The continuation inverse problem revisited

The non-uniqueness of the continuation of a finite collection of harmonic potential field data to a level surface in the source-free region forces its treatment as an inverse problem. A formalism is proposed for the construction of continuation functions which are extremal by various measures. The problem is cast in such a form that the inverse problem solution is the potential function on the lowest horizontal surface above all sources, serving as the boundary function for the Dirichlet problem in the upper half-plane. The desired continuation, at the higher level of interest, must then be in the range of the upward continuation operator acting on this boundary function, rather than being allowed the full freedom of itself being part of a Dirichlet problem boundary function. Extremal solutions minimize non-linear functionals of the continuation function, which are re-expressed as different functionals of the boundary function. A crux of the method is that there is no essential distinction between the upward and downward continuation inverse problems to levels above or below data locations. Casting the optimization as a Lagrange multiplier problem leads to an integral equation for the boundary function, which is readily solved in the Fourier domain for a certain class of functionals. The desired extremal continuation is then given by upward continuation. It is found that for some functionals, application of the Lagrange multiplier theorem requires a further restriction on the set of allowable boundary functions: bandlimitedness is a natural choice for the continuation problem. With this imposition, the theory is developed in detail for semi-norm functionals penalizing departure from a constant potential, in the 2-norm and Sobelev norm senses, and illustrated by application for a small synthetic Deep Tow magnetic field data set.     

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.     

Solution of the inverse problem of seismology for laterally inhomogeneous media

Summary. The Backus-Gilbert method has been extended to the estimation of the seismic wave velocity distribution in 2-D or 3-D inhomogeneous media from a finite set of travel-time data. The method may be applied to the inversion of body wave as well as surface wave data. The problem of determining a local average of the unknown velocity corrections may be reduced to a choice of a suitable δ-ness criterion for the averaging kernel. For 2-D and 3-D inhomogeneous media the simplest criterion is to minimize a sum of 'spreads' over all the coordinates. The use of this criterion requires the solution (the averaged velocity corrections) to be represented as a sum of functions, each of which depends only on one coordinate. This is a basic restriction of the method. In practice it is possible to achieve good agreement between the solution and a real velocity distribution by a reasonable choice of the coordinate system.
Numerical tests demonstrate the efficiency of the method. Some examples of the application of the method to the inversion of real seismological data for body and surface waves are given.     

The inverse problem of geomagnetic induction at a fixed frequency

Summary. A method for the determination of the electrical conductivity of the Earth is developed when the components of the response function on the basis of spherical harmonics for a fixed frequency are known. By writing the differential equation for the field inside a spherically symmetric conductor as a finite difference equation, it is shown that the formal solution of the latter for the response function has a Thiele representation in the degree of the harmonics. This property enables one to calculate the conductivity at a finite number of points using a continued-fraction expansion of the response function.     

Time-domain electromagnetic migration in the solution of inverse problems

Time-domain electromagnetic (TDEM) migration is based on downward extrapolation of the observed field in reverse time. In fact, the migrated EM field is the solution of the boundary-value problem for the adjoint Maxwell's equations. The important question is how this imaging technique can be related to the solution of the geoelectrical inverse problem. In this paper we introduce a new formulation of the inverse problem, based on the minimization of the residual-field energy flow through the surface or profile of observations. We demonstrate that TDEM migration can be interpreted as the first step in the solution of this specially formulated TDEM inverse problem. However, in many practical situations this first step produces a very efficient approximation to the geoelectrical model, which makes electromagnetic migration so attractive for practical applications. We demonstrate the effectiveness of this approach in inverting synthetic and practical TDEM data.     

The potential of GIS modelling of gravity flows and slope instabilities

Abstract

The component of gravitational acceleration parallel to the slope of the local surface partly determines the state of slope stability and the kinematics of flow under gravity on that slope. Geographical information systems based on digital elevation models offer the potential to be able to map this variable and permit the modelling of a variety of stability criteria and surface processes including landslides, rock avalanches, pyroclastic flows and lava flows. Three types of models and the basic map operations required to run them are discussed. The models are as follows: (i) sites of potential shallow slope failure (e.g. landslides), (ii) maps of flow deposition based on energy balance calculations (e.g. rock avalanches) and (iii) finite difference, initial value type simulations of dynamic flow (e.g. lava flows). The potential value of these models to hazard assessment is great but their application in specific cases must be assessed with reference to the accuracy of the digital elevation model used.     

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.