\) are currently estimated. To investigate whether a better separation between \(C_{20}\) and the higher-degree terms could be achieved, several combinations of additional SLR satellites were investigated. In addition, a series of monthly gravity field solutions (\(60\times 60\)) were estimated from a combination of GRACE and SLR data. The results indicate that the combination of GRACE and SLR data might benefit the resonant orders in the GRACE-derived gravity fields, but it appears to degrade the recovery of the \(C_{20}\) variations. In fact, the results suggest that the poorer recovery of \(C_{40}\) by GRACE, where the annual variation is significantly underestimated, may be affecting the estimates of \(C_{20}\). Consequently, it appears appropriate to continue using the SLR-based estimates of \(C_{20}\), and possibly also \(C_{40}\), to augment the existing GRACE mission.     

Error analysis of the NGS’ surface gravity database

Are the National Geodetic Survey’s surface gravity data sufficient for supporting the computation of a 1 cm-accurate geoid? This paper attempts to answer this question by deriving a few measures of accuracy for this data and estimating their effects on the US geoid. We use a data set which comprises ${\sim }1.4$ million gravity observations collected in 1,489 surveys. Comparisons to GRACE-derived gravity and geoid are made to estimate the long-wavelength errors. Crossover analysis and $K$ -nearest neighbor predictions are used for estimating local gravity biases and high-frequency gravity errors, and the corresponding geoid biases and high-frequency geoid errors are evaluated. Results indicate that 244 of all 1,489 surface gravity surveys have significant biases ${>}2$  mGal, with geoid implications that reach 20 cm. Some of the biased surveys are large enough in horizontal extent to be reliably corrected by satellite-derived gravity models, but many others are not. In addition, the results suggest that the data are contaminated by high-frequency errors with an RMS of ${\sim }2.2$  mGal. This causes high-frequency geoid errors of a few centimeters in and to the west of the Rocky Mountains and in the Appalachians and a few millimeters or less everywhere else. Finally, long-wavelength ( ${>}3^{\circ }$ ) surface gravity errors on the sub-mGal level but with large horizontal extent are found. All of the south and southeast of the USA is biased by +0.3 to +0.8 mGal and the Rocky Mountains by $-0.1$ to $-0.3$  mGal. These small but extensive gravity errors lead to long-wavelength geoid errors that reach 60 cm in the interior of the USA.     

A note on adjustment of free networks

The present paper deals with the least-squares adjustment where the design matrix (A) is rank-deficient. The adjusted parameters \(\hat x\) as well as their variance-covariance matrix ( \(\sum _{\hat x} \) ) can be obtained as in the “standard” adjustment whereA has the full column rank, supplemented with constraints, \(C\hat x = w\) , whereC is the constraint matrix andw is sometimes called the “constant vector”. In this analysis only the inner adjustment constraints are considered, whereC has the full row rank equal to the rank deficiency ofA, andAC ^T =0. Perhaps the most important outcome points to the three kinds of results

1. A general least-squares solution where both \(\hat x\) and \(\sum _{\hat x} \) are indeterminate corresponds tow=arbitrary random vector.
2. The minimum trace (least-squares) solution where \(\hat x\) is indeterminate but \(\sum _{\hat x} \) is detemined (and trace \(\sum _{\hat x} \) corresponds tow=arbitrary constant vector.
3. The minimum norm (least-squares) solution where both \(\hat x\) and \(\sum _{\hat x} \) are determined (and norm \(\hat x\) , trace \(\sum _{\hat x} \) corresponds tow?0

     

Self-consistent treatment of tidal variations in the geocenter for precise orbit determination

We show that the current levels of accuracy being achieved for the precise orbit determination (POD) of low-Earth orbiters demonstrate the need for the self-consistent treatment of tidal variations in the geocenter. Our study uses as an example the POD of the OSTM/Jason-2 satellite altimeter mission based upon Global Positioning System (GPS) tracking data. Current GPS-based POD solutions are demonstrating root-mean-square (RMS) radial orbit accuracy and precision of \({<}1\)  cm and 1 mm, respectively. Meanwhile, we show that the RMS of three-dimensional tidal geocenter variations is \({<}6\)  mm, but can be as large as 15 mm, with the largest component along the Earth’s spin axis. Our results demonstrate that GPS-based POD of Earth orbiters is best performed using GPS satellite orbit positions that are defined in a reference frame whose origin is at the center of mass of the entire Earth system, including the ocean tides. Errors in the GPS-based POD solutions for OSTM/Jason-2 of \({<}4\)  mm (3D RMS) and \({<}2\)  mm (radial RMS) are introduced when tidal geocenter variations are not treated consistently. Nevertheless, inconsistent treatment is measurable in the OSTM/Jason-2 POD solutions and manifests through degraded post-fit tracking data residuals, orbit precision, and relative orbit accuracy. For the latter metric, sea surface height crossover variance is higher by \(6~\hbox {mm}^{2}\) when tidal geocenter variations are treated inconsistently.     

M-estimation with probabilistic models of geodetic observations

The paper concerns \(M\) -estimation with probabilistic models of geodetic observations that is called \(M_{\mathcal {P}}\) estimation. The special attention is paid to \(M_{\mathcal {P}}\) estimation that includes the asymmetry and the excess kurtosis, which are basic anomalies of empiric distributions of errors of geodetic or astrometric observations (in comparison to the Gaussian errors). It is assumed that the influence function of \(M_{\mathcal {P}}\) estimation is equal to the differential equation that defines the system of the Pearson distributions. The central moments \(\mu _{k},\, k=2,3,4\) , are the parameters of that system and thus, they are also the parameters of the chosen influence function. The \(M_{\mathcal {P}}\) estimation that includes the Pearson type IV and VII distributions ( \(M_{\mathrm{PD(l)}}\) method) is analyzed in great detail from a theoretical point of view as well as by applying numerical tests. The chosen distributions are leptokurtic with asymmetry which refers to the general characteristic of empirical distributions. Considering \(M\) -estimation with probabilistic models, the Gram–Charlier series are also applied to approximate the models in question ( \(M_{\mathrm{G-C}}\) method). The paper shows that \(M_{\mathcal {P}}\) estimation with the application of probabilistic models belongs to the class of robust estimations; \(M_{\mathrm{PD(l)}}\) method is especially effective in that case. It is suggested that even in the absence of significant anomalies the method in question should be regarded as robust against gross errors while its robustness is controlled by the pseudo-kurtosis.