A fast inversion technique for the interpretation of data from resistivity tomography surveys has been developed for operation on a microcomputer. This technique is based on the smoothness-constrained least-squares method and it produces a two-dimensional subsurface model from the apparent resistivity pseudosection. In the first iteration, a homogeneous earth model is used as the starting model for which the apparent resistivity partial derivative values can be calculated analytically. For subsequent iterations, a quasi-Newton method is used to estimate the partial derivatives which reduces the computer time and memory space required by about eight and twelve times, respectively, compared to the conventional least-squares method. Tests with a variety of computer models and data from field surveys show that this technique is insensitive to random noise and converges rapidly. This technique takes about one minute to invert a single data set on an 80486DX microcomputer.

Rapid least-squares inversion of apparent resistivity pseudosections by a quasi-Newton method1

Rapid least-squared inversion of apparent resisitivity pseudosections by a quasi-Newton method

Responding to the latest developments in rock physics research, this popular reference book has been thoroughly updated while retaining its comprehensive coverage of the fundamental theory, concepts, and laboratory results. It brings together the vast literature from the field to address the relationships between geophysical observations and the underlying physical properties of Earth materials - including water, hydrocarbons, gases, minerals, rocks, ice, magma and methane hydrates. This third edition includes expanded coverage of topics such as effective medium models, viscoelasticity, attenuation, anisotropy, electrical-elastic cross relations, and highlights applications in unconventional reservoirs. Appendices have been enhanced with new materials and properties, while worked examples (supplemented by online datasets and MATLAB® codes) enable readers to implement the workflows and models in practice. This significantly revised edition will continue to be the go-to reference for students and researchers interested in rock physics, near-surface geophysics, seismology, and professionals in the oil and gas industries.

The Rock Physics Handbook

Tutorial : 2-D and 3-D electrical imaging surveys

We use new interior models of cold planets to investigate the mass-radius relationships of solid exoplanets, considering planets made primarily of iron, silicates, water, and carbon compounds. We find that the mass-radius relationships for cold terrestrial mass planets of all compositions we considered follow a generic functional form that is not a simple power law: log10 Rs = k1 + log10(Ms) - k2M for up to Mp ≈ 20 M⊕, where Ms and Rs are scaled mass and radius values. This functional form arises because the common building blocks of solid planets all have equations of state that are well approximated by a modified polytrope of the form ρ = ρ0 + cPn. We find that highly detailed planet interior models, including temperature structure and phase changes, are not necessary to derive solid exoplanet bulk composition from mass and radius measurements. For solid exoplanets with no substantial atmosphere we have also found the following: with 5% fractional uncertainty in planet mass and radius it is possible to distinguish among planets composed predominantly of iron or silicates or water ice but not more detailed compositions; with ~5% uncertainty water ice planets with 25% water by mass may be identified; the minimum plausible planet size for a given mass is that of a pure iron planet; and carbon planet mass-radius relationships overlap with those of silicate and water planets due to similar zero-pressure densities and equations of state. We propose a definition of "super-Earths" based on the clear distinction in radii between planets with significant gas envelopes and those without.

http://militzer.berkeley.edu/papers/34_seager_kuchner_hier-majumder_militzer_2007.pdf

Mass-Radius Relationships for Solid Exoplanets

Geophysical inversion involves the estimation of the parameters of a postulated earth model from a set of observations. Since the associated model responses can be nonlinear functions of the model parameters, nonlinear least-squares techniques prove to be useful for performing the inversion. A common type of inversion applies iterative damped linear least squares through use of the Marquardt-Levenberg method. Traditionally, this method has been implemented by solving the associated normal equations in conventional ways. However, Singular Value Decomposition (SVD) produces significant improvements in computational precision when applied to the same system of normal equations. Iterative least-squares modeling finds application in a wide variety of geophysical problems. Two examples illustrate the approach: (1) seismic wavelet deconvolution, and (2) the location of a buried wedge from surface gravity data. More generally, nonlinear least-squares inversion can be used to estimate earth models for any set of geophysical observations for which an appropriate mathematical description is available.

A review of least-squares inversion and its application to geophysical problems*

This analysis draws together basic rock physics, amplitude variations with offset (AVO), and seismic amplitude inversion to discuss how fluid‐factor discrimination can be performed using prestack seismic data. From both Biot and Gassmann theories for porous, fluid‐saturated rocks, a general formula is first derived for fluid‐factor discrimination given that both the P and S impedances are available. In essence, the two impedances are transformed so that they better differentiate between the fluid and rock matrix of the porous medium. This formula provides a more sensitive discriminator of the pore‐fluid saturant than the acoustic impedance and is especially applicable in hard‐rock environments. The formulation can be expressed with either the Lame constants and density, or the bulk and shear moduli and density. Numerical and well‐log examples illustrate the applicability of this approach. AVO inversion results are then incorporated to show how this method can be implemented using prestack seismic data. Fi...

/pdf/fluid-property-discrimination-with-avo-a-biot-gassmann-44yjqoqf5i.pdf

Fluid-property discrimination with AVO: A Biot-Gassmann perspective

Geophysical inversion by iterative modeling involves fitting observations by adjusting model parameters. Both seismic and potential-field model responses can be influenced by the adjustment of the parameters of the rock properties. The objective of this "cooperative in­ version" is to obtain a model which is consistent with all available surface and borehole geophysical data. Al­ though inversion of geophysical data is generally non­ unique and ambiguous, we can lessen the ambiguities by inverting all available surface and borehole data. This paper illustrates this concept with a case history in which surface seismic data, sonic logs, surface gravity data, and borehole gravity meter (BHGM) data are ad­ equately modeled by using least-squares inversion and a series offorward modeling steps. The traditional analysis of various geophysical data sets has been termed "integrated interpretation." Such analysis has traditionally involved much processing, modeling, and subjec­ tive interpretation in order to develop a geologic model which is consistent with various surface and borehole geophysical data. A method which automates some of the above pro­ cedures is least-squares inversion. This procedure perturbs an initial set of model parameters in order to fit the observed data. Least-squares inversion is described in many papers, in­ cluding those of Jackson (1972), Wiggins (1972), Jupp and Vozoff (1975), and Lines and Treitel (1984). This inversion technique could also be termed "automated iterative mod­ eling." In terms of modeling different geophysical data sets, there are many possible approaches which allow inversion of a given data set, and it is doubtful whether one single recipe will prove suitable for all problems. Nevertheless, by jointly invert­ ing various types of geophysical observations, we may lessen the extent of some of the ambiguities inherent in individual data sets. LaFehr (1984) gives a germane discussion of several reasons for this type of coordinated geophysical effort. In the present study, we define "cooperative inversion" as the esti­ mation of a subsurface model which is consistent with various independent geophysical data sets. In the case discussed in this paper, these data sets include surface and borehole observa­ tions of the seismic and gravity responses. In our definition, cooperative inversion may include either "joint inversion" or "sequential inversion."

Cooperative inversion of geophysical data

Summary. This paper describes the application of tomography to seismic travel-time inversion. There are various implementations of travel-time tomography. In reflection tomography, sources and receivers are on the surface of the Earth and the principal seismic events are reflections from subsurface velocity discontinuities. In transmission tomography, sources and/or receivers may be buried beneath the surface and the events correspond to direct, or unreflected, arrivals; this is the analogue of medical tomography. There are also cases in which both direct as well as reflected arrivals are important, such as in Vertical Seismic Profiling. The latter is a direct application of the first two, but is not discussed in any detail here. It is also shown how the iterative use of travel-time tomography and depth migration can produce much enhanced subsurface images. Examples of both transmission tomography and reflection tomography combined with depth migration illustrate the methods.

/pdf/applications-of-seismic-travel-time-tomography-54revx2060.pdf

Applications of seismic travel-time tomography

/pdf/introduction-to-the-physics-of-the-earth-s-interior-1zryailyju.pdf

Larry Lines

Papers

A review of least-squares inversion and its application to geophysical problems*

Fluid-property discrimination with AVO: A Biot-Gassmann perspective

Cooperative inversion of geophysical data

Applications of seismic travel-time tomography

Introduction to the Physics of the Earth's Interior