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Table Of Integrals Series And Products

On robust estimation of the location parameter

Interpretation theory in applied geophysics: Grant, F S ... Interpretation theory in applied geophysics Unknown Binding – January 1, 1965 5.0 out of 5 stars 1 rating. See all formats and editions Hide other formats and editions. The Amazon Book Review Book recommendations, author interviews, editors' picks, and more. Read it now. Enter your mobile number or email address below and we'll send you a ...

Interpretation Theory in Applied Geophysics Grant (McGraw-Hill, 140s.)

We present a method for inverting surface magnetic data to recover 3-D susceptibility models. To allow the maximum flexibility for the model to represent geologically realistic structures, we discretize the 3-D model region into a set of rectangular cells, each having a constant susceptibility. The number of cells is generally far greater than the number of the data available, and thus we solve an underdetermined problem. Solutions are obtained by minimizing a global objective function composed of the model objective function and data misfit. The algorithm can incorporate a priori information into the model objective function by using one or more appropriate weighting functions. The model for inversion can be either susceptibility or its logarithm. If susceptibility is chosen, a positivity constraint is imposed to reduce the nonuniqueness and to maintain physical realizability. Our algorithm assumes that there is no remanent magnetization and that the magnetic data are produced by induced magnetization only. All minimizations are carried out with a subspace approach where only a small number of search vectors is used at each iteration. This obviates the need to solve a large system of equations directly, and hence earth models with many cells can be solved on a deskside workstation. The algorithm is tested on synthetic examples and on a field data set.

3-D inversion of magnetic data

Electrical resistivity of the soil can be considered as a proxy for the spatial and temporal variability of many other soil physical properties (i.e. structure, water content, or fluid composition). Because the method is non-destructive and very sensitive, it offers a very attractive tool for describing the subsurface properties without digging. It has been already applied in various contexts like: groundwater exploration, landfill and solute transfer delineation, agronomical management by identifying areas of excessive compaction or soil horizon thickness and bedrock depth, and at least assessing the soil hydrological properties. The surveys, depending on the areas heterogeneities can be performed in one-, two- or three-dimensions and also at different scales resolution from the centimetric scale to the regional scale. In this review, based on many electrical resistivity surveys, we expose the theory and the basic principles of the method, we overview the variation of electrical resistivity as a function of soil properties, we listed the main electrical device to performed one-, two- or three-dimensional surveys, and explain the basic principles of the data interpretation. At least, we discuss the main advantages and limits of the method.

/pdf/electrical-resistivity-survey-in-soil-science-a-review-52naoumd0b.pdf

Electrical resistivity survey in soil science: a review .

We present two methods for inverting surface gravity data to recover a 3-D distribution of density contrast. In the first method, we transform the gravity data into pseudomagnetic data via Poisson's relation and carry out the inversion using a 3-D magnetic inversion algorithm. In the second, we invert the gravity data directly to recover a minimum structure model. In both approaches, the earth is modeled by using a large number of rectangular cells of constant density, and the final density distribution is obtained by minimizing a model objective function subject to fitting the observed data. The model objective function has the flexibility to incorporate prior information and thus the constructed model not only fits the data but also agrees with additional geophysical and geological constraints. We apply a depth weighting in the objective function to counteract the natural decay of the kernels so that the inversion yields depth information. Applications of the algorithms to synthetic and field data produce density models representative of true structures. Our results have shown that the inversion of gravity data with a properly designed objective function can yield geologically meaningful information.

https://www.eoas.ubc.ca/ubcgif/pubs/papers/geop63-109.pdf

3-D inversion of gravity data

In this paper, the term “depth of investigation” refers generically to the depth below which surface data are insensitive to the value of the physical property of the earth. Estimates of this depth for dc resistivity and induced polarization (IP) surveys are essential when interpreting models obtained from any inversion because structure beneath that depth should not be interpreted geologically. We advocate carrying out a limited exploration of model space to generate a few models that have minimum structure and that differ substantially from the final model used for interpretation. Visual assessment of these models often provides answers about existence of deeper structures. Differences between the models can be quantified into a depth of investigation (DOI) index that can be displayed with the model used for interpretation. An explicit algorithm for evaluating the DOI is presented. The DOI curves are somewhat dependent upon the parameters used to generate the different models, but the results are robust...

/pdf/estimating-depth-of-investigation-in-dc-resistivity-and-ip-310j7lwt7v.pdf

Estimating depth of investigation in DC resistivity and IP surveys

The magnetic method, perhaps the oldest of geophysical exploration techniques, blossomed after the advent of airborne surveys in World War II. With improvements in instrumentation, navigation, and platform compensation, it is now possible to map the entire crustal section at a variety of scales, from strongly magnetic basement at regional scale to weakly magnetic sedimentary contacts at local scale. Methods of data filtering, display, and interpretation have also advanced, especially with the availability of low-cost, high-performance personal computers and color raster graphics. The magnetic method is the primary exploration tool in the search for minerals. In other arenas, the magnetic method has evolved from its sole use for mapping basement structure to include a wide range of new applications, such as locating intrasedimentary faults, defining subtle lithologic contacts, mapping salt domes in weakly magnetic sediments, and better defining targets through 3D inversion. These new applications have increased the method’s utility in all realms of exploration — in the search for minerals, oil and gas, geothermal resources, and groundwater, and for a variety of other purposes such as natural hazards assessment, mapping impact structures, and engineering and environmental studies.

/pdf/the-historical-development-of-the-magnetic-method-in-2o9psruy4m.pdf

The historical development of the magnetic method in exploration

We present an algorithm for inverting induced polarization (IP) data acquired in a 3-D environment. The algorithm is based upon the linearized equation for the IP response, and the inverse problem is solved by minimizing an objective function of the chargeability model subject to data and bound constraints. The minimization is carried out using an interior-point method in which the bounds are incorporated by using a logarithmic barrier and the solution of the linear equations is accelerated using wavelet transforms. Inversion of IP data requires knowledge of the background conductivity. We study the effect of different approximations to the background conductivity by comparing IP inversions performed using different conductivity models, including a uniform half-space and conductivities recovered from one-pass 3-D inversions, composite 2-D inversions, limited AIM updates, and full 3-D nonlinear inversions of the dc resistivity data. We demonstrate that, when the background conductivity is simple, reasonable IP results are obtainable without using the best conductivity estimate derived from full 3-D inversion of the dc resistivity data. As a final area of investigation, we study the joint use of surface and borehole data to improve the resolution of the recovered chargeability models. We demonstrate that the joint inversion of surface and crosshole data produces chargeability models superior to those obtained from inversions of individual data sets.

/pdf/inversion-of-induced-polarization-data-20m2tksm2h.pdf

Yaoguo Li

Papers

3-D inversion of magnetic data

3-D inversion of gravity data

Estimating depth of investigation in DC resistivity and IP surveys

The historical development of the magnetic method in exploration

Inversion of induced polarization data