We have formulated a 3-D inverse solution for the magnetotelluric (MT) problem using the non-linear conjugate gradient method. Finite difference methods are used to compute predicted data efficiently and objective functional gradients. Only six forward modelling applications per frequency are typically required to produce the model update at each iteration. This efficiency is achieved by incorporating a simple line search procedure that calls for a sufficient reduction in the objective functional, instead of an exact determination of its minimum along a given descent direction. Additional efficiencies in the scheme are sought by incorporating preconditioning to accelerate solution convergence. Even with these efficiencies, the solution's realism and complexity are still limited by the speed and memory of serial processors. To overcome this barrier, the scheme has been implemented on a parallel computing platform where tens to thousands of processors operate on the problem simultaneously. The inversion scheme is tested by inverting data produced with a forward modelling code algorithmically different from that employed in the inversion algorithm. This check provides independent verification of the scheme since the two forward modelling algorithms are prone to different types of numerical error.

A 3D frequency-domain EM modelling code has been implemented for helicopter electromagnetic (HEM) simulations. A vector Helmholtz equation for the electric fields is employed to avoid convergence problems associated with the first-order Maxwell's equations when air is present. Additional stability is introduced by formulating the problem in terms of the scattered electric fields. Vith this formulation the impressed dipole source is replaced with an equivalent source, which for the airborne configuration possesses a smoother spatial dependence and is easier to model. In order to compute this equivalent source, a primary field arising from dipole sources of either a whole space or a layered half-space must be calculated at locations where the conductivity is different from that of the background.The Helmholtz equation is approximated using finite differences on a staggered grid. After finite-differencing, a complex-symmetric matrix system of equations is assembled and preconditioned using Jacobi scaling before it is solved using the quasi-minimum residual (QMR) method. The modelling code has been compared with other lD and 3D numerical models and is found to produce results in good agreement.\7e have used the solution to simulate novel HEM responses that are computationally intractable using integral equation (IE) solutions. These simulations include a 2D conductor residing at a fault contact with and without topography. Our simulations show that the quadrature response is a very good indicator of the faulted background, while the in-phase response indicates the presence of the conductor. However when interpreting the in-phase response, it is possible erroneously to infer a dipping conductor due to the contribution of the faulted background.

by a large rectangular loop is substantial when host currents are strong near the conductor. The more con ductive the host, the longer the galvanic responses will The three-dimensional (3-D) electromagnetic scatter persist. Large galvanic responses occur if a 3-D conduc ing problem is tirst formulated in the frequency domain tor is in contact with a conductive overburden. For a in terms of an electric field volume integral equation.thin vertical dike embedded within a conductive host, Three-dimensional responses are then Fourier trans the 3-D response is similar in form but differs in mag formed with sine and cosine digital filters or with the nitude and duration from the 2-D response generated decay spectrum. The digital filter technique is applied to by two infinite line sources positioned parallel to the a sparsely sampled frequency sounding, which is re strike direction of the 2-D structure.placed by a cubic spline interpolating function prior to We have used the 3-D solution to study the appli convolution with the digital filters. Typically, 20 to 40 cation of the central-loop method to structural interpre frequencies at five to eight points per decade are re tation. The results suggest variations of thickness of quired for an accurate solution. A calculated transient is conductive overburden and depth to sedimentary struc usually in error after it has decayed more than six ture beneath volcanics can be mapped with one orders in magnitude from early to late time. The decay dimensional inversion. Successful 1-D inversions of 3-D spectrum usually req uires ten frequencies for a satisfac transient soundings replace a 3-D conductor by a con tory solution. However, the solution using the decay ducting layer at a similar depth. However, other pos spectrum appears to be less accurate than the solution sibilities include reduced thickness and resistivity of the using the digital filters, particularly after early times.(-0 host containing the body. Many different l-D Checks on the 3-D solution include reciprocity and con models can be fit to a transient sounding over a 3-D vergence checks in the frequency domain, and a com structure. Near-surface, 3-D geologic noise will not per parison of Fourier-transformed responses with results manently contaminate a central-loop apparent resistivi from a direct time-domain integral equation solution. ty sounding. The noise is band-limited in time and even The galvanic response of a 3-D conductor energized tually vanishes at late times.

Linearized methods are presented for appraising resolution and parameter accuracy in images generated with 2-D and 3-D nonlinear electromagnetic (EM) inversion schemes. When direct matrix inversion is used, the model resolution and a posteriori model covariance matrices can be calculated readily. By analyzing individual columns of the model resolution matrix, the spatial variation of the resolution in the horizontal and vertical directions can be estimated empirically. Plotting the diagonal of the model covariance matrix provides an estimate of how errors in the inversion process, such as data noise and incorrect a priori assumptions, map into parameter error and thus provides valuable information about the uniqueness of the resulting image. Methods are also derived for image appraisal when the iterative conjugate gradient technique is applied to solve the inverse. An iterative statistical method yields accurate estimates of the model covariance matrix as long as enough iterations are used. Although determining the entire model resolution matrix in a similar manner is computationally prohibitive, individual columns of this matrix can be determined. Thus, the spatial variation in image resolution can be determined by calculating the columns of this matrix for key points in the image domain and then interpolating between. Examples of the image analysis techniques are provided on 2-D and 3-D synthetic cross‐well EM data sets as well as a field data set collected at Lost Hills oil field in central California.

S U M M A R YNew techniques for improving both the computational and imaging performance of the threedimensional (3-D) electromagnetic inverse problem are presented. A non-linear conjugate gradient algorithm is the framework of the inversion scheme. Full wave equation modelling for controlled sources is utilized for data simulation along with an efficient gradient computation approach for the model update. Improving the modelling efficiency of the 3-D finite difference (FD) method involves the separation of the potentially large modelling mesh, defining the set of model parameters, from the computational FD meshes used for field simulation. Grid spacings and thus overall grid sizes can be reduced and optimized according to source frequencies and source-receiver offsets of a given input data set. Further computational efficiency is obtained by combining different levels of parallelization. While the parallel scheme allows for an arbitrarily large number of parallel tasks, the relative amount of message passing is kept constant. Image enhancement is achieved by model parameter transformation functions, which enforce bounded conductivity parameters and thus prevent parameter overshoots. Further, a remedy for treating distorted data within the inversion process is presented. Data distortions simulated here include positioning errors and a highly conductive overburden, hiding the desired target signal. The methods are demonstrated using both synthetic and field data.