Nonlinear avo inversion based on a novel exact pp reflection. Seismic data analysis is the most important and most commonly used tool used in the oil and gas. Avo analysis is a technique that geophysicists can execute on seismic data to determine a rocks fluid content, porosity, density or. Beyond this point, the equations do not hold due to rapid variations in reflected and transmitted p and swave amplitude data. Linearization of zoeppritz equations and practical utilization. Workflow for zoeppritz avo inversion to estimate seismic. The amplitudes of the transmitted and reflected waves can be calculated from zoeppritz s equations, which have a more complicated form than those listed in c1. P d s s vs p vp r v v r v v r 1 tan, 8 sin, 1 4 sin, and. In this study, we will formulate our joint inversion based on the exact zoeppritz equations. Specifically, we wish to infer or possibly estimate elastic parameters of.
Zoeppritz equations of reflected and refracted energy partitioning in a physical model are generally stable until the incident angle approaches the critical angle 28. Zoeppritz equations muskat et al 1940, koefod 1955 and bortfeld 1961. The inversion objective function is then constructed by utilising bayesian theory. Gas phase calculations may be carried either by using mole fraction input and employing the pengrobinson equation of state, or by using the gas density ratio as input employing empirical relations. The zoeppritz 1919 equations allow geophysicists to derive the exact plane wave amplitudes of a reflected p wave as a function of angle, but they do not give an intuitive understanding of how these amplitudes relate to the various physical parameters. The equations relevant to an introduction to electrical resistivity studies in geophysics are derived and related to the qualitative discussions of sec.
We propose an efficient algorithm for nonlinear avo inversion of precritical reflections using the exact zoeppritz equations in multichannel and multiinterface form for simultaneous estimation of. To overcome the weaknesses of conventional prestack amplitude variation with angle inversion based on various linear or quasilinear approximations, we have conducted a nonlinear inversion method using the exact zoeppritz matrix ezai. A computer program for the application of zoeppritz s amplitude equations and knotts energy equations, bulletin of the seismological society of america, vol. This website beta version contains information on geophysical methods, references to geophysical citations, and a glossary of geophysical terms related to environmental applications. Equationthe first term gives the amplitude at normal incidence theta 0, the second term characterizes rtheta at intermediate angles, and the third term describes the approach to critical angle. Accepted 21 june, 20 the theory and practical utilization of simplification of the original general expressions for the reflection. Linearization of zoeppritz equations and practical utilization oladapo, michael ilesanmi department of applied geophysics, federal university of technology, akure, nigeria.
The attempt is to recapture results used quoted, not derived by levin and keys, and to extend these to include the case of an incident swave. Machine learning applications to geophysical data analysis. Zhou et al proposed a generalized linear avo inversion method with the a priori constraint of a modified trivariate cauchy distribution based on the zoeppritz equations, which improved the stability and quality of the joint inversion. The zoeppritz equations, information theory, and support. Karl bernhard zoeppritz 22 october 1881 20 july 1908 was a german geophysicist who made important contributions to seismology, in particular the formulation of the zoeppritz equations. In geophysics and reflection seismology, the zoeppritz equations are a set of equations that describe the partitioning of seismic wave energy at an interface, typically a boundary between two different layers of rock. Aug, 2018 the analysis of amplitude variation with offset avo plays a significant role in fluid detection and lithology discrimination in hydrocarbon reservoirs. Zhu and mcmechan 2012 propose a new algorithm using the exact zoeppritz equations to obtain four independent parameters a density ratio and three velocity ratios for pp reflections. The akirichards equation the akirichards equation is a linearized form of the zoeppritz equations which is written. Reflection seismology or seismic reflection is a method of exploration geophysics that uses the principles of seismology to estimate the properties of the earths subsurface from reflected seismic waves.
The compressional wave reflection coefficient rtheta given by the zoeppritz equations is simplified to the following. Approximations to the zoeppritz equations and their use in avo analysis article pdf available in geophysics 646 november 1999 with 798 reads how we measure reads. Using the exact zoeppritz equations in pseudolinear form. It consists essentially of solving a cubic equation for the density contrast, the coefficients of which are dependent upon amplitudes, velocities, and angles of incidence. Joint pp and ps ava seismic inversion using exact zoeppritz. Nonlinear pp and ps joint inversion based on the exact. Investigations in geophysics series 3 geophysical developments series 6 geophysical references series 10 geophysical monograph series 14 geophysics reprint series 18. This technology started with connolly, who, basing it on linearization of the zoeppritz equations, defined the elastic impedance ei as the equivalent of acoustic impedance ai at nonzero angles. Calculation of a synthetic gather using the akirichards approximation to the zoeppritz equations a thesis submitted to the graduate faculty of the university of new orleans in partial fulfillment of the requirements for the degree master of science in applied physics by graham r. However, these equations only reflect the relation between reflection coefficients, incidence angles, and elastic parameters on either side of the interface, which means that wavepropagation effects, such as spherical spreading, attenuation, transmission loss.
Zoeppritzs equations with ray tracing and full elastic wave equation with finite difference method fdm are two most commonly used techniques in avo modeling. Prestack amplitude versus angle inversion for youngs. In addition, special attention is focused on the sensitive analysis of controlling the stability of simultaneous. Many geophysical modeling programs require an enormous number of parameters to describe a 2 or 3. The avo equations derived in this section are all expressed in terms of angle of incidence equations 24, 31, and 36. The amplitudes of the transmitted and reflected waves can be calculated from zoeppritzs equations, which have a more complicated form than those listed in c1. Consider the two elastic halfspace layers in figure 11. This page links to crewes interactive java explorers, plotters and calculators. The analysis of amplitude variation with offset avo plays a significant role in fluid detection and lithology discrimination in hydrocarbon reservoirs. Avo modeling in seismic processing and interpretation ii.
Zoeppritz approximations crewes research report volume 16 2004 1 modelling of linearized zoeppritz approximations arnim b. In this paper, we directly used the full zoeppritz equation to calculate r pp rather than its approximations, and we attempted to incorporate the epr with the mrf to solve the illposed problem of prestack seismic inversion. A section on the factors that affect seismic amplitudes and a. Accurate jacobian matrix using the exact zoeppritz equations. Geophysics a set of equations that describes the partitioning of energy in a wavefield relative to its angle of incidence at a boundary across which the properties of the rock and fluid content changes. In this paper, based on the assumption of isotropy, we perform several substitutions to convert the zoeppritz equations from the classical form to a new form containing the chosen elastic constants of youngs modulus, poissons ratio, and density. Then, the early seismic observations and how they led to the birth of avo analysis are presented. The main difference between these two techniques is the former calculates the primaryonly reflectivities and the latter calculates. A twostage procedure article pdf available in journal of geophysics and engineering 152. Amplitude variation with angle inversion using the exact.
The akirichards equation any other angle is modelled with the akirichards equation, a linearized form of the zoeppritz equations which is written and is the basis of virtually all avo methods. Eagle ford case study, seg technical program expanded abstracts 2018. If the address matches an existing account you will receive an email with instructions to reset your password. Seismology is a way of studying the interior of the earth by creating sound waves, and recording them after they have traveled through the earth. The zoeppritz equations zoeppritz and erdbebnenwellen, 1919 are the foundation of avo modeling and can describe the relationship between the amplitudes of the pwave incident waves and the pp or ps reflected waves. The present study pertains to the development of a computer program for zoeppritz energy partition equations to calculate the reflection coefficient curves of the productive zones from seismic and well log data of the missakeswal area. The various approximations for the zoeppritz equations are examined, and the assumptions and limitations of each approximation are clearly identified. Approximations to the zoeppritz equations and their use in avo analysis yanghua wang.
Pdf approximations to the zoeppritz equations and their. Zoeppritz based avo inversion using an improved markov chain monte carlo method xinpeng pan1 guangzhi zhang1,2 jiajia zhang1,2 xingyao yin1,2 received. The zoeppritz equations can be solved for the reflected and refracted p and swave amplitudes, a 1, b 1, a 2, and b 2. Zoeppritz s equations with ray tracing and full elastic wave equation with finite difference method fdm are two most commonly used techniques in avo modeling. In other words, its just a load of functions that implement important equations in mostly seismic geophysics, from akirichards to zoeppritz. The angles of incidence, reflection and refraction. Zoeppritz equations schlumberger oilfield glossary. Geophysics letters datadriven selection of the minimumgradient support parameter in timelapse focused electric imaging frederic nguyen, andreas kemna, tanguy robert, and thomas hermans.
Pdf approximations to the zoeppritz equations and their use. Amplitude variation with offset avo inversion is an important component of prestack seismic inversion and utilizes more formation information than poststack inversion chen et al2007, yin et al20, 2014, 2015, zong et al2015. Borehole geophysics a sequential dynamic bayesian network for porepressure estimation with uncertainty quantification rachel h. The zoeppritz equations describe the relationship between seismic properties that are useful for the interpretation of lithology and fluid properties. Nonlinear pp and ps joint inversion based on the exact zoeppritz equations. Accurate jacobian matrix using the exact zoeppritz equations and effects on the inversion of reservoir properties in porous media article pdf available in pure and applied geophysics 1761. The zoeppritz equation zoeppritz 1919 is the foundation of avo inversion, which illustrates the energy distribution of incident and. Summary timelapse avo analysis is used to interpret the amplitude difference and map fluid variation in a reservoir. Avo inversion based on inverse operator estimation in.
Law which relates the angles of incidence, reflection and refraction to the velocities of the medium. Karl bernhard zoeppritz 22 october 1881 20 july 1908 was a german geophysicist who made important contributions to seismology, in particular the formulation of the zoeppritz equations these equations relate the amplitudes of pwaves and swaves at each side of an interface, between two arbitrary elastic media, as a function of the angle of incidence and are largely used in reflection. In geophysics and reflection seismology, amplitude versus offset avo or amplitude variation with offset is the general term for referring to the dependency of the seismic attribute, amplitude, with the distance between the source and receiver the offset. Pdf constrained nonlinear avo inversion using zoeppritz. Reflection and refraction university of california, berkeley. Stability study of prestack seismic inversion based on. An earlier linear approximation to the zoeppritz equations was given by bortfeld. Pdf nonlinear pp and ps joint inversion based on the. Pdf joint pp and ps ava seismic inversion using exact.
They are named after their author, the german geophysicist karl bernhard zoeppritz, who died before they were published in 1919. Pdf accurate jacobian matrix using the exact zoeppritz. Nonlinear estimation of rj from avo intercept and gradient. An appropriate and feasible method for inverting parameters is significant due to the difficulty to find the global optimum solution.
A full derivation of these equations can be found in most exploration geophysics text books, such as. The zoeppritz equations are very useful in exploration geophysics. An introduction to electrical resistivity in geophysics. Avo intercept and gradient refer either to terms in the linearized avo equations or to the equivalent measurements from seismic data. The cartoon below illustrates the ray geometry for a pwave incident on the boundary between media of velocity v1 and v2. The zoeppritz equations are part of the basic theory of avo analysis which describes the relationship between seismic reflection and transmission coefficients and elastic rock properties e. A simplification of the zoeppritz equations geophysics.
R ar vp br vs cr d, the akirichards equation says that the reflectivity at angle is the weighted sum of the v p, v s and density reflectivities. Seismic attribute for hydrocarbon expressions in stack section. Calculation of a synthetic gather using the akirichards. Today, personal computers can generate synthetics based on the full zoeppritz formalism, but approximations are still used to gain physical insight into the relative influence of velocity and density changes on. Development of a computer program for zoeppritz energy. Approximations to the zoeppritz equations and their use in. Most current prestack ava joint inversion methods are based on the exact zoeppritz equation and its various approximations. Interpretive geophysics, processing geophysics, avo analysis, synthetic seismogram, seismic gather, zoeppritz equations, akirichards approximation, hydrocarbon detection, drilling risk mitigation.
Matrix forms for the knott zoeppritz equations matrix forms for the knott zoeppritz equations kris innanen abstract in this note we derive convenient matrix forms for the knott zoeppritz equations. However, our interest in exploration seismology is largely the angledependency of the ptop reflections given by the coefficient a 1. The inversion of prestack seismic data using amplitude variation with offset avo has received increased attention in the past few decades because of its key role in estimating reservoir properties. The method requires a controlled seismic source of energy, such as dynamite or tovex blast, a specialized air gun or a seismic vibrator, commonly known by the trademark name vibroseis. Evolving geophysics through innovation 405 nonlinear estimation of rj from avo intercept and gradient chuck ursenbach, crewes, university of calgary, calgary, canada abstract two methods are derived for incorporating the quadratic shear reflectivity term into estimation of shear impedance reflectivity from avo intercept and gradient data. Zoeppritz equations, and may be applied to either conventional or convertedwave data.
Accurate jacobian matrix using the exact zoeppritz. Geophysical forward models can act as bandpass filters, allowing only some of the inherent structure of the earth to be reflected in geophysical data. Domenico 1974 shows that the bright spot can arise with less than a. This opened a new dimension of utilization of aiei pairs for lithology and fluid discrimination. Estimating petrophysical reservoir properties through. It contains mostly basic stuff, and is aimed at the interpreter rather than the weathered processor or numbercrunching seismic analyst. Zoeppritz equationbased prestack inversion and its. Zoeppritz equation based prestack inversion and its application in fluid identification. Haase abstract the aki and richards approximations to zoeppritz s equations as well as approximations by stewart, smith and gidlow, and fatti et al. Stacking process and approximations of zoeppritz equations. In practice, the mapping of amplitudes associated with a reflection event on a cmp gather from offset to angle of incidence needs to be performed. I have included shueys linear approximation of the zoeppritz equations.
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