Apparent resistivity is a useful concept for initial quickscan interpretation and quality checks in the field, because it represents the resistivity properties of the subsurface better than the raw data. For frequency‐domain soundings several apparent‐resistivity definitions exist. One definition uses an asymptote for the field of a magnetic dipole in a homogeneous half‐space and is useful only for low induction numbers. Another definition uses only the amplitude information of the total magnetic field, although this results in a non‐unique apparent resistivity. To overcome this non‐uniqueness, a complex derivation using two different source–receiver configurations and several magnetic field values for different frequencies or different offsets is derived in another definition. Using the latter theory, in practice, this means that a wide range of measurements have to be carried out, while commercial systems are not able to measure this wide range. In this paper, an apparent‐resistivity concept is applied beyond the low‐induction zone, for which the use of different source–receiver configurations is not needed. This apparent‐resistivity concept was formerly used to interpret the electromagnetic transients that are associated with the turn‐off of the transmitter current. The concept uses both amplitude and phase information and can be applied for a wide range of frequencies and offsets, resulting in a unique apparent resistivity for each individual (offset, frequency) combination. It is based on the projection of the electromagnetic field data on to the curve of the field of a magnetic dipole on a homogeneous half‐space and implemented using a non‐linear optimization scheme. This results in a fast and efficient estimation of apparent resistivity versus frequency or offset for electromagnetic sounding, and also gives a new perspective on electromagnetic profiling. Numerical results and two case studies are presented. In each case study the results are found to be comparable with those from other existing exploration systems, such as EM31 and EM34. They are obtained with a slight increase of effort in the field but contain more information, especially about the vertical resistivity distribution of the subsurface.
A reduced modeling technique for solving electromagnetic diffusion problems is the spectral Lanczos decomposition method (SLDM). In this paper an alternative approach of applying this method to transient electromagnetic diffusion problems is presented. The approach is based on Maxwell's equations as a system of first‐order partial differential equations as opposed to the standard SLDM that is based on a second‐order partial differential equation for either the electric or the magnetic field strength. By taking the system of first‐order equations as a starting point, it is possible to simultaneously construct approximations to the electric and magnetic field strength. Moreover, these approximations are highly structured. The structure of the approximations reflects the structure that is present in the original set of equations. Certain extensions of the present method are also given, and some numerical results for two‐dimensional configurations are presented.
In this paper the Rayleigh hypothesis in the theory of reflection by a cylindrical perturbation in a plane surface is investigated analytically. The hypothesis asserts that above the surface the scattered field may be expanded in terms of outward‐going wave functions. As such, it is analogous to the assumption made by Lord Rayleigh in his treatment of diffraction by a reflection grating. We show that the validity of the Rayleigh hypothesis is governed by the distribution of singularities in the analytic continuation of the exterior scattered field. Conditions are derived under which the Rayleigh hypothesis is rigorously valid. A procedure is presented that enables the validity of the Rayleigh hypothesis to be checked for a surface whose profile can be described by an analytic function. Numerical results are presented.
This paper reviews the collaborative research of Kleinman and Van den Berg with respect to the inverse scattering problem of the determination of the shape, the location and the constitutive parameters of a local inhomogeneity from measurements of the scattered field when a monochromatic wave is incident upon the inhomogeneity. Since the inverse scattering problem is nonlinear, an algorithm for its solution is iterative in nature and each iteration requires the solution of a forward or direct problem. In order to avoid a full solution of the forward problem in each iteration, the Modified Gradient method was developed, in which a cost functional was minimized such that the unknown fields and contrast are updated simultaneously. This cost functional consists of the superposition of the mismatch of the measured field data with the field scattered by an object with a particular contrast function and the error in satisfying consistency in the interior of the object. In these relations integral operators act on contrast sources being the products of the unknown fields and unknown material contrast. Further advantage of this structure has been taken by introducing the Contrast Source Inversion method that is based on a dual minimization of the cost functional by developing updates for the unknown contrast sources (instead of the fields) and the contrast. This inversion algorithm exhibits the best features of the modified gradient method, successfully reconstructing a variety of contrasts and fairly insensitive to noise. However, it exhibits additional properties which surpass the modified gradient method.
This paper describes some of the recent developments that have been made to MIKE 21, as part of the new MIKE Flood modelling system. These developments have been aimed at increasing the robustness of the flooding and drying routines, and at extending MIKE 21's capability to include modelling of high Froude Number flows. Through these improvements, it has been possible to carry out realistic simulations of flood wave propagation over an initially dry bed, and of a range of high Froude Number flow conditions, including super-critical flows, broad-crested weir flows, hydraulic jumps and even dam-break flows. The emphasis in making these changes has been on ensuring that the high accuracy and computational efficiency of the MIKE 21 numerical solution procedure has been maintained.
ABSTRACT Images of the subsurface are made for the detection of land‐mines using a bistatic stepped‐frequency continuous‐wave spiral‐antenna system. While the system moves along the surface, the emitted electromagnetic wavefields are scattered by objects in the subsurface and cause changes in the voltages measured at the receiver. These changes are formulated as a convolution of a sensitivity function and a complex contrast function. Within the Born approximation, this sensitivity function is equal to the inner product of the wavefield emitted by the transmitter and the field from the receiver operating in transmitting mode. For true amplitude imaging purposes, knowledge of the wavefields in the subsurface is needed. Since it is difficult to obtain a model which describes the radiation characteristics accurately, we measure the footprints of the antennae at one level in the near‐field region and propagate the emitted wavefields using Huygens’ principle. We use both synthetic and experimental data to localize objects in a homogeneous space. First, we apply time‐domain synthetic‐aperture‐radar (SAR) imaging in its most basic appearance. Next, we apply a single‐step inversion algorithm to the data, where we use the measured radiation characteristics of the antenna system. This results in an increase in resolution. We refer to this method as ‘minimized back‐propagation’.
