A data-driven approach to local gravity field modelling using spherical radial basis functions

We propose a methodology for local gravity field modelling from gravity data using spherical radial basis functions. The methodology comprises two steps: in step 1, gravity data (gravity anomalies and/or gravity disturbances) are used to estimate the disturbing potential using least-squares techniques. The latter is represented as a linear combination of spherical radial basis functions (SRBFs). A data-adaptive strategy is used to select the optimal number, location, and depths of the SRBFs using generalized cross validation. Variance component estimation is used to determine the optimal regularization parameter and to properly weight the different data sets. In the second step, the gravimetric height anomalies are combined with observed differences between global positioning system (GPS) ellipsoidal heights and normal heights. The data combination is written as the solution of a Cauchy boundary-value problem for the Laplace equation. This allows removal of the non-uniqueness of the problem of local gravity field modelling from terrestrial gravity data. At the same time, existing systematic distortions in the gravimetric and geometric height anomalies are also absorbed into the combination. The approach is used to compute a height reference surface for the Netherlands. The solution is compared with NLGEO2004, the official Dutch height reference surface, which has been computed using the same data but a Stokes-based approach with kernel modification and a geometric six-parameter “corrector surface” to fit the gravimetric solution to the GPS-levelling points. A direct comparison of both height reference surfaces shows an RMS difference of 0.6 cm; the maximum difference is 2.1 cm. A test at independent GPS-levelling control points, confirms that our solution is in no way inferior to NLGEO2004.     

Integration of local–global upscaling and grid adaptivity for simulation of subsurface flow in heterogeneous formations

We propose a methodology, called multilevel local–global (MLLG) upscaling, for generating accurate upscaled models of permeabilities or transmissibilities for flow simulation on adapted grids in heterogeneous subsurface formations. The method generates an initial adapted grid based on the given fine-scale reservoir heterogeneity and potential flow paths. It then applies local–global (LG) upscaling for permeability or transmissibility [7], along with adaptivity, in an iterative manner. In each iteration of MLLG, the grid can be adapted where needed to reduce flow solver and upscaling errors. The adaptivity is controlled with a flow-based indicator. The iterative process is continued until consistency between the global solve on the adapted grid and the local solves is obtained. While each application of LG upscaling is also an iterative process, this inner iteration generally takes only one or two iterations to converge. Furthermore, the number of outer iterations is bounded above, and hence, the computational costs of this approach are low. We design a new flow-based weighting of transmissibility values in LG upscaling that significantly improves the accuracy of LG and MLLG over traditional local transmissibility calculations. For highly heterogeneous (e.g., channelized) systems, the integration of grid adaptivity and LG upscaling is shown to consistently provide more accurate coarse-scale models for global flow, relative to reference fine-scale results, than do existing upscaling techniques applied to uniform grids of similar densities. Another attractive property of the integration of upscaling and adaptivity is that process dependency is strongly reduced, that is, the approach computes accurate global flow results also for flows driven by boundary conditions different from the generic boundary conditions used to compute the upscaled parameters. The method is demonstrated on Cartesian cell-based anisotropic refinement (CCAR) grids, but it can be applied to other adaptation strategies for structured grids and extended to unstructured grids.     

Accurate local upscaling with variable compact multipoint transmissibility calculations

We propose a new single-phase local upscaling method that uses spatially varying multipoint transmissibility calculations. The method is demonstrated on two-dimensional Cartesian and adaptive Cartesian grids. For each cell face in the coarse upscaled grid, we create a local fine grid region surrounding the face on which we solve two generic local flow problems. The multipoint stencils used to calculate the fluxes across coarse grid cell faces involve the six neighboring pressure values. They are required to honor the two generic flow problems. The remaining degrees of freedom are used to maximize compactness and to ensure that the flux approximation is as close as possible to being two-point. The resulting multipoint flux approximations are spatially varying (a subset of the six neighbors is adaptively chosen) and reduce to two-point expressions in cases without full-tensor anisotropy. Numerical tests show that the method significantly improves upscaling accuracy as compared to commonly used local methods and also compares favorably with a local–global upscaling method.     

New basin modelling results from the Polish part of the Central European Basin system: implications for the Late Cretaceous–Early Paleogene structural inversion

Combined subsidence and thermal 1D modelling was performed on six well-sections located in the north-western Mid-Polish Trough/Swell in the eastern part of the Central European Basin system. The modelling allowed constraining quantitatively both the Mesozoic subsidence and the magnitude of the Late Cretaceous–Paleocene inversion and erosion. The latter most probably reached 2,400 m in the Mid-Polish Swell area. The modelled Upper Cretaceous thickness did not exceed 500 m, and probably corresponded to 200–300 m in the swell area as compared with more than 2,000 m in the adjacent non-inverted part of the basin. Such Upper Cretaceous thickness pattern implies early onset of inversion processes, probably in the Late Turonian or Coniacian. Our modelling, coupled with previous results of stratigraphic and seismic studies, demonstrates that the relatively low sedimentation rates in the inverted part of the basin during the Late Cretaceous were the net result of several discrete pulses of non-deposition and/or erosion that were progressively more pronounced towards the trough axis. The last phase of inversion started in the Late Maastrichtian and was responsible for the total amount of erosion, which removed also the reduced Upper Cretaceous deposits. According to our modelling results, a Late Cretaceous heat-flow regime which is similar to the present-day conditions (about 50 mW/m²) was responsible for the observed organic maturity of the Permian-Mesozoic rocks. This conclusion does not affect the possibility of Late Carboniferous–Permian and Late Permian–Early Triassic thermal events.     

The Glueckstadt Graben of the North-German Basin: new insights into the structure from 3D and 2D gravity analyses

The structure of the Glueckstadt Graben has been investigated by use of 3D gravity backstripping technique and by 2D gravity and magnetic modelling. Subtracting the gravity effects of the Meso-Cenozoic sediments together with Permian salt reveals a positive residual anomaly within the Glueckstadt Graben. This anomaly includes two local maxima over the Westholstein and Eastholstein Troughs. The 2D gravity models point to the presence of a high-density body within the lower crust of the Glueckstadt Graben. In addition, the results of 2D magnetic modelling indicate that the central part of the high-density body is overlain by an area with high susceptibility. Most probable, the formation of this high-density body is a result of complex poly-phase tectonic history of the study area. Finally, the results of gravity modelling indicate that Permian salt is not homogeneous. 3D gravity analysis and, especially, 2D gravity modelling have distinguished the differences in degree of salt saturation in salt-rich bodies, and elucidate the proportion of Rotliegend salt.     

New basis for the constitutive modelling of aggregated soils

Natural and compacted soils are usually characterized by aggregation of particles. The mechanical behaviour of these materials depends on soil structure. The oedometric compression tests performed on aggregated samples presented here showed that these materials exhibit a yield limit depending not only on stress history and stress state but also on soil structure. Evidence is provided using the neutron tomography technique. These results revealed that soil structure modification occurs together with plastic deformations. The experimental results are used to propose a new state parameter to quantify the soil structure. Based on pore-scale experimental observations, an evolution law for this parameter is proposed as a function of associated plastic strains. Considering both soil fabric and inter-particle bonding effects, a new yield limit depending on stress state, stress history and soil structure is introduced for the aggregated soils. Accordingly, a new constitutive framework consistent with strain hardening plasticity is proposed to consider soil structure effects in the modelling of aggregated soils.     

Changing grading of soil: effect on critical states

Examples of situations are presented where the grading of a soil changes during its lifetime either by crushing of particles leading to an increase of fine material or by slow transport of fine particles with seepage leading to a decrease of fine material. Such grading changes influence the basic constitutive properties of the soil, in particular properties such as critical states which are dependent on the available range of densities of packing. Discrete element modelling is used to show the dependence of critical state conditions on grading and the way in which the particle assembly seeks out new critical state conditions as the grading changes.     

Using numerical modelling to evaluate tsunami hazard near the Kuril Island

A sequence of computer experiments is used to study questions concerning the tsunami problem as a quantitative estimate of tsunami danger, detailed geographical tsunami classification, determination of the parameters of critical tsunami waves, and the conditions of their development. We call a wave critical, if its impact on the coast is most hazardous.Using the Middle Kuril Island as an example, we present the results of a computer experiment which includes determining the wavefields on the shelf and estimating the effects connected with the deep-water Bussol and Diana Straits.Numerical simulation of tsunami waves of different sources permits the assessment of the extent of tsunami danger in different areas of the coastal zone of Simushir Island, depending on the location of the focus zone and their geometry.The major singularities of the wavefield arise in the zones of the deep-water straits. The distribution of the amplification factors is determined by both the global parameters of the wavefields and the local properties of individual harbours. The results obtained for a particular harbour in the northern part of Simushir Island, formed the basis for the quantitative estimate of tsunami danger for this area.     

Numerical study of droplet size dependent chemistry in oceanic,wintertime stratus cloud at southern mid-latitudes

Cloud droplet chemistry is modelled for the first 150 m of rise in a wintertime, mid-latitude, marine stratus cloud using observations made at and near the Cape Grim Baseline Station as a source of input parameters. The emphasis in this work was to study the variation in droplet chemistry as a function of both droplet size and nucleus composition, with a particular focus on the way in which oxidation of dissolved sulfur dioxide varied.At 150 m above the condensation level, solute concentration as a function of droplet size was found to increase by as much as 2 to 3 orders of magnitude for only a factor of 2 increase in droplet radius, primarily as a consequence of the 1/r dependence in the droplet growth equation. This type of size dependence exists at all levels in the model cloud, and has a significant influence on oxidation rate of sulfur dioxide in droplets growing on sulfate nuclei, oxidation by ozone being favoured in the smallest droplets, but oxidation by hydrogen peroxide being favoured in the larger droplets. Oxidation by ozone is favoured at all sizes in droplets formed on sea-salt nuclei as a result of the initially high alkalinity of these droplets, and in the cloud overall is calculated to be the more important oxidation pathway. Although based on a simplified chemical scheme, these results suggest that both size-dependent and nucleus-dependent chemistry of cloud droplets may need to be considered explicitly in cloud modelling work.Volume-weighted mean pH values in the range 5 to 6 were predicted from sensitivity studies in which input variables were varied over reasonable ranges, in agreement with two sets of bulk cloud-water pH data obtained by aircraft near Cape Grim.     

Estimation of runoff distribution within river systems: Application to the Lena Basin (Siberia)

A channel account approach is proposed to estimate longitudinal changes in runoff along large river systems. This new method provides a quantitative basis for describing the fluvial transport of suspended particulate material and dissolved substances. This method includes an evaluation of basic elements of water balance in separate sections of the river network and subsequent correction of channel accounting equations for the entire system using a maximum likelihood principle. To calculate water discharges of tributaries that have no hydrological information, structural analysis of river network is performed. This approach provides less error in comparison with traditional methods of estimating lateral inflow. The method is used to trace water discharge with increasing distance along the Lena river basin and to evaluate the contribution of geologically and lithologically uneven sub-basins in water discharge formation during a summer low water period.