Radiometric measurements for pearl millet crop

A simple method for determining leaf area index (LAI), absorbed photosynthetically active radiation (APAR), dry biomass and yield in pearl millet (Pennisetum glaucum (L.) R. Br.) crop from remote sensing data is presented. The dry matter and yield can be simulated with maximum accuracy by normalised difference (ND) data during milking stage using the following models: $$\begin{array}{l} DM = - 435.45 + 786.86ND R^2 = 0.90 \\ Yield = - 95.43 + 180.00ND R^2 = 0.62 \\ \end{array}$$      

Evaluation of isotropic covariance functions of torsion balance observations

Torsion balance observations in spherical approximation may be expressed as second-order partial derivatives of the anomalous (gravity) potential,T, $$T_{13} = \frac{{\partial ^2 T}}{{\partial x_1 \partial x_3 }}, T_{23} = \frac{{\partial ^2 T}}{{\partial x_2 \partial x_3 }}, T_{12} = \frac{{\partial ^2 T}}{{\partial x_1 \partial x_2 }}, T_\Delta = \frac{{\partial ^2 T}}{{\partial x_1^2 }} - \frac{{\partial ^2 T}}{{\partial x_1^2 }},$$ wherex ₁ ,x ₂ andx ₃ are local coordinates withx ₁ “east”,x ₂ “north” andx ₃ “up.” Auto- and cross-covariances for these quantities derived from an isotropic covariance function for the anomalous potential will depend on the directions between the observation points. However, the expressions for the covariances may be derived in a simple manner from isotropic covariance functions of torsion balance measurements. These functions are obtained by transforming the torsion balance observations in the points to local (orthogonal) horizontal coordinate systems with first axes in the direction to the other observation point. If the azimuth of the direction from one point to the other point is a, then the result of this transformation may be obtained by rotating the vectors $$\left\{ \begin{gathered} T_{13} \hfill \\ T_{23} \hfill \\ \end{gathered} \right\}and\left\{ \begin{gathered} T_\Delta \hfill \\ 2T_{12} \hfill \\ \end{gathered} \right\}$$ the angles a?90° and 2 (a?90°) respectively. The reverse rotations applied on the 2×2 matrices of covariances of these quantities will produce all the direction dependent covariances of the original quantities.     

Contemporary deformation in the Kashmir–Himachal,Garhwal and Kumaon Himalaya: significant insights from 1995–2008 GPS time series

We present new insights on the time-averaged surface velocities, convergence and extension rates along arc-normal transects in Kumaon, Garhwal and Kashmir–Himachal regions in the Indian Himalaya from 13 years of high-precision Global Positioning System (GPS) time series (1995–2008) derived from GPS data at 14 GPS permanent and 42 campaign stations between $29.5{-}35^{\circ }\hbox {N}$ and $76{-}81^{\circ }\hbox {E}$ . The GPS surface horizontal velocities vary significantly from the Higher to Lesser Himalaya and are of the order of 30 to 48 mm/year NE in ITRF 2005 reference frame, and 17 to 2 mm/year SW in an India fixed reference frame indicating that this region is accommodating less than 2 cm/year of the India–Eurasia plate motion ( ${\sim }4~\hbox {cm/year}$ ). The total arc-normal shortening varies between ${\sim }10{-}14~\hbox {mm/year}$ along the different transects of the northwest Himalayan wedge, between the Indo-Tsangpo suture to the north and the Indo-Gangetic foreland to the south indicating high strain accumulation in the Himalayan wedge. This convergence is being accommodated differentially along the arc-normal transects; ${\sim } 5{-}10~\hbox {mm/year}$ in Lesser Himalaya and 3–4 mm/year in Higher Himalaya south of South Tibetan Detachment. Most of the convergence in the Lesser Himalaya of Garhwal and Kumaon is being accommodated just south of the Main Central Thrust fault trace, indicating high strain accumulation in this region which is also consistent with the high seismic activity in this region. In addition, for the first time an arc-normal extension of ${\sim }6~\hbox {mm/year}$ has also been observed in the Tethyan Himalaya of Kumaon. Inverse modeling of GPS-derived surface deformation rates in Garhwal and Kumaon Himalaya using a single dislocation indicate that the Main Himalayan Thrust is locked from the surface to a depth of ${\sim }15{-}20~\hbox {km}$ over a width of 110 km with associated slip rate of ${\sim }16{-}18~\hbox {mm/year}$ . These results indicate that the arc-normal rates in the Northwest Himalaya have a complex deformation pattern involving both convergence and extension, and rigorous seismo-tectonic models in the Himalaya are necessary to account for this pattern. In addition, the results also gave an estimate of co-seismic and post-seismic motion associated with the 1999 Chamoli earthquake, which is modeled to derive the slip and geometry of the rupture plane.     

Global empirical model for mapping zenith wet delays onto precipitable water

We can map zenith wet delays onto precipitable water with a conversion factor, but in order to calculate the exact conversion factor, we must precisely calculate its key variable $T_\mathrm{m}$ . Yao et al. (J Geod 86:1125–1135, 2012. doi:10.1007/s00190-012-0568-1) established the first generation of global $T_\mathrm{m}$ model (GTm-I) with ground-based radiosonde data, but due to the lack of radiosonde data at sea, the model appears to be abnormal in some areas. Given that sea surface temperature varies less than that on land, and the GPT model and the Bevis $T_\mathrm{m}$ – $T_\mathrm{s}$ relationship are accurate enough to describe the surface temperature and $T_\mathrm{m}$ , this paper capitalizes on the GPT model and the Bevis $T_\mathrm{m}$ – $T_\mathrm{s}$ relationship to provide simulated $T_\mathrm{m}$ at sea, as a compensation for the lack of data. Combined with the $T_\mathrm{m}$ from radiosonde data, we recalculated the GTm model coefficients. The results show that this method not only improves the accuracy of the GTm model significantly at sea but also improves that on land, making the GTm model more stable and practically applicable.     

A technique for routinely updating the ITU-R database using radio occultation electron density profiles

Well credited and widely used ionospheric models, such as the International Reference Ionosphere or NeQuick, describe the variation of the electron density with height by means of a piecewise profile tied to the F2-peak parameters: the electron density, $N_m \mathrm{F2}$ N m F 2 , and the height, $h_m \mathrm{F2}$ h m F 2 . Accurate values of these parameters are crucial for retrieving reliable electron density estimations from those models. When direct measurements of these parameters are not available, the models compute the parameters using the so-called ITU-R database, which was established in the early 1960s. This paper presents a technique aimed at routinely updating the ITU-R database using radio occultation electron density profiles derived from GPS measurements gathered from low Earth orbit satellites. Before being used, these radio occultation profiles are validated by fitting to them an electron density model. A re-weighted Least Squares algorithm is used for down-weighting unreliable measurements (occasionally, entire profiles) and to retrieve $N_m \mathrm{F2}$ N m F 2 and $h_m \mathrm{F2}$ h m F 2 values—together with their error estimates—from the profiles. These values are used to monthly update the database, which consists of two sets of ITU-R-like coefficients that could easily be implemented in the IRI or NeQuick models. The technique was tested with radio occultation electron density profiles that are delivered to the community by the COSMIC/FORMOSAT-3 mission team. Tests were performed for solstices and equinoxes seasons in high and low-solar activity conditions. The global mean error of the resulting maps—estimated by the Least Squares technique—is between $0.5\times 10^{10}$ 0.5 × 10 10 and $3.6\times 10^{10}$ 3.6 × 10 10 elec/m $^{-3}$ ? 3 for the F2-peak electron density (which is equivalent to 7 % of the value of the estimated parameter) and from 2.0 to 5.6 km for the height ( $\sim $ ～ 2 %).     

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.     

Optimal design of broadcast ephemeris parameters for a navigation satellite system

Computation of broadcast ephemerides is a fundamental task in satellite navigation and positioning. The GPS constellation is composed of medium-earth-orbit (MEO) satellites, and therefore can employ a uniform parameter set to produce broadcast ephemerides. However, other navigation satellite systems such as Compass and IRNSS may include a mixture of inclined-geosynchronous-orbit (IGSO), geostationary-earth-orbit (GEO) and MEO satellites, requiring different parameter sets for each type of orbit. We analyze the variational characteristics of satellite ephemerides with respect to orbital elements; then present a method to design an optimal parameter set for broadcast ephemerides, and derive the parameter sets for IGSO, GEO, and MEO satellites. The computational complexities of the user algorithms for the optimal parameter sets are equivalent to that of the standard GPS user algorithm. Simulation and statistical analyses indicate that the optimal parameter set is $ \left\{ {\sqrt {A_{0} } ,e_{0} ,i_{0} ,\Upomega_{0} ,M_{0} ,\omega_{0} ,\dot{\Upomega },\dot{u},\dot{i},C_{\Upomega c3} ,C_{\Upomega s3} ,C_{uc2} ,C_{us2} ,C_{rc2} ,C_{rs2} } \right\} $ for IGSO and GEO satellites, and $ \left\{ {\sqrt {A_{0} } ,e_{0} ,i_{0} ,\Upomega_{0} ,M_{0} ,\omega_{0} ,\dot{\Upomega },\dot{u},\dot{i},C_{uc2} ,C_{us2} ,C_{rc2} ,C_{rs2} ,C_{ic2} ,C_{is2} } \right\} $ for MEO satellites.     

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

     

Effect of the processing methodology on satellite altimetry-based global mean sea level rise over the Jason-1 operating period

Determining how the global mean sea level (GMSL) evolves with time is of primary importance to understand one of the main consequences of global warming and its potential impact on populations living near coasts or in low-lying islands. Five groups are routinely providing satellite altimetry-based estimates of the GMSL over the altimetry era (since late 1992). Because each group developed its own approach to compute the GMSL time series, this leads to some differences in the GMSL interannual variability and linear trend. While over the whole high-precision altimetry time span (1993–2012), good agreement is noticed for the computed GMSL linear trend (of $3.1\pm 0.4$  mm/year), on shorter time spans (e.g., ${<}10~\hbox {years}$ ), trend differences are significantly larger than the 0.4 mm/year uncertainty. Here we investigate the sources of the trend differences, focusing on the averaging methods used to generate the GMSL. For that purpose, we consider outputs from two different groups: the Colorado University (CU) and Archiving, Validation and Interpretation of Satellite Oceanographic Data (AVISO) because associated processing of each group is largely representative of all other groups. For this investigation, we use the high-resolution MERCATOR ocean circulation model with data assimilation (version Glorys2-v1) and compute synthetic sea surface height (SSH) data by interpolating the model grids at the time and location of “true” along-track satellite altimetry measurements, focusing on the Jason-1 operating period (i.e., 2002–2009). These synthetic SSH data are then treated as “real” altimetry measurements, allowing us to test the different averaging methods used by the two processing groups for computing the GMSL: (1) averaging along-track altimetry data (as done by CU) or (2) gridding the along-track data into $2^{\circ }\times 2^{\circ }$ meshes and then geographical averaging of the gridded data (as done by AVISO). We also investigate the effect of considering or not SSH data at shallow depths $({<}120~\hbox {m})$ as well as the editing procedure. We find that the main difference comes from the averaging method with significant differences depending on latitude. In the tropics, the $2^{\circ }\times 2^{\circ }$ gridding method used by AVISO overestimates by 11 % the GMSL trend. At high latitudes (above $60^{\circ }\hbox {N}/\hbox {S}$ ), both methods underestimate the GMSL trend. Our calculation shows that the CU method (along-track averaging) and AVISO gridding process underestimate the trend in high latitudes of the northern hemisphere by 0.9 and 1.2 mm/year, respectively. While we were able to attribute the AVISO trend overestimation in the tropics to grid cells with too few data, the cause of underestimation at high latitudes remains unclear and needs further investigation.     

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.     

The integral formulas of the associated Legendre functions

A new kind of integral formulas for ${\bar{P}_{n,m} (x)}$ is derived from the addition theorem about the Legendre Functions when n ? m is an even number. Based on the newly introduced integral formulas, the fully normalized associated Legendre functions can be directly computed without using any recursion methods that currently are often used in the computations. In addition, some arithmetic examples are computed with the increasing degree recursion and the integral methods introduced in the paper respectively, in order to compare the precisions and run-times of these two methods in computing the fully normalized associated Legendre functions. The results indicate that the precisions of the integral methods are almost consistent for variant x in computing ${\bar{P}_{n,m} (x)}$ , i.e., the precisions are independent of the choice of x on the interval [0,1]. In contrast, the precisions of the increasing degree recursion change with different values on the interval [0,1], particularly, when x tends to 1, the errors of computing ${\bar{P}_{n,m} (x)}$ by the increasing degree recursion become unacceptable when the degree becomes larger and larger. On the other hand, the integral methods cost more run-time than the increasing degree recursion. Hence, it is suggested that combinations of the integral method and the increasing degree recursion can be adopted, that is, the integral methods can be used as a replacement for the recursive initials when the recursion method become divergent.     

Confirming regional 1 cm differential geoid accuracy from airborne gravimetry: the Geoid Slope Validation Survey of 2011

A terrestrial survey, called the Geoid Slope Validation Survey of 2011 (GSVS11), encompassing leveling, GPS, astrogeodetic deflections of the vertical (DOV) and surface gravity was performed in the United States. The general purpose of that survey was to evaluate the current accuracy of gravimetric geoid models, and also to determine the impact of introducing new airborne gravity data from the ‘Gravity for the Redefinition of the American Vertical Datum’ (GRAV-D) project. More specifically, the GSVS11 survey was performed to determine whether or not the GRAV-D airborne gravimetry, flown at 11 km altitude, can reduce differential geoid error to below 1 cm in a low, flat gravimetrically uncomplicated region. GSVS11 comprises a 325 km traverse from Austin to Rockport in Southern Texas, and includes 218 GPS stations ( $\sigma _{\Delta h }= 0.4$ cm over any distance from 0.4 to 325 km) co-located with first-order spirit leveled orthometric heights ( $\sigma _{\Delta H }= 1.3$ cm end-to-end), including new surface gravimetry, and 216 astronomically determined vertical deflections $(\sigma _{\mathrm{DOV}}= 0.1^{\prime \prime })$ . The terrestrial survey data were compared in various ways to specific geoid models, including analysis of RMS residuals between all pairs of points on the line, direct comparison of DOVs to geoid slopes, and a harmonic analysis of the differences between the terrestrial data and various geoid models. These comparisons of the terrestrial survey data with specific geoid models showed conclusively that, in this type of region (low, flat) the geoid models computed using existing terrestrial gravity, combined with digital elevation models (DEMs) and GRACE and GOCE data, differential geoid accuracy of 1 to 3 cm (1 $\sigma )$ over distances from 0.4 to 325 km were currently being achieved. However, the addition of a contemporaneous airborne gravity data set, flown at 11 km altitude, brought the estimated differential geoid accuracy down to 1 cm over nearly all distances from 0.4 to 325 km.     

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.     

Canadian gravimetric geoid model 2010

A new gravimetric geoid model, Canadian Gravimetric Geoid 2010 (CGG2010), has been developed to upgrade the previous geoid model CGG2005. CGG2010 represents the separation between the reference ellipsoid of GRS80 and the Earth’s equipotential surface of $W_0=62{,}636{,}855.69~\mathrm{m}^2\mathrm{s}^{-2}$ W 0 = 62 , 636 , 855.69 m 2 s ? 2 . The Stokes–Helmert method has been re-formulated for the determination of CGG2010 by a new Stokes kernel modification. It reduces the effect of the systematic error in the Canadian terrestrial gravity data on the geoid to the level below 2 cm from about 20 cm using other existing modification techniques, and renders a smooth spectral combination of the satellite and terrestrial gravity data. The long wavelength components of CGG2010 include the GOCE contribution contained in a combined GRACE and GOCE geopotential model: GOCO01S, which ranges from $-20.1$ ? 20.1 to 16.7 cm with an RMS of 2.9 cm. Improvement has been also achieved through the refinement of geoid modelling procedure and the use of new data. (1) The downward continuation effect has been accounted accurately ranging from $-22.1$ ? 22.1 to 16.5 cm with an RMS of 0.9 cm. (2) The geoid residual from the Stokes integral is reduced to 4 cm in RMS by the use of an ultra-high degree spherical harmonic representation of global elevation model for deriving the reference Helmert field in conjunction with a derived global geopotential model. (3) The Canadian gravimetric geoid model is published for the first time with associated error estimates. In addition, CGG2010 includes the new marine gravity data, ArcGP gravity grids, and the new Canadian Digital Elevation Data (CDED) 1:50K. CGG2010 is compared to GPS-levelling data in Canada. The standard deviations are estimated to vary from 2 to 10 cm with the largest error in the mountainous areas of western Canada. We demonstrate its improvement over the previous models CGG2005 and EGM2008.     

Fast error analysis of continuous GNSS observations with missing data

One of the most widely used method for the time-series analysis of continuous Global Navigation Satellite System (GNSS) observations is Maximum Likelihood Estimation (MLE) which in most implementations requires $\mathcal{O }(n^3)$ operations for $n$ observations. Previous research by the authors has shown that this amount of operations can be reduced to $\mathcal{O }(n^2)$ for observations without missing data. In the current research we present a reformulation of the equations that preserves this low amount of operations, even in the common situation of having some missing data.Our reformulation assumes that the noise is stationary to ensure a Toeplitz covariance matrix. However, most GNSS time-series exhibit power-law noise which is weakly non-stationary. To overcome this problem, we present a Toeplitz covariance matrix that provides an approximation for power-law noise that is accurate for most GNSS time-series.Numerical results are given for a set of synthetic data and a set of International GNSS Service (IGS) stations, demonstrating a reduction in computation time of a factor of 10–100 compared to the standard MLE method, depending on the length of the time-series and the amount of missing data.     

Non-linear station motions in epoch and multi-year reference frames

In the conventions of the International Earth Rotation and Reference Systems Service (e.g. IERS Conventions 2010), it is recommended that the instantaneous station position, which is fixed to the Earth’s crust, is described by a regularized station position and conventional correction models. Current realizations of the International Terrestrial Reference Frame use a station position at a reference epoch and a constant velocity to describe the motion of the regularized station position in time. An advantage of this parameterization is the possibility to provide station coordinates of high accuracy over a long time span. Various publications have shown that residual non-linear station motions can reach a magnitude of a few centimeters due to not considered loading effects. Consistently estimated parameters like the Earth Orientation Parameters (EOP) may be affected if these non-linear station motions are neglected. In this paper, we investigate a new approach, which is based on a frequent (e.g. weekly) estimation of station positions and EOP from a combination of epoch normal equations of the space geodetic techniques Global Positioning System (GPS), Satellite Laser Ranging (SLR) and Very Long Baseline Interferometry (VLBI). The resulting time series of epoch reference frames are studied in detail and are compared with the conventional secular approach. It is shown that both approaches have specific advantages and disadvantages, which are discussed in the paper. A major advantage of the frequently estimated epoch reference frames is that the non-linear station motions are implicitly taken into account, which is a major limiting factor for the accuracy of the secular frames. Various test computations and comparisons between the epoch and secular approach are performed. The authors found that the consistently estimated EOP are systematically affected by the two different combination approaches. The differences between the epoch and secular frames reach magnitudes of $23.6~\upmu \hbox {as}$ (0.73 mm) and $39.8~\upmu \hbox {as}$ (1.23 mm) for the x-pole and y-pole, respectively, in case of the combined solutions. For the SLR-only solutions, significant differences with amplitudes of $77.3~\upmu \hbox {as}$ (2.39 mm) can be found.     

Basic equations for constructing geopotential models from the gravitational potential derivatives of the first and second orders in the terrestrial reference frame

This research represents a continuation of the investigation carried out in the paper of Petrovskaya and Vershkov (J Geod 84(3):165–178, 2010) where conventional spherical harmonic series are constructed for arbitrary order derivatives of the Earth gravitational potential in the terrestrial reference frame. The problem of converting the potential derivatives of the first and second orders into geopotential models is studied. Two kinds of basic equations for solving this problem are derived. The equations of the first kind represent new non-singular non-orthogonal series for the geopotential derivatives, which are constructed by means of transforming the intermediate expressions for these derivatives from the above-mentioned paper. In contrast to the spherical harmonic expansions, these alternative series directly depend on the geopotential coefficients ${\bar{{C}}_{n,m}}$ and ${\bar{{S}}_{n,m}}$ . Each term of the series for the first-order derivatives is represented by a sum of these coefficients, which are multiplied by linear combinations of at most two spherical harmonics. For the second-order derivatives, the geopotential coefficients are multiplied by linear combinations of at most three spherical harmonics. As compared to existing non-singular expressions for the geopotential derivatives, the new expressions have a more simple structure. They depend only on the conventional spherical harmonics and do not depend on the first- and second-order derivatives of the associated Legendre functions. The basic equations of the second kind are inferred from the linear equations, constructed in the cited paper, which express the coefficients of the spherical harmonic series for the first- and second-order derivatives in terms of the geopotential coefficients. These equations are converted into recurrent relations from which the coefficients ${\bar{{C}}_{n,m}}$ and ${\bar{{S}}_{n,m}}$ are determined on the basis of the spherical harmonic coefficients of each derivative. The latter coefficients can be estimated from the values of the geopotential derivatives by the quadrature formulas or the least-squares approach. The new expressions of two kinds can be applied for spherical harmonic synthesis and analysis. In particular, they might be incorporated in geopotential modeling on the basis of the orbit data from the CHAMP, GRACE and GOCE missions, and the gradiometry data from the GOCE mission.     

Estimating crustal deformations from a combination of baseline measurements and geophysical models

The estimation of crustal deformations from repeated baseline measurements is a singular problem in the absence of prior information. One often applied solution is a free adjustment in which the singular normal matrix is augmented with a set of inner constraints. These constraints impose no net translation nor rotation for the estimated deformations X which may not be physically meaningful for a particular problem. The introduction of an available geophysical model from which an expected deformation vector \(\bar X\) and its covariance matrix \(\sum _{\bar X} \) can be computed will direct X to a physically more meaningful solution. Three possible estimators are investigated for estimating deformations from a combination of baseline measurements and geophysical models.