Effects of multi-scale velocity heterogeneities on wave-equation migration

Yong Ma and Paul Sava, Center for Wave Phenomena, Colorado School of Mines

ABSTRACTVelocity models used for wavefield-based seismic imagingrepresent approximations of the velocity characterizing the areaunder investigation. The real subsurface velocity can at best beapproximated by the combination of a known backgroundvelocity and unknown multi-scale heterogeneities. Here, wemodel the multi-scale heterogeneity assuming a fractal behaviorand compare this type of heterogeneity with conventionalcorrelated Gaussian random distributions. Data simulated for thevarious heterogeneity distributions are characterized by spectrawith different shapes when analyzed in the log-log domain. Forexample, Gaussian distributions are characterized by exponentialfunctions and fractal distributions are characterized by linearfunctions with fractional slopes. These properties hold for bothdata and migrated images after deconvolution of the sourcewavelet. Exploiting the distinctions between the various kinds ofheterogeneities, we can use least-squares fitting to ascertaincharacteristics and parameters of heterogeneity from the seismicdata and migrated images.

Gaussian heterogeneity

Correlated Gaussian random distributions can be constructedby convolution of uncorrelated random noise n(x) with aGaussian smoothing function g(x)

g(x) is the inverse Fourier transform of

Relate the wavenumber k with the wavelength λ:

Log-log spectrum of Gaussian randomness

The ln G -- ln λ dependence is exponential. (see GM)

)()()( xgxnxrg

4/22

)( kekG

222 /)( eG

ln222ln eG (Left) An example of correlated Gaussian random distributionwith 10 m correlation in depth. (Right) Log-log spectrum ofthe Gaussian random distribution and the least-squaresfitting curve in blue show an exponential trend.

(Left) An example of blocky model of the subsurface. (Right)Log-log spectrum of the blocky variation and the least-squares fitting result in blue show a linear trend with a slopeof 1.

Blocky Stratigraphy is conventionally represented by blocky models. It represents the k -1 background of Earth’s model. (see BM)

GM

BM

DATA HETEROGENEITY

Data carries the information about model heterogeneity.

Recovery of model heterogeneity from data using the

following steps:

• Deconvolution of the wavelet

• Log-Log spectrum analysis (similar to BM, GM, FM)

• Least-squares fitting of the spectrum

y=α1x+ α2 (Linear) y=α3 e-2x+ α4 (Exponential)

Fractal heterogeneity Fractal random distributions can be constructed byconvolution of uncorrelated random noise n(x) with a fractalseries f (x):

f (x) is the inverse Fourier transform of

β is fractal parameter

Relate the wavenumber k with the wavelength λ:

Log-log spectrum of fractal randomness ln F = β ln λ

The ln F -- ln λ dependence is linear. (see FM)

)()()( xfxnxrf kkF )(

2)( F

(Left) Data simulated in the Gaussian random model. (Right)The exponential least-squares fitting of the spectrum gives α3

= -0.001 (σ = 0.01 km).(compare with GM)

(Left) Data simulated in the fractal model. (Right) The linearleast-squares fitting of the spectrum gives the slope α1 = 0.49.(compare with FM)

(Left) An example of fractal random distribution with β = 0.5.

(Right) Log-log spectrum of the fractal randomness and theleast-squares fitting in blue show a linear trend.

(Left) Data simulated in the blocky model. (Right) The linearleast-squares fitting of the spectrum gives the slope α1 = 1.(compare with BM)

GD

FDBD

FM

IMAGE HETEROGENEITY

Migrated images reflect model heterogeneities.

Recover model heterogeneity from a migrated image using

the following steps:

• Deconvolution of the wavelet

• Remove the k -1 background from the spectrum

• Log-Log analysis of the residual spectrum

• Least-squares fitting of the residual spectrum

Well-log data of P-wave velocity from Rulison field, CO.

(Left) Log-log spectrum of the well-log data; least-squaresfitting in blue gives the slope α1 = 1. (Right) The log-logspectrum after removing k-1 background; least-squares fittingin blue gives the slope α1 = 0.60.

One synthetic common shot gather overlaid by thezero-offset trace.

Image of data simulated in the well-log model usingconventional imaging condition. The image trace at x =

2.0 km is superimposed on the image.

Simulate the wavefield of a single shot in the well-log

model (see WM) and record the data (see WD)

Obtain the migrated image of the recorded data (see WI)

Extract the model heterogeneity from the zero-offset

image trace (see IF)

Extract the model heterogeneity from the multi-offset

image traces (see IM)

WM

WD

WIWF

REFERENCESClaerbout, J. F., 1985, Imaging the Earth’s interior: BlackwellScientific Publications.Hoshiba, M., 2000, Large fluctuation of wave amplitudeproduced by small fluctuation of velocity structure: Physics ofthe earth and planetary interiors, 120, 201–217.Shtatland, E. S., 1991, Fractal stochastic models for acousticimpedance: An explanation of scaling or 1/f geology andstochastic inversion: 61th Annual International Meeting, SEG,Expanded Abstracts, 1598–1601.Stefani, J., and G. S. De, 2001, On the Power-Law Behavior ofSubsurface Heterogeneity: 71th Annual International Meeting,SEG, Expanded Abstracts, 2033–2036.Turcotte, D. L., 1997, Fractals and Chaos in Geology andGeophysics: Cambridge University Press.

CONCLUSIONSWe compare different types of multi-scale heterogeneitiesand recover information about the parameters characterizingsuch models from the seismic data and migrated images. Ouranalysis shows that various types of heterogeneity havedifferent character when analyzed in log-log plots. Assumingthat the subsurface model is a combination of a relativelysmooth background plus a few strong interfaces with a blockycharacter, we can attempt to infer the statistics of modelheterogeneities currently undetectable by conventionalseismic methodology. Heterogeneity information is availablein data, as well as migrated images, but accessing thisinformation requires reasonably good estimates of the sourcewavelet.

(Left) For the image trace at x = 2.0 km, least-squares fittingof its log-log spectrum indicates α1 = 1. (Right) Afterremoving k-1 background, least-squares fitting indicates α1 =

0.62. (compare with WF)

Dependence of extracted fractal parameter on thehorizontal position with respect to the source location.Results in the near offset are more precise than in the faroffset. (compare with WF and IF)

IFExtraction of the fractal parameter from

the zero-offset image trace

Extraction of the fractal parameter frommulti-offset image traces

IM

ACKONWLEDGMENTWe acknowledge the support of the sponsors of the Center for Wave Phenomena at Colorado School of Mines. The well-log datawas supplied by the Reservoir Characterization Project of the Colorado School of Mines.