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  • Prestack seismic amplitude analysis: An integrated overview

    Brian H. Russell1

    Abstract

    In this tutorial, I present an overview of the techniques that are in use for prestack seismic amplitude analysis,current and historical. I show that these techniques can be classified as being based on the computation andanalysis of either some type of seismic reflection coefficient series or seismic impedance. Those techniques thatare based on the seismic reflection coefficient series, or seismic reflectivity for short, are called amplitudevariation with offset methods, and those that are based on the seismic impedance are referred to as prestackamplitude inversion methods. Seismic reflectivity methods include: near and far trace stacking, intercept ver-sus gradient analysis, and the fluid factor analysis. Seismic impedance methods include: independent and si-multaneous P and S-impedance inversion, lambda-mu-rho analysis, Poisson impedance inversion, elasticimpedance, and extended elastic impedance inversion. The objective of this tutorial is thus to make senseof all of these methods and show how they are interrelated. The techniques will be illustrated using a 2D seismicexample over a gas sand reservoir from Alberta. Although I will largely focus on isotropic methods, the last partof the tutorial will extend the analysis to anisotropic reservoirs.

    IntroductionThe amplitude variation with offset (AVO) and pre-

    stack inversion techniques have grown to include amultitude of subtechniques, each with their own as-sumptions. In this tutorial, I make the assumption thatAVO techniques are based on the computation andanalysis of some type of seismic reflection coefficientseries, or reflectivity for short, and that prestack inver-sion techniques are based on the computation andanalysis of some type of seismic impedance. To under-stand the distinction between impedance and reflectiv-ity, refer to Figure 1, which shows the different ways inwhich geologists and geophysicists look at their data.

    As shown on the left side of Figure 1, the goal of allgeoscientists is to understand the subsurface of theearth, which consists of a series of layers of variablestructure, lithology, porosity, and fluid content. To dothis, the geologist generally analyzes borehole welllog measurements of some layer parameter P, whereasthe geophysicist, when using exploration seismology,analyzes changes in this layer parameter at the interfa-ces between successive layers, called the reflectivity(R). The fundamental idea in this tutorial is that thereis a relationship between P and R, and it can be written

    RPi Pi1 PiPi1 Pi

    Pi2Pi

    ; (1)

    where Pi Pi1 Pi; Pi Pi1Pi2 and subscript i re-fers to the ith layer. That is, the reflectivity is found

    by dividing the change in the parameter between twosuccessive layers by twice its average value. The sub-script P in the reflectivity RP indicates that the reflec-tivity is associated with parameter P. I will use thisnotation throughout the tutorial. This is, of course,an oversimplification, because the seismic trace alsocontains the source wavelet, amplitude scaling effects,noise contamination due to random noise and coherentnoise such as multiples, static time shifts due to topog-raphy and near-surface weathering, spatial misposition-ing due to structure and subsurface scattering at depth,and so on. A complete discussion of these effects andthe techniques of seismic processing which have beendeveloped to remove or ameliorate them is beyond thescope of this tutorial.

    Even if our processing is extremely good, we arealways left with a band-limited reflectivity which canbe modeled using the convolutional model, written inmatrix form as266666666664

    W 0 0 0

    ..

    .W0

    . .. ..

    .

    Wn1... . .

    .0

    0 Wn1. ..

    W 0... . .

    . . .. ..

    .

    0 0 Wn1

    377777777775

    264

    R0...

    Rm1

    375

    264

    S0...

    Snm2

    375;

    (2)

    1Hampson-Russell, A CGG Company, Calgary, Alberta, Canada. E-mail: [email protected] received by the Editor 10 August 2013; revised manuscript received 1 October 2013; published online 20 March 2014. This paper

    appears in Interpretation, Vol. 2, No. 2 (May 2014); p. SC19SC36, 25 FIGS., 1 TABLE.http://dx.doi.org/10.1190/INT-2013-0122.1. 2014 Society of Exploration Geophysicists and American Association of Petroleum Geologists. All rights reserved.

    t

    Special section: Interpreting AVO

    Interpretation / May 2014 SC19Interpretation / May 2014 SC19

  • or S WR, where R is an m-sample reflectivity (notethat subscript 0 represents zero time),W is the convolu-tional matrix in which each column contains an n-sam-ple wavelet shifted by one sample from the previouscolumn, and S is the nm 1 sample result, or seismictrace. This technique of convolution is illustrated picto-rially in Figure 2, where a reflectivity similar to that ofFigure 1 has been convolved with a symmetrical, orzero-phase, wavelet.

    We can think of convolution as shifting the waveletso that it is centered at each reflection coefficient, scal-ing the wavelet by the amplitude of each reflection co-efficient, and then summing the result. We say that thisresult is band limited because, although the spectrumof the reflectivity contains all frequencies from 0 Hz tothe Nyquist, the seismic wavelet is typically band lim-ited between approximately 1060 Hz (although someof the new seismic receivers and acquisition techniquesare pushing the low end much lower). It is thus impor-tant to note that in subsequent sections when I refer tothe particular reflectivity model used in an AVO analy-sis, this more precisely refers to the band-limited resultfound from applying equation 2.

    A key question is which parameter P we are inter-ested in. For the geologist, the parameter is usuallyone that enables him or her to infer the reservoir prop-erties, such as porosity, water saturation, TOC, etc.However, the geophysicist is usually limited to param-eters that are related to the propagation and reflectionof the seismic signal, such a P-impedance, the productof P-wave velocity and density, and S-impedance, theproduct of S-wave velocity and density. Once theseproperties have been estimated, the geophysicist caninfer the reservoir properties mentioned above, butin this tutorial we will focus more on the estimationof these impedances themselves and the analysis ofthe band-limited reflectivity from which we derivethe impedances.

    The geometry of an idealized common midpoint(CMP) seismic gather is shown in Figure 3, in whichtraces recorded at different offsets are grouped by

    increasing offset around a CMP point. Notice that eachoffset Xi corresponds to an angle of incidence i at thereflector. In Figure 3, the single reflector shown willcreate a seismic reflection due to the contrast betweenthe density (), P-wave velocity (VP), and S-wave veloc-ity (VS) in each layer. As we will discuss, the amplitudeof this reflection will change at each offset.

    Figure 4 shows a set of real gathers which were ac-quired and processed over a shallow gas field in centralAlberta. This data set will be used to explain many ofthe methods described in this tutorial. In particular, no-tice the amplitudes change as a function of offset on thetrough and peak of the seismic reflection event below620 ms. Our goal in this tutorial is a physical under-standing of this amplitude change, and how it relatesto the gas sand.

    Reflectivity methodsNormal incidence reflectivity

    Unlike the single incident and reflected raypathsshown in Figure 3, in an elastic earth, we actually

    Figure 1. Two different ways of analyzing the subsurfaceof the earth, where the geologist analyzes borehole well logmeasurements of some layer parameter P, whereas the geo-physicist measures changes in the layer parameter, the reflec-tivity R.

    Figure 2. Convolution of a reflectivity similar to that inFigure 1 with a zero-phase wavelet, where convolution canbe seen as shifting the wavelet so that it is centered at eachreflection coefficient, scaling the wavelet by the amplitude ofthe reflection coefficient, and then summing the result.

    Figure 3. An idealized CMP gather, where the Xi are the off-sets and the 0 0 represent the angles of incidence of eachseismic raypath (note that X0 0 and 0 0). The single re-flector shown will produce a seismic reflection caused by thecontrast between the density (), P-wave velocity (VP), and S-wave velocity (VS) in each layer.

    SC20 Interpretation / May 2014

  • observe mode converted reflected and transmittedP- and S-raypaths for an incident P-wave raypath at ar-bitrary incidence angle, as illustrated in Figure 5, whichshows the conversion of an incident P-wave at a non-zero angle of incidence into its reflected and transmit-ted P- and S-components. The P, or compressional,waves are elastic waves that travel by the alternatecompression and rarefaction of the earths subsurface,whereas S, or shear, waves are created by side-to-sideshearing of the earths subsurface. Shear waves in anisotropic medium with a polarization (particle motion)in a vertical plane are SV-waves, whereas those with apolarization in the horizontal plane are SH-waves.However, there are other directions of polarizationperpendicular to the direction of travel which produceparticle motion that is a combination of SV- and SH-waves. For SV-waves from a horizontal surface, the par-ticle motion is in the plane of Figure 5 but for SH-waves,think of the particle motion as coming out of the planeshown in Figure 5. In HTI-fractured media, like somecarbonates, a vertically traveling S-wave splits intotwo S-waves, one has polarization aligned paralleland the other perpendicular to the direction of the frac-tures with the faster S-wave having a polarization