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 Volume 49, Issue 4, 2001
Geophysical Prospecting  Volume 49, Issue 4, 2001
Volume 49, Issue 4, 2001

1D inversion of DC resistivity data using a quality‐based truncated SVD
Authors Elonio A. Muiuane and Laust B. PedersenMany DC resistivity inversion schemes use a combination of standard iterative least‐squares and truncated singular value decomposition (SVD) to optimize the solution to the inverse problem. However, until quite recently, the truncation was done arbitrarily or by a trial‐and‐error procedure, due to the lack of workable guidance criteria for discarding small singular values. In this paper we present an inversion scheme which adopts a truncation criterion based on the optimization of the total model variance. This consists of two terms: (i) the term associated with the variance of statistically significant principal components, i.e. the standard model estimate variance, and (ii) the term associated with statistically insignificant principal components of the solution, i.e. the variance of the bias term. As an initial model for the start of iterations, we use a multilayered homogeneous half‐space whose layer thicknesses increase logarithmically with depth to take into account the decrease of the resolution of the DC resistivity technique with depth. The present inversion scheme has been tested on synthetic and field data. The results of the tests show that the procedure works well and the convergence process is stable even in the most complicated cases. The fact that the truncation level in the SVD is determined intrinsically in the course of inversion proves to be a major advantage over other inversion schemes where it is set by the user.

Interval velocity and thickness estimate from wide‐angle reflection data
More LessA method to estimate interval velocities and thickness in a horizontal isotropic layered medium from wide‐angle reflection traveltime curves is presented. The method is based on a relationship between the squared reflection traveltime differences and the squared offset differences relative to two adjacent reflectors. The envelope of the squared‐time versus offset‐difference curves, for rays with the same ray parameter, is a straight line, whose slope is the inverse of the square of the interval velocity and whose intercept is the square of the interval time. The method yields velocity and thickness estimates without any knowledge of the overlying stratification. It can be applied to wide‐angle reflection data when either information on the upper crust and/or refraction control on the velocity is not available. Application to synthetic and real data shows that the method, used together with other methods, allows us to define a reliable 1D starting model for estimating a depth profile using either ray tracing or another technique.

On removing the primary field from fixed‐wing time‐domain airborne electromagnetic data: some consequences for quantitative modelling, estimating bird position and detecting perfect conductors
More LessIn the process of removing the primary field from fixed‐wing time‐domain airborne EM data, the response is decomposed into two parts, which are referred to here as the time‐domain ‘in‐phase’ and ‘quadrature’ components. The time‐domain in‐phase component is dominated by the primary field, which varies significantly as the transmitter–receiver separation changes. The time‐domain quadrature component comes solely from the secondary response associated with currents induced in the ground and this is the component that has traditionally been used in the interpretation of data from fixed‐wing towed‐bird time‐domain EM systems. In the off‐time, the quadrature response is very similar to the total secondary response. However, there are large differences in the on‐time and even some small differences in the off‐time.One consequence of these differences is that when airborne EM data are to be interpreted using a synthetic mathematical model, the synthetic data calculated should also be the quadrature component. A second consequence relates to the time‐domain in‐phase component which is sometimes used to estimate the receiver‐sensor (bird) position. The bird‐position estimation process assumes there is no secondary field in the in‐phase component. If the ground is resistive, the secondary contained in the in‐phase component is small, so the bird‐position estimate is accurate to about 30 cm, but in highly conductive areas the secondary contribution can be large and the position estimate can be out by as much as 5 m. A third consequence arises for highly conductive bodies, the response of which is predominantly in‐phase. This means that any response from these types of body is lost in the component that has been removed in the primary‐field extraction procedure. However, if the bird position is measured very accurately, the actual free‐space primary field can be estimated. If this is then subtracted from the estimated primary (actually free‐space primary plus secondary in‐phase response), then the residual is the secondary in‐phase response of the ground. Using this methodology, very conductive ore bodies could be detected. However, a sensitivity analysis shows that detection of a large vertically dipping very conductive body at 150 m depth would require that the bird position be measured to an accuracy of about 1.4 cm and the aircraft attitude to within about 0.01°. Such tolerances are very stringent and not easily attainable with current technology.

Inversion of DC resistivity data using neural networks
Authors Gad El‐Qady and Keisuke UshijimaThe inversion of geoelectrical resistivity data is a difficult task due to its non‐linear nature. In this work, the neural network (NN) approach is studied to solve both 1D and 2D resistivity inverse problems. The efficiency of a widespread, supervised training network, the back‐propagation technique and its applicability to the resistivity problem, is investigated. Several NN paradigms have been tried on a basis of trial‐and‐error for two types of data set. In the 1D problem, the batch back‐propagation paradigm was efficient while another paradigm, called resilient propagation, was used in the 2D problem. The network was trained with synthetic examples and tested on another set of synthetic data as well as on the field data. The neural network gave a result highly correlated with that of conventional serial algorithms. It proved to be a fast, accurate and objective method for depth and resistivity estimation of both 1D and 2D DC resistivity data. The main advantage of using NN for resistivity inversion is that once the network has been trained it can perform the inversion of any vertical electrical sounding data set very rapidly.

Porosity and permeability prediction from wireline logs using artificial neural networks: a North Sea case study
Authors Hans B. Helle, Alpana Bhatt and Bjørn UrsinEstimations of porosity and permeability from well logs are important yet difficult tasks encountered in geophysical formation evaluation and reservoir engineering. Motivated by recent results of artificial neural network (ANN) modelling offshore eastern Canada, we have developed neural nets for converting well logs in the North Sea to porosity and permeability. We use two separate back‐propagation ANNs (BP‐ANNs) to model porosity and permeability. The porosity ANN is a simple three‐layer network using sonic, density and resistivity logs for input. The permeability ANN is slightly more complex with four inputs (density, gamma ray, neutron porosity and sonic) and more neurons in the hidden layer to account for the increased complexity in the relationships. The networks, initially developed for basin‐scale problems, perform sufficiently accurately to meet normal requirements in reservoir engineering when applied to Jurassic reservoirs in the Viking Graben area. The mean difference between the predicted porosity and helium porosity from core plugs is less than 0.01 fractional units. For the permeability network a mean difference of approximately 400 mD is mainly due to minor core‐log depth mismatch in the heterogeneous parts of the reservoir and lack of adequate overburden corrections to the core permeability. A major advantage is that no a priori knowledge of the rock material and pore fluids is required. Real‐time conversion based on measurements while drilling (MWD) is thus an obvious application.

2D surface topography boundary conditions in seismic wave modelling
Authors Bent Ruud and Stig HestholmNew formulations of boundary conditions at an arbitrary two‐dimensional (2D) free‐surface topography are derived. The top of a curved grid represents the free‐surface topography while the grid's interior represents the physical medium. The velocity–stress version of the viscoelastic wave equations is assumed to be valid in this grid. However, the rectangular grid version attained by grid transformation is used to model wave propagation in this work in order to achieve the numerical discretization. We show the detailed solution of the particle velocities at the free surface resulting from discretizing the boundary conditions by second‐order finite‐differences (FDs). The resulting system of equations is spatially unconditionally stable. The FD order is gradually increased with depth up to eighth order inside the medium. Staggered grids are used in both space and time, and the second‐order leap‐frog and Crank–Nicholson methods are used for time‐stepping. We simulate point sources at the surface of a homogeneous medium with a plane free surface containing a hill and a trench. Applying parameters representing exploration surveys, we present examples with a randomly realized surface topography generated by a 1D von Kármán function of order 1. Viscoelastic simulations are presented using this surface with a homogeneous medium and with a layered, randomized medium realization, all generating significant scattering.

2D inversion of refraction traveltime curves using homogeneous functions
By V.B. PiipA method using simple inversion of refraction traveltimes for the determination of 2D velocity and interface structure is presented. The method is applicable to data obtained from engineering seismics and from deep seismic investigations. The advantage of simple inversion, as opposed to ray‐tracing methods, is that it enables direct calculation of a 2D velocity distribution, including information about interfaces, thus eliminating the calculation of seismic rays at every step of the iteration process. The inversion method is based on a local approximation of the real velocity cross‐section by homogeneous functions of two coordinates. Homogeneous functions are very useful for the approximation of real geological media. Homogeneous velocity functions can include straight‐line seismic boundaries. The contour lines of homogeneous functions are arbitrary curves that are similar to one another. The traveltime curves recorded at the surface of media with homogeneous velocity functions are also similar to one another. This is true for both refraction and reflection traveltime curves. For two reverse traveltime curves, non‐linear transformations exist which continuously convert the direct traveltime curve to the reverse one and vice versa. This fact has enabled us to develop an automatic procedure for the identification of waves refracted at different seismic boundaries using reverse traveltime curves. Homogeneous functions of two coordinates can describe media where the velocity depends significantly on two coordinates. However, the rays and the traveltime fields corresponding to these velocity functions can be transformed to those for media where the velocity depends on one coordinate. The 2D inverse kinematic problem, i.e. the computation of an approximate homogeneous velocity function using the data from two reverse traveltime curves of the refracted first arrival, is thus resolved. Since the solution algorithm is stable, in the case of complex shooting geometry, the common‐velocity cross‐section can be constructed by applying a local approximation. This method enables the reconstruction of practically any arbitrary velocity function of two coordinates. The computer program, known as godograf, which is based on this theory, is a universal program for the interpretation of any system of refraction traveltime curves for any refraction method for both shallow and deep seismic studies of crust and mantle. Examples using synthetic data demonstrate the accuracy of the algorithm and its sensitivity to realistic noise levels. Inversions of the refraction traveltimes from the Salair ore deposit, the Moscow region and the Kamchatka volcano seismic profiles illustrate the methodology, practical considerations and capability of seismic imaging with the inversion method.

Three‐dimensional imaging of subsurface structures using resistivity data
Authors Myeong‐Jong Yi, Jung‐Ho Kim, Yoonho Song, Seong‐Jun Cho, Seung‐Hwan Chung and Jung‐Hee SuhWe have developed a three‐dimensional inverse scheme for carrying out DC resistivity surveys, incorporating complicated topography as well as arbitrary electrode arrays. The algorithm is based on the finite‐element approximation to the forward problem, so that the effect of topographic variation on the resistivity data is effectively evaluated and incorporated in the inversion. Furthermore, we have enhanced the resolving power of the inversion using the active constraint balancing method. Numerical verifications show that a correct earth image can be derived even when complicated topographic variation exists. By inverting the real field data acquired at a site for an underground sewage disposal plant, we obtained a reasonable image of the subsurface structures, which correlates well with the surface geology and drill log data.
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