**Introduction **

**to **

**Seismic Inversion Methods **

**Brian H. Russell **

**Hampson-Russell **

**Software **

**Services, **

**Ltd. **

**Calgary, **

**Alberta **

**Course Notes Series, No. 2 **

**S. N. Domenico, Series Editor **

**These course notes are published without the normal SEG peer reviews. **
**They have not been examined for accuracy and clarity. Questions or **
**comments by the reader should be referred directly to the author. **

**ISBN 978-0-931830-48-8 ** **(Series) **
**ISBN 978-0-931830-65-5 (Volume) **

**Library of Congress Catalog Card Number 88-62743 **
**Society of Exploration Geophysicists **

**P.O. ** **Box 702740 **

**Tulsa, Oklahoma 74170-2740 **

**¸ 1988 by the Society of Exploration Geophysicists **

**All rights reserved. This book or portions hereof may not be reproduced in any form without permission **
**in writing from the publisher. **

**Reprinted 1990, 1992, 1999, 2000, 2004, 2006, 2008, 2009 **

**]: nl;roduc t1 on •o Selsmic I nversion •thods ** **Bri an Russell **

**Table ** **of Contents **

**PAGE **

**Part I ** **Introduction ** **1-2 **

**Part Z ** **The Convolution ** **Model ** **2-1 **

**Part ** **3 **

**Part ** **4 **

**Part ** **5 **

**P art ** **6 **

**P art ** **7 **

**2.1 Tr•e Sei smic Model **

**2.2 The Reflection ** **Coefficient ** **Series **

**2.3 The Seismic Wavelet **

**2.4 The Noise Component **
**Recursive Inversion ** **- Theory **

**3.1 Discrete ** **Inversion **

**3.2 Problems encountered ** **with real **

**3.3 Continuous ** **Inversion **

**data **

**Seismic Processing Consi derati ons **

**4. ! I ntroduc ti on **

**4.2 Ampl i rude recovery **

**4.3 Improvement of vertical **

**4.4 Lateral ** **resolution **
**4.5 ** **Noise attenuation **

**resolution **

**Recursive ** **Inversion ** **- Practice **

**5.1 The recursive ** **inversion ** **method **

**5.2 Information in the low frequency component **

**5.3 Seismically derived porosity **
**Sparse-spike Inversi on **

**6.1 ** **I ntroduc ti on **

**6.2 Maximum-likelihood aleconvolution and inversion **

**6.3 The L I norm method **
**6.4 Reef Problem **

**I nversi on appl ied to Thi n-beds **

**7.1 Thin bed analysis **

**7.Z Inversion compari son of thin beds **

**Model-based ** **Inversion **

**B. 1 I ntroducti on . **

**8.2 Generalized ** **linear ** **inversion **

**8.3 Seismic 1 ithologic roodell **

**ing (SLIM) **

**Appendix 8-1 Matrix applications in geophysics **
**Part ** **8 **
**2-2 **
**2-6 **
**2-12 **
**2-18 **
**3-1 **
**3-2 **
**3-4 **
**3-8 **
**4-1 **
**4-2 **
**4-4 **
**4-6 **
**4-12 **
**4-14 **
**5-1 **
**5-2 **
**5-10 **
**5-16 **
**6-1 **
**6-2 **
**6-4 **
**6-22 **
**6-30 **
**7-1 **
**7-2 **
**7-4 **
**8-1 **
**8-2 **
**8-4 **
**8-10 **
**8-14 **

**Introduction ** **to Seismic Inversion ** **Methods ** **Brian Russell **

**Part ** **9 ** **Travel-time ** **Inversion **

**g. 1. I ntroducti ** **on **

**9.2 Numerical examples of traveltime inversion **

**9.3 Seismic Tomography **

**Part 10 Amplitude versus offset (AVO) Inversion **

**10.1 AVO theory **

**10.2 AVO inversion by GLI **
**Part 11 Velocity Inversion **

**I ntroduc ti on **

**Theory and Examples **
**Part 12 Summary **
**9-1 **
**9-2 **
**9-4 **
**9-10 **
**10-1 **
**10-2 **
**10-8 **
**11-1 **
**11-2 **
**11-4 **
**12-1 **

**Introduction ** **to Seismic •nversion ** **Methods ** **Brian Russell **

**PART I - INTRODUCTION **

**Introduction ** **to Seismic Inversion ** **Methods ** **Brian Russell **

**I NTRODUCT ION TO SE I SMI C INVERSION METHODS **

**, ** **__ ** **_• ** **i i ** **_ ** **, ** **. , ** **, ** **! ** **• _, ** **l_ ** **, ** **, ** **i.,. ** **_ **

**Part ** **i - ** **Introduction _ ** **. ** **. **

**This ** **course is intended as an overview of the current techniques used in **

**the inversion of seismic data. It ** **would therefore seem appropriate to begin **

**by defining what is meant by seismic inversion. The most general definition **

**is as fol 1 ows' **

**Geophysical inversion involves mapping the physical structure and **

**properties of the subsurface of the earth using measurements made on **

**the surface of the earth. **

**The above definition ** **is so broad that it encompasses virtually ** **all ** **the **

**work that is done in seismic analysis ** **and interpretation. ** **Thus, in this **

**course we shall primarily 'restrict ** **our discussion to those inversion ** **methods **

**which attempt to recover a broadband pseudo-acoustic impedance log from a **

**band-1 imi ted sei smic trace. **

**Another way to look at inversion is to consider it as the technique for **

**creating a model of the earth using the seismic data as input. ** **As such, it **

**can be considered as the opposite of the forwar• modelling technique, which **

**involves creating a synthetic seismic ** **section ** **based on a model of the earth **

**(or, in the simplest case, using a sonic log as a one-dimensional model). The **
**relationship between forward and inverse modelling is shown in Figure 1.1. **

**To understand seismic inversion, we must first understand the physical **

**processes involved in the creation of seismic data. ** **Initially, ** **we will **

**therefore ** **look at the basic ** **convolutional ** **model of the seismic trace in the **

**time and frequency domains, **

**considering the thre e components **

**of this model: **

**reflectivity,**

**seismic wavelet, and noise.**

**Part ** **I - Introduction **

**_ ** **m i ** **--. **

**Introduction to Seismic InverSion Methods ** **Brian Russell **

**FORWARD MODELL I NG **

**i ** **m ** **ß **

**INVERSE MODELLING (INVERSION) _ **

**, ** **ß ** **ß ** **_ **
**Input' **
**Process: **
**Output' **

**EARTH **

**MODEL **

**,**

**MODELLING**

**ALGORITHM**

**SEISMIC RESPONSE**

**i**

**m**

**mlm**

**ii**

**INVERSION**

**ALGORITHM**

**EARTH **

**MODEL **

**i**

**ii**

**Figure 1.1 Fo.•ard **

**' andsInverse **

**Model,ling **

**Introduction. to Seismic Inversion Methods ** **Brian l•ussel 1 **

**Once we have an understanding of these concepts and the problems which **

**can occur, we are in a position to look at the methods which are currently **

**ß **

**used to invert seismic data. These methods are summarized in Figure 1.2. ** **The **

**primary emphasis of the course will ** **be **

**the ultimate ** **resul.t, ** **as was previously **

**on poststack seismic inversion where **

**o **

**Oiscussed, is a pseudo-impeaance **

**section. **

**We will start by looking at the ** **most contanon methods of poststack **

**inversion, ** **which are based on single trace recursion. To better ** **unUerstand **

**these recurslye inversion ** **procedures, ** **it ** **is ** **important to look at the **

**relationship ** **between aleconvolution anU inversion, and how Uependent each **

**method is on the deconvolution scheme Chosen. ** **Specifically, ** **we will consider **

**classical ** **"whitening" aleconvolution methods, wavelet extraction methods, and **

**the newer sparse-spike deconvolution methods such as Maximum-likelihood **

**deconvolution ** **and the L-1 norm metboa. **

**Another important type of inversion method which will be aiscussed is **

**model-based inversion, where a geological moael is iteratively ** **upUated to finU **

**the best fit ** **with the seismic data. ** **After this, traveltime ** **inversion, ** **or **

**tomography, will be discussed along with several illustrative ** **examples. **

**After the discussion on poststack inversion, we shall move into the realm **

**of pretstack. ** **These methoUs, still ** **fairly new, allow us to extract parameters **

**other than impedance, such as density and shear-wave velocity. **

**Finally, ** **we will aiscuss the geological aUvantages anU limitations ** **of **

**each seismic inversion roethoU, looking at examples of each. **

**Introduction to Selsmic Inversion Methods ** **Brian Russell **

**SE **

**I SMI **

**C I NV **

**ERSI **

**ON **

**.MET•OS ,,, **

**POSTSTACK**

**INVERSION**

**PRESTACK**

**INVERSION**

**MODEL-BASED **

**I RECURSIVE **

**INVERSION **

**• ,INVE **

**SION **

**- "NARROW **

**BAND**

**TRAVELTIME**

**INVERSION**

**!TOMOGRAPHY) **

**SPARSE- **

**SPIKE **

**WAV **

**EF **

**I EL **

**D **

**NVERSIOU **

**i**

**LINEAR**

**METHODS**

**,,**

**i**

**i**

**--**

**I METHODS **

**] **

**Figure 1.2 ** **A summary of current inversion techniques. **

**Introduction to Seismic Inversion Methods ** **Brtan Russell **

**PART **

**2 - THE CONVOLUTIONAL **

**MODEL **

**Introduction ** **to Seismic Inversion Methods ** **Brian Russell **
**Part 2 - ** **The Convolutional Mooel **

**2.1 Th'e Sei smi c Model **

**The most basic and commonly **

**used one-Oimensional **

**moael for the seismic **

**trace is referreU to as the convolutional moOel, which states that the seismic**

**trace is simply the convolution of the earth's reflectivity with a seismic **

**source function with the adUltion of a noise component. In equation form, **

**where * implies convolution, **

**s(t) : w(t) * r(t) + n(t)s **

**where **

**and **

**s (t) = the sei smic trace, **
**w(t) : a seismic wavelet, **
**r (t) : earth refl ecti vi ty, **
**n(t) ** **: additive ** **noise. **

**An even simpler assumption **

**is to consiUer the noise component **

**to be zero, **

**in which case the seismic tr•½e is simply the convolution of a seismic wavelet **

**with t•e earth ' s refl ecti vi ty, **

**s(t) = w{t) * r(t). **

**In seismic processing **

**we deal exclusively with digital data, that is, **

**data sampled **

**at a constant **

**time interval. **

**If we consiUer **

**the relectivity to **

**consist of a reflection coefficient at each time sample (som• of which can be **

**zero), and the wavelet to be a smooth function in time, convolution can be **

**thought of as "replacing" each reflection. coefficient with a scaled version of **

**the wavelet and summing **

**the result. The result of this process **

**is illustrated **

**in Figures 2.1 and 2.Z for both a "sparse" **

**and a "dense" **

**set of reflection **

**coefficients. ** **Notice that convolution with ** **the wavelet tends to "smear" the **

**reflection coefficients. That is, there is a total loss of resolution, which **

**is the ability to resolve closely spaced reflectors. **

**Introduction ** **to Seismic Inversion Nethods ** **Brian Russell **
**WAVELET: **

**(a) ' * • • **

**: -' ':' **

**REFLECTIVITY**

**Figure 2.1**

**TRACE:**

**Convolution **

**of a wavelet with a **

**(a) •avelet. (b) Reflectivit.y. **

**sparse" reflectivity. **

_{(c) Resu }_{1 ting Sei smic Trace. }**(a) **
**(b') **
**! **
**. **
**i **
**: ** **! **

**! **

**: **

**: **

**i**

**i**

**,**

**ß**

**:**

**i**

**!**

**i**

**i**

**'?t**

*****

**c**

**o**

**o**

**o**

**o**

**o**

**Fi õure 2.2 ** **Convolution of a wavelet with a sonic-derived "dense" **

**reflectivity. ** **(a) Wavelet. (b) Reflectivity. ** **(c) Seismic Trace **

**, ** **i ** **, ß .... ** **! , ** **m ** **i ** **i ** **L _ ** **- ** **' **

**Introduction ** **to Seismic Inver'sion ** **Methods ** **Brian Russell **

**An alternate, but equivalent, way of looking at the seismic trace is in **

**the frequency domain. If we take the Fourier transform of the previous **

**ß **

**equati on, we may write **

**S(f) = W(f) x R(f), **

**where **

**S(f) = Fourier transform **

**of s(t), **

**W(f) = Fourier transform of w(t), **

**R(f) = Fourier transform of r(t), ** **ana f = frequency. **

**In the above equation we see that ** **convolution becomes multiplication in **

**the frequency domain. However, the Fourier transform is a complex function, **

**and it is normal to consiUer the amplitude and phase spectra of the individual **

**components. The spectra of S(f) may then be simply expressed **

**esCf) = e **

**w **

**where **

**(f) + er(f), **

**I •ndicates **

**amplitude **

**spectrum, **

**and **

**0 indicates phase spectrum. ** **. **

**In other words, convolution involves multiplying the amplitude spectra **

**and adding the phase spectra. ** **Figure 2.3 ** **illustrates ** **the convolutional model **

**in the frequency domain. Notice that the time Oomain problem of loss of **

**resolution becomes one of loss of ** **frequency content in the frequency domain. **

**Both the high and low frequencies of the reflectivity ** **have been severely **

**reOuceo by the effects of the seismic wavelet. **

**Introduction to Seismic Inversion Methods ** **Brian Russell **

**AMPLITUDE ** **SPECTRA ** _{PHASE SPECTRA }

**w (f) **
**I I **
**-t- **
**R (f) **
**i i ** **, ** **I ** **! **
**i. ** **iit ** **|11 **
**loo **
**s (f) **
**I ** **i! **
**I **
**i ** **i **

**Figure 2.3 ** **Convolution in the frequency domain for **
**the time series shown in Figure 2.1. **

**Introduction ** **to Seismic ** **Inversion ** **Methods ** **Brian ** **Russell **

**2.g The Reflection ** **Coefficient ** **Series **

**l_ ** **_ ** **,m i ** **_ _ ** **, ** **_ _ ** **m_ ** **_,• ** **, ** **_ _ ** **ß ** **_ ** **el **

**of ** **as ** **the res **
**within ** **the ** **ear **

**compres si onal **
**i ropedance to re **
**impedances by **

**coefficient ** **at **
**fo11 aws: **

**'The reflection coefficient series (or reflectivity, ** **as it is also called) **

**described **

**in the previous **

**section **

**is one **

**of the fundamental **

**physical **

**concepts **

**in the seismic method. Basically, each reflection coefficient may be thought **

**ponse of the seismic wavelet to an acoustic impeUance change **

**th, where acoustic impedance is defined as the proUuct of **
**velocity and Uensity. Mathematically, converting from acoustic **

**flectivity ** **involves dividing ** **the difference in the acoustic **

**the sum of the acoustic impeaances. This gives t•e ** **reflection **

**the boundary between the two layers. ** **The equation is as **

**•i+lVi+l - iVi **

**Zi+l- Z **

**i **

**i • ** **i+1 **
**where **
**and **
**r = reflection coefficient, **

**/o__ density, **

**V -- compressional velocity,**

**Z -- acoustic impeUance,**

**Layer i overlies Layer i+1.**

**We must also convert from depth to time by integrating the sonic log **
**transit ** **times. ** **Figure •.4 ** **shows a schematic sonic log, density log, ** **anU **
**resulting acoustic impedance for a simplifieU earth moael. Figure 2.$ shows **

**the result of converting **

**to the reflection **

**coefficient **

**series **

**and **

**integrating **

**to time. **

**It ** **should be pointed out that this formula is true only for the normal **
**incidence case, that is, for a seismic wave striking the reflecting interface **
**at right angles to the beds. Later in this course, we shall consider the case **
**of nonnormal ** **inciaence. **

**Introduction to Seismic Inversion Methods ** **Brian Russell **

**STRATIGRAPHIC ** **SONIC LOG **

**SECTION ** **•T (•usec./mette) **
**4OO **
**SHALE ... ** **DEPTH **
**ß ** **ß ß ** **ß ** **ß ß **
**SANOSTONE ** **. . - .. , **

**' **

**I**

**!**

**!_1**

**! ! !**

**UMESTONE I I I ! I ! I 1**

**LIMESTONE**

**2000111**

**30O**

**200**

**I **

**3600 m/s**

**_**

**v-- I **

**V--3600 **

**J **

**V= 6QO0**

**I **

**loo**

**2.0**

**,**

**3.0**

**OENSITY LOG.**

**ß •**

**Fig. 2.4. Borehole **

**Log Measurements. **

**mm ** **mm ** **rome ** **m ** **.am **
**,mm ** **mm ** **m ** **--- ** **mm **
**SHALE ... ** **OEPTH **

**•---'- **

**[ **

**SANDSTONE . . ...**

**,**

**!**

**I**

**!11**

**I1**

**UMESTONE I I 1 I I I II**

**i ! I 1 i I i 1000m**

**SHALE •.--._--.---- •**

**•.'•**

**LIMESTONE**

**2000 m**

**ACOUSTIC**

**IMPED,M•CE (2•**

**(Y•ocrrv x OEaSn•**

**REFLECTWrrY**

**V$ OEPTH**

**VS TWO.WAY**

**TIME**

**20K -.25**

**O**

**Q.2S**

**-.25**

**O**

**+ .2S**

**I**

**I**

**v**

**'**

**I**

**-**

**1000 m**

**--**

**NO**

**,•**

**, ..**

**- 20o0 m**

**I SECOND**

**Fig. ** **2.5. ** **Creation of Reflectivity ** **Sequence. **

**IntroductJ ** **on 1:o Sei stoic Inversion ** **Herhods ** **Bri an Russell **

**Our best method of observing seJsm•c impedance and reflectivity ** **is •o **

**derlye ** **them from well log curves. Thus, we may create an impedance curve by **

**multiplying together •he sonic and density logs from a well. We may •hen **

**compute the reflectivlty by using •he formula shown earlier. ** **Often, we do not **

**have the density log available• to us and must make do with only the sonJc. The **

**approxJmatJon of velocJty to •mpedance 1s a reasonable approxjmation, and **

**seems **

**to hold well for clas;cics and carbonates (not evaporltes, however). **

**Figure 2.6 shows the sonic and reflectJv•ty traces from a typJcal Alberta well **

**after they have been Jntegrated to two-way tlme. **

**As we shall see later, ** **the type of aleconvolution and inversion used is **

**dependent on the statistical ** **assumptions which are made about the seismic **

**reflectivity ** **and wavelet. Therefore, how can we describe the reflectivity ** **seen **

**in ** **a well? ** **The traditional ** **answer has always been that ** **we consider ** **the **

**reflectivity ** **to be a perfectly ** **random sequence and, from Figure •.6, this **

**appears to be a good assumption. A ranUom sequence has the property that ** **its **

**autocorrelation is a spike at zero-lag. ** **That is, all the components of the **

**autocorrelation are zero except the zero-lag value, as shown in the following **

**equati on- **

**t(Drt = ( 1 , 0 , 0 , ... **

**) **

**t **

**zero-lag. **

**Let ** **us test this idea on a theoretical ** **random sequence, shown in ** **Figure **

**2.7. ** **Notice that the autocorrelation ** **of this sequence has a large spike at **

**ß **

**the zeroth lag, but that there is a significant noise component at nonzero **
**lags. To have a truly random sequence, it must be infinite ** **in extent. ** **Also **
**on ** **this ** **figure is shown the autocorrelation of a well ** **log ** **•erived **
**reflectivity. ** **We see that it ** **is even less "random" than the random spike **
**sequence. We will discuss this in more detail on the next page. **

**IntroductJon ** **to Se•.s=•c Inversion ** **Methods ** **Br•an Russell **

**RFC **

**F•g. 2.6. Reflectivity **

**sequence **

**derived **

**from **

**sonJc **

**.log. **

**RANDOM SPIKE SEQUENCE **

**WELL LOG DERIVED REFLECT1vrrY **

**AUTOCORRE•JATION OF RANDOM SEQUENCE **

**AUTOCORRELATION ** **OF REFLECTIVITY **

**Fig. ** **2.7. ** **Autocorrelat4ons of random and well log **

**der4ved **

**spike sequences. **

**Introductlon ** **to Sei smic Inversion Methods ** **Brian Russel 1 **

**Therefore, the true earth reflectivity ** **cannot be considered as being **

**truly random. For a typical Alberta well we see a number **

**of large spikes **

**(co•responding **

**to major lithol ogic change) **

**sticking up above **

**the crowd. A good **

**way to describe this statistically is as a Bernoulli-Gaussian **

**sequence. The **

**Bernoulli part of this term implies a sparseness in the positions of the **

**spikes and the Gaussian **

**implies a randomness **

**in their amplitudes. When we **

**generate such a sequence, there is a term, lambda, which controls the **

**sparseness of the spikes. For a lambda **

**of 0 there are no spikes, and for a **

**lambda **

**of 1, the sequence **

**is perfectly Gaussian in distribution. **

**Figure 2.8 **

**shows a number of such series for different**

**values of lambda. Notice that**

**a**

**typical Alberta well log reflectivity **

**would have a lambda **

**value in the 0.1 to **

**0.5 range.**

**I ntroducti on to Sei smic I nversi on Methods ** **Brian Russell **
**It **
**tl ** **I ** **I ** **I **
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**OF REFLECTIVITIES **

**Fig. ** **2.8. ** **Examples of reflectivities ** **using lambda **

**factor ** **to be discussed ** **in Part 6. **

**, ** **, ** **m ** **i ** **ß ** **i **

**Introduction to Seismic Inversion ,Methods ** **Brian Russell **

**2.3 ** **The Seismic ** **Wavelet **

**-- ** **_ ß ** **• ** **, **

**Zero Phase and Constant Phase Wavelets **

**m _ ** **m _ ** **m ** **ß ** **m ** **u ** **, L ** **m ** **_ ** **J **

**The assumption tha.t there is a single, well-defined wavelet which is **
**convolved with the reflectivity ** **to produce the seismic trace is overly **
**simplistic. ** **More realistically, ** **the wavelet is both time-varying and complex **
**in shape. However, the assumption of a simple wavelet is reasonable, and in **

**this ** **section ** **we shall ** **consider ** **several ** **types ** **of wavelets ** **and their **

**characteristics. **

**First, ** **let us consider the Ricker wavelet, which consists of a peak and **

**two troughs, or side lobes. ** **The Ricker wavelet is dependent only on its **
**dominant frequency, that is, the peak frequency of its a•litude ** **spectrum or **
**the inverse of the dominant period in the time domain (the dominant period is **

**found by measuring **

**the time from trough to trough). Two Ricker wave'lets are **

**shown in Figures 2.9 and 2.10 of frequencies 20 and 40 Hz. Notice that as the **
**anq•litude spectrum of a wavelet .is broadened, the wavelet gets narrower in the **

**time domain, **

**indicating **

**an increase **

**of resolution. Our ultimate **

**wavelet **

**would **

**be a spike, with a flat amplitude spectrum. Such a wavelet is an unrealistic **
**goal in seismic processing, but one that is aimed for. **

**The Rtcker wavelets of Figures 2.9 and 2.10 are also zero-phase, or **

**perfectly ** **symmetrical. This is a desirable character. tstic of wavelets since **

**the energy is then concentrated at a positive peak, and the convol'ution of the **

**wavelet with a reflection ** **coefficient ** **will ** **better ** **resolve ** **that reflection. ** **To **

**get an idea of non-zero-phase wavelets, consider Figure 2.11, where a Ricker **

**wavelet has been rotated by 90 degree increments, and Figure 2.12, where the **

**same wavelet has been shifted by 30 degree increments. ** **Notice that the 90 **

**degree rotation ** **displays perfect antis•nmnetry, whereas a 180 degree shift **

**simply inverts the wavelet. The 30 degree rotations are asymetric. **

**Introduction to Seismic Inversion Methods ** **Brian Russell **

**Fig. **

**Fig. **

**2.9. ** **20 Hz Ricker Wavelet'. **

**•.10. ** **40 Hz Ricker ** **wavelet. **

**Fig. ** **2.11. ** **Ricker wavelet ** **rotated **
**by 90 degree increments **

**Fig. **

**Part 2 - The Convolutional ** **Model **

**2.12. ** **Ricker wavelet ** **rotated **
**by 30 degree increments **

**Introduction ** **to Seismic Inversion ** **Methods ** **Brian Russell **

**Of course, a typical ** **seismic wavelet contains a larger ** **range of **

**frequencies than that shown on the Ricker wavelet. Consider the banapass **

**fil•er shown **

**in Figure 2.13, where we have passed a bana of frequencies **

**between 15 and 60 Hz. The filter ** **has also had cosine tapers applied between 5 **

**and 15 Hz, and between 60 and 80 Hz. ** **The taper reduces the "ringing" effect **

**that would be noticeable if the wavelet amplitude spectrum was a simple **

**box-car. The wavelet of Figure 2.13 is zero-phase, and would be excellent as **

**a stratigraphic wavelet. It is often referred to as an Ormsby wavelet. **

**Minimum Phase Wavelets **

**The concept of minimum-phase is one that ** **is ** **vital ** **to aleconvolution, but **

**is also a concept that is poorly understood. ** **The reason for this lack of **

**understanding is that most discussions of ** **the ** **concept stress the mathematics **

**at the expense of the physical ** **interpretation. ** **The ** **definition ** **we **

**use of minimum-phase is adapted from Treitel ** **and Robinson (1966): **

**For a given set of wavelets, all with the same amplitude spectrum, **

**the minimum-phase **

**wavelet **

**is the one **

**which **

**has the sharpest **

**leading **

**edge. That is, only wavelets which have positive time values.**

**The reason that minimum-phase concept is important to us is ** **that ** **a **

**typical wavelet in dynamite work is close to minimum-phase. Also, the wavelet **

**from the seismic instruments is ** **also minimum-phase. The minimum-phase **

**equivalent of the 5/15-60/80 ** **zero-phase wavelet is shown in Figure 2.14. ** **As **

**in the aefinition used, notice that the minimum-phase wavelet has no component **

**prior to time zero and has its energy concentrated ** **as close to the origin as **

**possible. ** **The phase spectrum of the minimum-wavelet is also shown. **

**I•troduct•on to Sei stoic !nversion **

**Nethods. **

**Br•an Russell **

**ql ** **Re• R ** **Zero Phase I•auel•t ** **5/15-68Y88 ** **{• **

**0.6 **

**f1.38 ** **- ** **Trace ** **1 **

**iii **

**- e.3e **

**... , **

**• ... **

**' **

**2be **

**1 ** **Trace ** **I **

**Fig. ** **2.13. Zero-phase bandpass **

**wavelet. **

**Reg 1) min,l• ** **wavelet ** **•/15-68/88 hz **

**18.00 p ** **Trace I **

**Reg E ** **wayel ** **Speetnm **

**'188.88 **

**• **

**Trace 1 **

**0.8 **

**188 **

**Fig. ** **2.14. Minim•-phase equivalent **

**of zero-phase wavelet **

**shown in Fig. ** **2.13. **

**_ **

**! ** **m,m, ** **i ** **m **

**Part 2 -Th 'e Convolutional Model **

**i **

**Introduction ** **to Seismic Inversion Methods ** **Brian Russell **

**Let us now look at the effect of different wavelets on the reflectivity **

**function itself. ** **Figure 2.15 a anU b shows a number of different wavelets **

**conv6lved with the reflectivity (Trace 1) from the simple blocky model shown **

**in Figure Z.5. ** **The following wavelets have been used- high **

**zero-phase (Trace •), **

**low frequency **

**zero-phase **

**(Trace ½), high **

**minimum phase (Trace 3), ** **low frequency minimum phase (Trace 5). **

**figure, we can make the fol 1 owing observations: **

**frequency **
**frequency **

**From the **

**(1) Low freq. zero-phase wavelet: (Trace 4) **

**- Resolution of reflections ** **is poor. **

**- Identification ** **of onset of reflection ** **is good. **

**(Z) High freq. zero-phase wavelet: (Trace Z) **

**- Resolution of reflections ** **is good. **

**- Identification ** **of onset of reflection ** **is good. **

**(3) Low freq. min. p•ase wavelet- (Trace 5) **

**- Resolution of reflections ** **i s poor. **

**- Identification of onset of reflection is poor. **
**(4) High freq. min. phase wavelet: (Trace 3) **

**- Resolution of refl ec tions is good. **

**- Identification ** **of onset of reflection ** **is poor. **

**Based on the above observations, we would have to consider the high **

**frequency, **

**zero-phase **

**wavelet **

**the best, and the low-frequency, **

**minimum **

**phase **

**wavelet ** **the worst. **

**(a) **

**Introduction ** **to Seismic Inversion ** **Methods ** **Brian Russell **

**!ql Reg R Zer• Phase Ua•elet ** **•,'1G-•1• ** **14z **

**F **

**- •.• [' **

**' **

**•,3 Recj B miniilium phue ** **' ** **' **

**17 .• **

**q2 Reg C Zero Phase 14aue16(' ' •'le-3•4B Hz **

**e **

**q• Reg 1) 'minimum phase " ** **•,leJ3e/4e h• ** **' **

**8 **

**e.e •/••/'•-•"v--,._,, **

**-r **

**e.• **

**' ' " s•e **

**,m**

**,,**

**'' **

**Tr'oce**

**[b)**

**Fig.**

**700**

**2.15. Convolution of four different ** **wavelets shown **
**in (a) with trace I of (b). ** **The results ** **are **
**shown on traces 2 to 5 of (b). **

**Introduction ** **to Seismic Inversion Methods ** **Brian Russell **

**g.4 Th•N. **

**oi se. C **

**o. **

**mp.o•ne **

**nt **

**- **

**The situation that has been discussed so far is the ideal case. ** **That is, **

**. **

**we have interpreted every reflection wavelet on a seismic trace as being an **

**actual ** **reflection ** **from a lithological ** **boundary. ** **Actually, ** **many of the **

**"wiggles" on a trace are not true reflections, but are actually the result of **

**seismic noise.**

**Seismic noise can be grouped under two categories-**

**(i) Random Noise - noise which is uncorrelated from trace to trace and is **

**•ue mainly to environmental factors. **

**(ii) Coherent Noise - noise which is predictable on the seismic trace but **
**is unwanted. An example is multiple reflection interference. **

**Random noise can be thought of as the additive component n(t) which was **

**seen in the equation on page 2-g. **

**Correcting for this term is the primary **

**reason for stacking our •ata. **

**Stacking actually uoes an excellent job of **

**removing ranUom noise.**

**Multiples, one of the major sources of coherent noise, are caused **

**by **

**multiple "bounces" **

**of the seismic signal within the earth, as shown **

**in Figure **

**2.16. They may be straightforward, as in multiple seafloor bounces **

**or **

**"ringing", or extremely **

**complex, as typified by interbed multiples. Multiples **

**cannot be thought of as additive noise and must be modeled **

**as a convolution **

**with the reflecti vi ty.**

**Figure **

**generated by the simple blocky model **

**this ** **data, ** **it ** **is ** **important that **

**Multiples may be partially ** **removed **

**powerful elimination ** **technique. **

**aleconvolution, f-k filter.ing, **
**wil 1 be consi alered in Part 4. **

**2.17 **

**shown on Figure •. 5. **
**the ** **multiples ** **be **

**by stacking, ** **but **

**Such ** **techniques **
**and inverse velocity stacking. **

**shows the ** **theoretical ** **multiple ** **sequence which would be **

**If ** **we are to invert **

**effectively ** **removed. **
**often require a more **
**include ** **predictive **

**These techniques **

**Introduction to Seismic Inversion Methods ** **Brian Russell **

**Fig. ** **2.16. Several multiple generating mechanisms. **

**TIME ** **TIME **

**[sec) ** **[sec) **

**0.7 ** **0.7 **

**REFLECTION ** **R.C.S. **

**COEFFICIENT ** **WITH ALL **

**SERIES ** **MULTIPLES **

**Fig. ** **2.17. **

_{Refl ectivi ty sequence }

_{Refl ectivi ty sequence }

_{of Fig. }

_{of Fig. }

**and without mul tipl es. **

**Part 2 - The Convolutional Model **

**2.5. ** **with **

**. **

**PART 3 - RECURS IVE INVERSION - THEORY **

**m•mmm•---' ** **.• ** **,- ** **- - ** **' •- ** **- ** **_ ** **- ** **- _- ** **_ **

**•ntroduct•on ** **to SeJsmic Znversion Methods ** **Brian Russell **

**PART 3 - RECURSIVE INVERSION - THEORY **

**3.1 Discrete ** **Inversion **

**, ** **! ** **ß ** **, ** **, • **

**In section 2.2, we saw that reflectivity ** **was defined in terms of **

**acoustic impedance changes. The formula was written: **

**Y•i+lV•+l **

**' •iV! 2i+ **

**1' Z **

**i **

**ri-- yoi'+lVi+l+ **

**Y•iVi **

**-- -Zi..+l **

**+ Z **

**i **

**where ** **r -- refl ecti on coefficient, **

**/0-- density, **

**V -- compressional velocity, **
**Z -- acoustic impedance, **

**and ** **Layer i overlies Layer i+1. **

**If ** **we have the true reflectivity ** **available to us, it is possible to **
**recover the a.coustic impedance by inverting the above formula. Normally, the **
**inverse' formulation is simply written down, but here we will ** **supply the **

**missing steps for completness. First, ** **notice that: **

**Also **

**Ther'efore **

**Zi+l+ **

**Z **

**i **

**Zi+ **

**1- Z **

**t **

**2 Zi+ **

**1 **

**I + ri- Zi+l **

**+ Zi + Zi+l **

**+ 2i Zi+l **

**+ Zi **

**I- ri-- **

**Zi+l+ **

_{Zi+l+ Z i }

_{Zi+l+ Z i }

**Z **

**i **

**Zi+ **

_{Zi+l+ Z i }

_{Zi+l+ Z i }

**1- Z **

**i **

_{Zi+l+ Z i }

_{Zi+l+ Z i }

**2 Zf[ **

**Zi+l **

**Z i **

**l+r. **

**1 **

**1 **

**Part 3 - Recursive Inversion- Theory **

**ill, ** **ß , ** **I **

**Introduction to Seismic Invers-•on Methods ** **Brian Russell **
**pv-e- **
**TIME **
**(sec] **
**0.7 **
**REFLECTION **
**COEFFICIENT **
**SERIES **
**RECOVERED **
**ACOUSTIC **
**IMPEDANCE **

**Fig. ** **3.1, **

**Applying **

**the recursive **

**inversion **

**formula **

**to a **

**simple, and exact, reflectivity. **

**, ! ** **ß **

**!ntroductt ** **on to Se1 smJc ! nversi on Methods ** **Brian ** **Russell **
**•9r• ;• • •;• • • •-•• ** **9rgr•t-k'k9r9r• •-;• ;• ** **... **

**Or, the final ** **•esult- **

**Zi+[= Z **

**ß **

**l+r i **

**. **

**This is called ** **the discrete ** **recursive ** **inversion ** **formula and is the basis **

**of many current inversion techniques. The formula tells us that if we know **
**the acoustic impedance of a particular layer and the reflection coefficient at **

**the base of that layer, we may recover the acoustic impedance of the next **

**layer. ** **Of course we need an estimate of the first layer impedance to start us **

**off. ** **Assume we can estimate this value for layer one. Then **

**l+rl **

**, **

**Z2: **

**Zl i r 1 **

**Z3= **

**Z **

**2 11 **

**+ r 2 **

**-**

**r**

**and so on ... **

**To find the nth impedance from the first, ** **we simply write the formula as **

**Figure 3.1 shows the application ** **of the recursive formula to the " **

**reflection ** **coefficients ** **derived ** **in ** **section ** **2.2. ** **As expected, ** **the full **
**acoustic impedance was recovered. **

**Problems ** **encountered ** **with ** **real ** **data **

**• ** **ß ** **, ** **m ** **i ** **i ** **• ** **i ** **! ** **m **

**When the ** **recursive inversion formula is applied to real data, ** **we find **

**that two serious problems are encountered. These problems are as follows- **

**(i) Frequency Bandl imi ti ng _ ** **ß **

**Referring back to Figure 2.2 we see that the reflectivity ** **is severely **

**bandlimited ** **when it ** **is convolved with the seismic wavelet. ** **Both ** **the **

**low frequency components and the high frequency components are lost. **

**Introduction to Seismic Inversion Methods ** **Brian Russell **

**0.2 **

**0 **

**V•) 'V,• **

**•R**

**R = +0.2**

**V **

**o: 1000 **

**m **

**Where: **

**--• V,• = 1000 i-o.t **

**- 1500 m**

**-**

**•ec'.**

**(a)**

**- 0.1**

**'•0.2**

**R• **

**R= **

**{ASSUME **

**j•: l) **

**R•= **

**-0.1 **

**R =+0.2**

**R: -0.1**

**V **

**o= 1000 m **

**-'+ ¾1 **

**= 818 m **

**ii•.**

**Figure 3.2 Effect of banUlimiting on reflectivity, where (a) shows **

**single reflection coefficient, anU (b) shows bandlimited **

**refl ecti on coefficient. **

**i ** **i ** **m ** **i ** **m ** **I **

**I ** **__ ** **___ ** **i ** **_ **

**Introduction ** **to Seismic ** **Inversion ** **Methods ** **Brian ** **Russell **

**(ii) ** **Noise **

**The ** **inclusion ** **of coherent ** **or random noise into ** **the seismic ** **'trace ** **will **

**make the estimate• reflectivity ** **deviate from the true reflectivity. **

**To get a feeling for the severity ** **of the above limitations on recursire **

**inversion, ** **let ** **us first use simple models. To illustrate ** **the effect ** **of **

**bandlimiting, consider Figure 3.Z. ** **It ** **shows the inversion of a single spike **
**(Figure 3.2 ** **(a)) ** **anU the inversion of this spike convolved with a Ricker **

**wavelet (Figure 3.2 ** **(b)). ** **Even with ** **this ** **very high frequency banUwidth **

**wavelet, we have totally lost our abil.ity to recover the low frequency **

**component of the acoustic impedance. **

**In Figure 3.3 the model derived in ** **section Z.2 has been convolved with a **

**minimum-phase wavelet. ** **Notice that the inversion of the data again shows a **

**loss of the low frequency component. The loss of the low frequency component **
**is the most severe problem facing us in the inversion of seismic data, for it **

**is extremely ** **Oifficult ** **to ** **directly ** **recover ** **it. ** **At the ** **high end of the ß **

**spectrum, we may recover much of the original ** **frequency content using **

**deconvolution techniques. In part 5 we will address the problem of recovering **

**the low frequency component. **

**Next, ** **consider ** **the problem of noise. ** **This noise may be from many **
**sources, but will always tend to ** **interfere ** **with ** **our recovery of the true **

**reflectivity. **

**Figure 3.4 shows the effect of adding the full multiple **

**reflection train ** **(including transmission losses) to the model reflectivity. **

**As we can see on the diagram, the recovered acoustic impedance has the same **

**basic shape as the true acoustic impedance, but becomes increasingly incorrect **

**with depth. This problem of accumulating error is compoundeU by the amplitude **

**problemns introduced by the transmission losses. **

**Introduction to Seismic Invers,ion Methods ** **Brian Russell **
**TIME **
**Fig. **
**TIME **
**(see) **
**Fig. **
**0.? **
**RECOVERED **
**ACOUSTIC **
**IMPEDANCE **
**REFLECTION ** **SYNTHETIC **
**COEFFICIENT ** **(MWNUM-PHASE **
**SERIES ** **WAVELET) **
**pv-•, **
**INVERSION **
**OF SYNTHETIC **

**3.3. ** **The effect of bandlimiting on recurslye inversion. **

**0.7 **

**TIME **

**(re.c) **

**REFLECTION ** **RECOVERED ** **R.C.S. ** **RECOVERED **

**COEFFICIENT ** **ACOUSTIC ** **WITH ALL ** **ACOUSTIC **

**SERIES ** **IMPEDANCE ** **MULTIPLES ** **IMPEDANCE **

**3.4. ** **The effect ** **of noise on recursive inversion. **

**Introduction ** **to Seismic Inversion Methods ** **Brian Russell **

**3.3 Continuous Inversion **

**A logarithmic relationship is often used to approximate the above **
**formulas. This is derived by noting that we can write r(t) as a continuous **
**function in the following way: **

**Or **

**r(t) - Z(t+dt) **

**- Z{t) _ 1 d Z(t) **

**ß ** **- Z(t+dt) + Z(•) - •' z'(t) **

**! d In Z(t) **

**r(t) = • **

**dt **

**The inverse ** **formula ** **is thus- **

**t **

**Z(t) **

**= Z(O) **

**exp **

**2y r(t) dt. **

**0 **

**The preceding **

**approximation **

**is valid if r(t) <10.3• which is usually the **

**case. ** **A paper by Berteussen and Ursin (1983), goes into much more detail ** **on **

**the continuous versus discrete approximation. ** **Figures 3.5 and 3.6 from their **

**paper show that the accuracy of the continuous inversion algorithm is within **

**4% of the correct value between reflection ** **coefficients ** **of -0.5 and +0.3. **

**If ** **our reflection coefficients are in the order of + or - 0.1, an even **

**simpler **

**approximation **

**may **

**be made **

**by dropp'ing **

**the logarithmic **

**relationship: **

**t **

**1 d Z(t) **

**•_==• **

**Z(t) **

**--2'Z(O) **

**fr(t) dt **

**r(t) **

**--• -dr **

**VO **

**Introduction to Seismic Inversion Methods ** **Brian Russell **
**Fig. 3.5 **
**m ** **i ** **,, ** **,m ** **I ** **I ** **IIIII **

**I + gt **

**½xp **

**(26•) **

**Difference **

**-1.0**

**0.0**

**0.14**

**-0.14**

**-0.9**

**0.05**

**0. I?**

**-0.12**

**-0.8**

**0.11**

**0.20**

**-0.09**

**-0.7**

**0.18**

**0.25**

**-0.07**

**-0.6**

**0.25**

**0.30**

**-0.05**

**-0.5**

**0.33**

**0.37**

**-0.04**

**'**

**-0.4**

**0.43**

**0.45**

**--0.02**

**-0.3**

**0.•**

**0.•5**

**--0.01**

**-0.2**

**0.667**

**0.670**

**-0.003**

**-0.1**

**0.8182**

**0.8187**

**--0.0005**

**0.0**

**1.0**

**1.0**

**0.0**

**0.1**

**1.222**

**1.221**

**0.001**

**0.2**

**1.500**

**1.492**

**0.008**

**0.3**

**1.86**

**1.82**

**0.04**

**0.4**

**2.33**

**2.23**

**o.1**

**0.5**

**3.0**

**2.7**

**0.3**

**0.6**

**4.0**

**3.3**

**0.7**

**0.7**

**5.7**

**4.1**

**1.6**

**0.8**

**9.0**

**5.0**

**4.0**

**0.9**

**19.0**

**6.0**

**13.0**

**1.0**

**co**

**7.4**

**•o**

**Numerical c•pari son of discrete and continuous **
**i nversi on. **

**(Berteussen and Ursin, 1983) **

**Fig. 3.6 **

**$000 **

**O**

**} m **

**MPEDANCE **

**(O **

**I SCR. **

**) **

**r-niL **

**${300 **

**o**

**-• **

**O **

**I FFERENCE **

**SO0 ** **O I FFERENCE ( SCALED UP ) **

**T •'•E t SECONOS **

**C•pari son between **

**impedance **

**c•putatins based **

**on a **

**discrete and a continuous **

**seismic •del. **

**(Berteussen and Ursin, 1983) **

**Introduction'to ** **Seismic Inversion Methods ** **Brian Russell **

**PART 4 - SEISMIC PROCESSING CONSIDERATIONS **

**•ntroduction ** **to Seismic •nvers•on Methods ** **B.r. ian Russell **

**4.1 ** **Introduction **

**Having looked at a simple model'of the seismic trace, anu at the **

**recursire inversion alogorithm in theory, we will now **

**look at the problem of **

**processing **

**real seismic eata in order to get the best results from seismic **

**inversion. **

**We may group the key processing **

**problems into the following **

**categories:**

**( i ) Amp 1 i tu de rec o very. **

**(i i) Vertical ** **resolution improvement. **

**(i i i ) Horizontal resol uti on improvement. **

**(iv) Noise elimination. **

**Amplitude problems are a major **

**consideration **

**at the early processing **

**stages **

**and we will look at both deterministic amplitude **

**recovery **

**and surface **

**consistent ** **residual static time corrections. ** **Vertical resolution improvement **

**will involve a discussion of aleconvolution and wavelet processing techniques. **
**In our discussion of horizontal resolution we will look at the resolution **

**improvement **

**obtained in migration, using a 3-D example. Finally, we will **

**consider several approaches **

**to noise elimination, especially the elimination **

**of multi pl es. **

**Simply stateu, ** **to invert ** **our **

**one-dimensional model given in the **

**approximation of this model (that **

**band-limited reflectivity ** **function) **

**these considerations in minU. Figure 4.1 **

**be useU to do preinversion processing. **

**seismic ** **data ** **we usually ** **assume the **

**previous section. ** **And to arrive at an **

**is, ** **that ** **each trace is a vertical, **

**we must carefully process our data with **
**shows a processing flow which could **

**Introduction ** **to Seismic Inversion Methods ** **Brian Russell **

**INPUT RAW DATA **

**DETERMINISTIC **
**AMPLITUDE **
**CORRECTIONS **
**,. ** **_•m **
**mlm **
**SURFACE-CONS ISTENT **

**DECONVOLUTIO, **

**N FOLLOWED **

**BY HI GH RESOIJUTI.ON DECON**

**i**

**i**

**SURFACE-CONS I STENT**

**AMPt:ITUDE ANAL'YSIS**

**SURFACE-CONSI STENT**

**STATI CS ANAIJY SIS**

**VELOCITY ANAUYS IS **

**APPbY STATICS AND VEUOCITY **

**MULTIPLE ATTENUATION **

**STACK **

**ß ** **• **

**MI GRATI ON **

**, **

**Fig. ** **4.1. **

**Simpl **

**i fied i nversi **

**on **

**processing **

**flow. **

**ll ** **, ** **ß ** **' ** **ß ** **I ** **, ** **_ ** **i ** **11 ** **, ** **m ** **- ** **-- ** **m ** **_ • ** **• ** **,11 **

**Inl;roducl:ion 1:o SeJ smlc Invers1 on Nethods ** **BrJ an Russell **

**4.2 Am.p'l i tu. de.. P,.ecovery **

**The most dJffJcult job in the p•ocessing of any seismic line is **

**ß**

**•econst•ucting **

**the amplJtudes **

**of the selsmJc **

**t•aces as they would **

**have been **

**Jf **

**the•e **

**were no dJs[urbJng inf'luences present. We normally make the **

**simplJfication **

**that the distortion of the seJsmic **

**amplJtudes **

**may **

**be put into **

**three main categories' sphe•Jcal **

**divergence, absorptJon, and t•ansmJssion **

**loss. ** **Based on a consideration of these three factors, we may wrJte aown an **

**approximate **

**functJon for the total earth attenuation- **

**Thus, **
**data, the **

**formula. **

**At: AO* **

**( **

**b / t) * exp(-at),**

**where ** **t ** **= time, **

**A **

**t = recorded **

**amplitude, **

**A **

**0 = true ampl **

**i tude, **

**anU **

**a,b = constants. **

**if we estimate the constants in the above equation from the seismic **

**true amplitudes **

**of the data coulU be recovered by using the inverse **

**The deterministic amplitude correction and trace to trace mean **

**scaling will account **

**for the overall gross changes in amplitude. However, **

**there may still **

**be subtle (or even not-so-subtle) amplitude problems **

**associated **

**with poor surface conditions or other factors. To compensate **

**for **

**these effects, it is often advisable to compute and apply surface-consistent **

**gain corrections. This correction involves computing **

**a total gain value for **

**each trace and then decomposing this single value in the four components **

**Aij= **

**Six **

**Rj **

**x G **

**k x MkX **

**•j, **

**where A = Total amplitude factor, **

**S = Shot component, **
**R: Receiver component, **
**G = CDP component, and **

**M = Offset component, **
**X = Offset ** **distance, **

**i,j ** **= shot,receiver ** **pos., **
**k = CDP position. **

**Introduction to Seismic .Inversion Methods ** **Brian Russell **

**SURFACE **

**SUEF'A•**

**CONS **

**Ib'TEh[O{ **

**AND**

**T |tV•E :**

**,Ri L-rE R ß**

**Fig. ** **4.2. ** _{Surface and sub-surface geometry and }

**surface-consistent decomposition. (Mike Graul). **

**, ** **, **

**Introduction ** **to Seismic Inversion Methods ** **Brian Russell **

**Figure 4.g ** **(from Mike Graul's unpublished course notes) shows the **

**geometry **

**used for this analysis. Notice that the surface-consistent **

**statics **

**anti aleconvolution **

**problem **

**are similar. For the statics problem, the averaging **

**can be •1one by straight summation. For the amplitude problem we must **

**transform the above equation into additive form using the logarithm: **

**In Aij= **

**In S **

**i + In Rj **

**+ In G **

**k + lnkMijX•. **

**The problem can then be treated exactly the same way as in the statics **

**case. Figure 4.3, from Taner anti Koehler (1981), shows **

**the effect of doing **

**surface consistent amplitude and statics corrections.**

**4.3 I•mp. **

**rov. **

**ement_ **

**o.[_Ver. **

**t.i.ca.1..Resoluti **

**on **

**Deconvol ution ** **is ** **a process by which an attempt is made to ** **remove the **

**seismic wavelet from the seismic trace, leaving an estimate of reflectivity. **

**Let us first discuss the "convolution" part of "deconvolution" starting with **

**the equation for the convolutional model**

**In ** **the **

**st-- wt* r t **

**where **

**frequency domain **

**st = the sei smic **

**trace, **

**wt= the seismic **

**wavelet, **

**rt= reflection coefficient series, **

*** = convol ution operation. **

**S(f) ** **• W(f) x R(f) . **

**The ** **deconvol ution **

**procedure and consists **

**reflection ** **coefficients. **

**fol 1 owl ng equati on- **

**rt: st* o **

**process is ** **simply the reverse ** **of the convolution **

**of "removing" the wavelet shape to reveal the **

**We must design an operator to do this, as in the **

**where Or-- **

**operator **

**-- inverse **

**of w **

**t . **

**Part 4 - Seismic Processing Considerations **

**Introduction ** **to Seismic ** **Inversion ** **Methods ** **Brian ** **Russell **
**ii ** **11 **
**ß ** **1' **
**i **
**ii **
**'..,•' •, ," ** **" ** **" ** **ß d. **

**Preliminary stack bet'ore surface consistent static and ompli- **
**lude corrections. **

**ß Stock with surface consistent static and amplitude cor- **
**rections. **

**Fig. ** **4.3. ** **Stacks with and without surface-consi stent **

**corrections. (Taner anu Koehler, 1981). **

**Part 4 - Seismic Processing Considerations **

**ß ** **, **

**Introduction ** **to Seismic Inversion Methods ** **Brian Russell **

**In the frequency domain, this becomes **
**R(f) = W(f) x 1/W(f) . **

**After this extremely simple introduction, it may appear that the **

**deconvolution **

**problem should be easy to solve. This is not the case, and the **

**continuing research into the problem testifies **

**to this. **

**There are two main **

**problems. Is our convolutional **

**model **

**correct, and, if the model is correct, **

**can we derive the**

**true**

**wavelet**

**from the data?**

**The answer to the first**

**question is that the convolutional **

**model **

**appears **

**to be the best model **

**we have **

**come **

**up with so far. The main problem is in assuming **

**that the wavelet does **

**not vary with time. In our discussion we will assume that the time varying **

**problem **

**is negligible within the zone of interest. **

**The second **

**problem is much more severe, since it requires solving the **

**ambiguous **

**problem **

**of separating a wavelet and reflectivity sequence **

**when only **

**the seismic trace is known. To get around this problem, all deconvolution or **

**wavelet estimation programs **

**make certain restrictive assumptions, **

**either about **

**the wavelet or the reflectivity. ** **There are two classes of deconvolution **

**methods: those which make restrictive ** **phase assumptions and can be considered , **

**true wavelet processing techniques only when these phase assumptions are met, **

**and those which do not make restrictive ** **phase assumptions and can be **

**considered as true wavelet processing methods. In the first category are **
**(1) Spiking deconvolution, **

**(2) Predictive deconvolution, **

**(3) Zero phase deconvoluti on, and **

**(4) Surface-consi stent deconvoluti on. **

**Introduction ** **to Seismic Inversion Methods ** **Brian Russell **

**(a) **

**Fig. 4.4 ** **A comparison of non surface-consistent ** **and surface-consistent **

**decon on pre-stack data. ** **{a) Zero-phase deconvolution. **

**{b) Surface-consistent soikinB d•convolution. **

**(b), **

**Fig. 4.5 Surface-consistent decon comparison after stack. **
**(a) Zero-phase aleconvolution. (b) Surface-consistent **

**deconvol ution. **

**'--'- ** **, ** **ß , ** **,• ** **,t ** **ß ß ** **_ ** **, ** **, ** **_ _ ** **,, , ** **,_ ** **, ** **, **

**Introduction ** **to Seismic Invers. ion Methods ** **Brian Russell **

**In the second category are found **

**(1) Wavelet estimation using a well **

**(Hampson and Galbraith 1981) **

**1 og (Strat Decon). **

**(2) Maximum-1 ikel ihood aleconvolution. **
**(Chi et al, lg84) **

**Let ** **us **
**surface-consi ** **stent **
**surface-consi ** **stent **

**components. ** **We **

**di recti ** **ons- ** **common **

**illustrate ** **the ** **effectiveness ** **of ** **one ** **of. ** **the methods, **
**aleconvolution. ** **Referring to Figure 4.•, ** **notice ** **that ** **a **

**scheme involves ** **the ** **convolutional ** **proauct of four **

**must therefore ** **average over four ** **different ** **geometry **

**source, common receiver, ** **common depth point (CDP), and **

**con, non offset (COS). The averaging must be performed iteratively ** **and there **

**are several different ways to perform it. ** **The example in Figures 4.4 ana 4.5 **

**shows an actual surface-consi stent case study which was aone in the following **

**way' **

**(a) Compute the autocorrelations of each trace, **

**(b) average the autocorrelations in each geometry eirection to get four **

**average autocorrel ati OhS, **

**(c) derive and apply the minimum-phase inverse of each waveform, and **

**(•) iterate through this procedure to get an optimum result. **

**Two points to note when you are looking ** **at ** **the case study are the **

**consistent definition of the waveform **

**in the surface-consistent **

**approach an• **

**the subsequent improvement of the stratigraphic ** **interpretability ** **of the stack. **

**We can compare all of the above techniques using Table 4-1 on the next **

**page. ** **The two major facets of the techniques which will be compared are the **

**wavelet estimation procedure and the wavelet shaping procedure. **

**Introduction to Seismic Inversion Methods ** **Brian Russell **

**Table ** **4-1 ** _{Comparison of Deconvol ution MethoUs }

**m ** **m ** **ß ** **ß ** **m **
**METHOD **
**Spiking **
**Deconvol ution **
**Predi cti ve **
**Deconvol uti on **
**Zero Phase **
**Deconvol utton **
**Surface-cons. **
**Deconvolution **
**Stratigraphic **
**Deconvol ution **
**Maximum- **
**L ik el i hood **
**deconvol ution **
**WAVELET ESTIMATION **

**Min.imum phase assumption **

**Random refl ecti vi ty **

**assumptions. **

**No assumptions about **

**wavelet• **

**Zero phase assumption. **

**Random refl ectt vi ty **

**assumption. **

**Minimum or zero phase. **

**Random reflecti vi ty **

**assumption. **

**No phase assumption. **

**However, well must match **

**sei smi c. **

**No phase assumption. **

**Sparse-spike assumption. **

**WAVELET SHAPING **

**Ideally shaped **

**to spike. **

**In practice, shaped **

**to minimum **

**phase, higher frequency **

**output. **

**Does not whiten data well. **

**Removes **

**short and long period **

**multiples. ** **Does not affect **

**phase **

**of wayel **

**et for long lags. **

**..1_,**

**m**

**Phase is not altered. **

**Amplitude spectrum i$ **

**whi tened. **

**Can shape **

**to desired output. **

**Phase **

**character i s improved. **

**Ampl i rude spectrum i s **

**whitened **

**less than in single **

**trace ** **methods. **

**Phase of wavelet is zeroed. **

**Amplitude **

**spectrum **

**not **

**whi tened. **

**Phase of wavelet is zeroed• **

**Amp 1 i rude spectrum i s **

**whi tened. **