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.     

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     

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     

Sediment transfer in an extremely low-gradient,low-relief and highly buffered system: Darwin Harbour catchment,northern Australia

Abstract

Sediment yields from and sediment transfer within catchments of very low relief and gradient, which make up about 50% of Earth’s surface, are poorly documented and their internal sediment dynamics are poorly known. Sediment sources, their proportionate contributions to valley floors and sediment yield, and storage are estimated using fallout radionuclides ²¹⁰Pb_(ex) and ¹³⁷Cs in the catchments that drain into Darwin Harbour, northern Australia, an example of this understudied catchment type that appears to be globally at the extreme end of this category of catchments. Unchannelled grassy valley floors (dambos, or seasonal wetlands) trap ～90% of the sediment delivered from hillslopes by sheet and rill erosion. Further down valley, small channels transport ～10% of the sediment that escapes from the dambos, and the remaining sediment comes from erosion of the channels. In this case, the fractional sediment storage is very high as a result of the existence of dambos, a landform that depends for its existence on low gradients.     

Sensitivity of soil parameters in unsaturated zone modelling and the relation between effective,laboratory and in situ estimates

Simulation of soil moisture content requires effective soil hydraulic parameters that are valid at the modelling scale. This study investigates how these parameters can be estimated by inverse modelling using soil moisture measurements at 25 locations at three different depths (at the surface, at 30 and 60 cm depth) on an 80 by 20 m hillslope. The study presents two global sensitivity analyses to investigate the sensitivity in simulated soil moisture content of the different hydraulic parameters used in a one‐dimensional unsaturated zone model based on Richards' equation. For estimation of the effective parameters the shuffled complex evolution algorithm is applied. These estimated parameters are compared to their measured laboratory and in situ equivalents. Soil hydraulic functions were estimated in the laboratory on 100 cm³ undisturbed soil cores collected at 115 locations situated in two horizons in three profile pits along the hillslope. Furthermore, in situ field saturated hydraulic conductivity was estimated at 120 locations using single‐ring pressure infiltrometer measurements. The sensitivity analysis of 13 soil physical parameters (saturated hydraulic conductivity (K_s), saturated moisture content (θ_s), residual moisture content (θ_r), inverse of the air‐entry value (α), van Genuchten shape parameter (n), Averjanov shape parameter (N) for both horizons, and depth (d) from surface to B horizon) in a two‐layer single column model showed that the parameter N is the least sensitive parameter. K_s of both horizons, θ_s of the A horizon and d were found to be the most sensitive parameters. Distributions over all locations of the effective parameters and the distributions of the estimated soil physical parameters from the undisturbed soil samples and the single‐ring pressure infiltrometer estimates were found significantly different at a 5% level for all parameters except for α of the A horizon and K_s and θ_s of the B horizon. Different reasons are discussed to explain these large differences. Copyright © 2004 John Wiley & Sons, Ltd.