A transformational approach to geophysical inverse problems

A set of coordinate transformations is used to linearize a general geophysical inverse problem. Statistical and analytic techniques are employed to estimate the parameters of such linearization transformations. In the transformed space, techniques from linear inverse theory may be utilized. Consequently, important concepts, such as model parameter covariance, model parameter resolution and averaging kernels, may be carried over to non-linear inverse problems. I apply the approach to a set of seismic cross-borehole traveltimes gathered at the Conoco Borehole Test Facility. the seismic survey was conducted within the Fort Riley formation, a limestone with thin interbedded shales. Between the boreholes, the velocity structure of the Fort Riley formation consists of a high-velocity region overlying a section of lower velocity. It is found that model parameter resolution is poorest and spatial averaging lengths are greatest in the underlying low-velocity region.     

Maximum entropy regularization of the geomagnetic core field inverse problem

The maximum entropy technique is an accepted method of image reconstruction when the image is made up of pixels of unknown positive intensity (e.g. a grey-scale image). The problem of reconstructing the magnetic field at the core–mantle boundary from surface data is a problem where the target image, the value of the radial field B_r , can be of either sign. We adopt a known extension of the usual maximum entropy method that can be applied to images consisting of pixels of unconstrained sign. We find that we are able to construct images which have high dynamic ranges, but which still have very simple structure. In the spherical harmonic domain they have smoothly decreasing power spectra. It is also noteworthy that these models have far less complex null flux curve topology (lines on which the radial field vanishes) than do models which are quadratically regularized. Problems such as the one addressed are ubiquitous in geophysics, and it is suggested that the applications of the method could be much more widespread than is currently the case.     

实施分水方案后对黑河下游地下水影响的分析

由于黑河中上游的水土资源近年来的大规模开发利用,加之缺乏有效的水资源管理,使得下游河道来水量逐年减少,进而产生了一系列严重的生态问题,为了遏制这种恶化,以及加以恢复和重建下游的生态环境,自2000年7月开始在黑河下游实施国家分水方案。本文根据近五年来的黑河分水资料以及对分水前后地下水监测区的地下水位观测资料进行分析,探讨了分水过程及分水量在各年的变化,以及分水前后地下水位的时空变化,同时依据前人的一些研究结果,探讨了分水前后地下水位与生态环境之间的关系,尤其是地下水位与植被生长之间的关系。进而为下游分水以及生态环境建设提供科学依据。     

The inverse problem for heat flow data in the presence of thermal conductivity variations

Summary. We demonstrate a method of performing linear programming optimizations of functionals of subsurface temperature, when thermal conductivity is a known piecewise-constant function. Data comprise heat flow measurements on the flat isothermal surface of this structure, within which heat transfer is by steady-state conduction. Two-dimensionality is assumed. The approach involves establishing constraints which demand the continuity of temperature and the normal component of heat flow across all internal boundaries. These unknown functions are expanded as truncated Fourier series whose coefficients become unknowns of the linear programming solution vector; linear relations are established between these coefficients which guarantee harmonicity of temperature in each region of uniform conductivity, as well as the continuity requirements. Variations of the formalism are detailed for three simple types of geometry. As an example the method is applied to a heat flow data set from Sass, Killeen & Mustonen over the Quirke Lake Syncline of Ontario, Canada.     

A continued fraction approach to the inverse problem of electrical conductivity

Summary. The problem of determination of the electrical conductivity of the Earth from the geomagnetic induction data is formulated as that of finding the coefficients of the continued fraction expansion of a certain rational fraction representation of the total response (or impedance) of the medium at the surface when this quantity is given for N frequencies. The coefficients of expansion are related to the conductivities of N layers of constant attenuation, where, within each layer the conductivity is assumed to be constant. Thus in this approach the conductivity profile resulting from the inversion of the response function is given as a series of step functions.     

Application of the boundary element method to a terrestrial heat flow problem

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In this paper, we apply the boundary element method (BEM) to the 2-D steady state heat flow problem of what would be the perturbation to the regional temperature gradient, and hence heat flow density, determined from temperatures measured in a borehole that passes close to, but does not penetrate, a body of anomalous thermal properties. This type of problem with an infinity boundary is particularly well suited to the BEM.
The results have been compared with those obtained from analytical solutions for bodies of simple shape; it is found that for the worst case of a close approach to a boundary of small radius of curvature, a numerical modelling error of less than 1 per cent can still be obtained provided the length of each element is less than the shortest distance between the calculation point and the object.     

A model to predict ordinal suitability using sparse and uncertain data

We describe the development of the algorithms that comprise the Spatial Decision Support System (SDSS) CaNaSTA (Crop Niche Selection in Tropical Agriculture). The system was designed to assist farmers and agricultural advisors in the tropics to make crop suitability decisions. These decisions are frequently made in highly diverse biophysical and socioeconomic environments and must often rely on sparse datasets.The field trial datasets that provide a knowledge base for SDSS such as this are characterised by ordinal response variables. Our approach has been to apply Bayes’ formula as a prediction model.This paper does not describe the entire CaNaSTA system, but rather concentrates on the algorithm of the central prediction model. The algorithm is tested using a simulated dataset to compare results with ordinal regression, and to test the stability of the model with increasingly sparse calibration data. For all but the richest input datasets it outperforms ordinal regression, as determined using Cohen’s weighted kappa. The model also performs well with sparse datasets. Whilst this is not as conclusive as testing with real world data, the results are encouraging.     

Joint three-dimensional inversion of coupled groundwater flow and heat transfer based on automatic differentiation: sensitivity calculation, verification, and synthetic examples

Inverse methods are useful tools not only for deriving estimates of unknown parameters of the subsurface, but also for appraisal of the thus obtained models. While not being neither the most general nor the most efficient methods, Bayesian inversion based on the calculation of the Jacobian of a given forward model can be used to evaluate many quantities useful in this process. The calculation of the Jacobian, however, is computationally expensive and, if done by divided differences, prone to truncation error. Here, automatic differentiation can be used to produce derivative code by source transformation of an existing forward model. We describe this process for a coupled fluid flow and heat transport finite difference code, which is used in a Bayesian inverse scheme to estimate thermal and hydraulic properties and boundary conditions form measured hydraulic potentials and temperatures. The resulting derivative code was validated by comparison to simple analytical solutions and divided differences. Synthetic examples from different flow regimes demonstrate the use of the inverse scheme, and its behaviour in different configurations.