Non-stationary Approaches for Mapping Terrain and Assessing Prediction Uncertainty

It is well known that terrain may vary markedly over small areas and that statistics used to characterise spatial variation in terrain may be valid only over small areas. In geostatistical terminology, a non-stationary approach may be considered more appropriate than a stationary approach. In many applications, local variation is not accounted for sufficiently. This paper assesses potential benefits in using non-stationary geostatistical approaches for interpolation and for the assessment of uncertainty in predictions with implications for sampling design. Two main non-stationary approaches are employed in this paper dealing with (1) change in the mean and (2) change in the variogram across the region of interest. The relevant approaches are (1) kriging with a trend model (KT) using the variogram of residuals from local drift and (2) locally-adaptive variogram KT, both applied to a sampled photogrammetrically derived digital terrain model (DTM). The fractal dimension estimated locally from the double-log variogram is also mapped to illustrate how spatial variation changes across the data set. It is demonstrated that estimation of the variogram of residuals from local drift is worthwhile in this case for the characterisation of spatial variation. In addition, KT is shown to be useful for the assessment of uncertainty in predictions. This is shown to be true even when the sample grid is dense as is usually the case for remotely-sensed data. In addition, both ordinary kriging (OK) and KT are shown to provide more accurate predictions than inverse distance weighted (IDW) interpolation, used for comparative purposes.     

Integer estimation in the presence of biases

 Carrier phase ambiguity resolution is the key to fast and high-precision GNSS (Global Navigation Satellite System) kinematic positioning. Critical in the application of ambiguity resolution is the quality of the computed integer ambiguities. Unsuccessful ambiguity resolution, when passed unnoticed, will too often lead to unacceptable errors in the positioning results. Very high success rates are therefore required for ambiguity resolution to be reliable. Biases which are unaccounted for will lower the success rate and thus increase the chance of unsuccessful ambiguity resolution. The performance of integer ambiguity estimation in the presence of such biases is studied. Particular attention is given to integer rounding, integer bootstrapping and integer least squares. Lower and upper bounds, as well as an exact and easy-to-compute formula for the bias-affected success rate, are presented. These results will enable the evaluation of the bias robustness of ambiguity resolution. Received: 28 September 2000 / Accepted: 29 March 2001     

Maximum-likelihood ambiguity resolution based on Bayesian principle

 Based on the Bayesian principle and the fact that GPS carrier-phase ambiguities are integers, the posterior distribution of the ambiguities and the position parameters is derived. This is then used to derive the maximum posterior likelihood solution of the ambiguities. The accuracy of the integer ambiguity solution and the position parameters is also studied according to the posterior distribution. It is found that the accuracy of the integer solution depends not only on the variance of the corresponding float ambiguity solution but also on its values. Received: 27 July 1999 / Accepted: 22 November 2000     

A model for adjustment of differential gravity measurements with simultaneous gravimeter calibration

 A mathematical model is proposed for adjustment of differential or relative gravity measurements, involving simultaneously instrumental readings, coefficients of the calibration function, and gravity values of selected base stations. Tests were performed with LaCoste and Romberg model G gravimeter measurements for a set of base stations located along a north–south line with 1750 mGal gravity range. This line was linked to nine control stations, where absolute gravity values had been determined by the free-fall method, with an accuracy better than 10 μGal. The model shows good consistence and stability. Results show the possibility of improving the calibration functions of gravimeters, as well as a better estimation of the gravity values, due to the flexibility admitted to the values of the calibration coefficients. Received: 15 November 1999 / Accepted: 31 October 2000     

Local geoid determination combining gravity disturbances and GPS/levelling: a case study in the Lake Nasser area, Aswan, Egypt

 The use of GPS for height control in an area with existing levelling data requires the determination of a local geoid and the bias between the local levelling datum and the one implicitly defined when computing the local geoid. If only scarse gravity data are available, the heights of new data may be collected rapidly by determining the ellipsoidal height by GPS and not using orthometric heights. Hence the geoid determination has to be based on gravity disturbances contingently combined with gravity anomalies. Furthermore, existing GPS/levelling data may also be used in the geoid determination if a suitable general gravity field modelling method (such as least-squares collocation, LSC) is applied. A comparison has been made in the Aswan Dam area between geoids determined using fast Fourier transform (FFT) with gravity disturbances exclusively and LSC using only the gravity disturbances and the disturbances combined with GPS/levelling data. The EGM96 spherical harmonic model was in all cases used in a remove–restore mode. A total of 198 gravity disturbances spaced approximately 3 km apart were used, as well as 35 GPS/levelling points in the vicinity and on the Aswan Dam. No data on the Nasser Lake were available. This gave difficulties when using FFT, which requires the use of gridded data. When using exclusively the gravity disturbances, the agreement between the GPS/levelling data were 0.71 ± 0.17 m for FFT and 0.63 ± 0.15 for LSC. When combining gravity disturbances and GPS/levelling, the LSC error estimate was ±0.10 m. In the latter case two bias parameters had to be introduced to account for a possible levelling datum difference between the levelling on the dam and that on the adjacent roads. Received: 14 August 2000 / Accepted: 28 February 2001     

Atmospheric, oceanic and hydrological contributions to seasonal variations in length of day

 The annual and semiannual residuals derived in the axial angular momentum budget of the solid Earth–atmosphere system reflect significant signals. They must be caused by further excitation sources. Since, in particular, the contribution for the wind term from the atmospheric layer between the 10 and 0.3 hPa levels to the seasonal variations in length of day (LOD) is still missing, it is necessary to extend the top level into the upper stratosphere up to 0.3 hPa. Under the conservation of the total angular momentum of the entire Earth, variations in the oceanic angular momentum (OAM) and the hydrological angular momentum (HAM) are further significant excitation sources at seasonal time scales. Focusing on other contributions to the Earth's axial angular momentum budget, the following data are used in this study: axial atmospheric angular momentum (AAM) data derived for the 10–0.3 hPa layer from 1991 to 1997 for computing the missing wind effects; axial OAM functions as generated by oceanic general circulation models (GCMs), namely for the ECHAM3 and the MICOM models, available from 1975 to 1994 and from 1992 to 1994, respectively, for computing the oceanic contributions to LOD changes, and, concerning the HAM variations, the seasonal estimates of the hydrological contribution as derived by Chao and O'Connor [(1988) Geophys J 94: 263–270]. Using vector representation, it is shown that the vectors achieve a close balance in the global axial angular momentum budget within the estimated uncertainties of the momentum quantities on seasonal time scales. Received: 6 April 2000 / Accepted: 13 December 2000     

New analytical and numerical approaches for geopotential modeling

 The standard analytical approach which is applied for constructing geopotential models OSU86 and earlier ones, is based on reducing the boundary value equation to a sphere enveloping the Earth and then solving it directly with respect to the potential coefficients _n,m . In an alternative procedure, developed by Jekeli and used for constructing the models OSU91 and EGM96, at first an ellipsoidal harmonic series is developed for the geopotential and then its coefficients _n,m ^e are transformed to the unknown _n,m . The second solution is more exact, but much more complicated. The standard procedure is modified and a new simple integral formula is derived for evaluating the potential coefficients. The efficiency of the standard and new procedures is studied numerically. In these solutions the same input data are used as for constructing high-degree parts of the EGM96 models. From two sets of _n,m (n≤360,|m|≤n), derived by the standard and new approaches, different spectral characteristics of the gravity anomaly and the geoid undulation are estimated and then compared with similar characteristics evaluated by Jekeli's approach (`etalon' solution). The new solution appears to be very close to Jekeli's, as opposed to the standard solution. The discrepancies between all the characteristics of the new and `etalon' solutions are smaller than the corresponding discrepancies between two versions of the final geopotential model EGM96, one of them (HDM190) constructed by the block-diagonal least squares (LS) adjustment and the other one (V068) by using Jekeli's approach. On the basis of the derived analytical solution a new simple mathematical model is developed to apply the LS technique for evaluating geopotential coefficients. Received: 12 December 2000 / Accepted: 21 June 2001