GEOPHYSICS, VOL. 64, NO. 5 (SEPTEMBER-OCTOBER 1999); P. 1512–1523, 13 FIGS.

The whiteness hypothesis: Reflectivity,inversion, chaos, and Enders

Tadeusz J. Ulrych∗

ABSTRACT

The earth’s reflectivity is one of the principal targets ofseismic exploration for oil and gas. It exhibits some fasci-nating properties. It is blue. It is fractal. In all probability,it reflects the output of a chaotic generator. I examinesome of these characteristics here. Specifically, I look atthe statistical properties of a canonical chaotic model,and I examine the effect of multiples on the statisticalproperties of seismograms generated from primary re-flectivities.

This article is a tribute to Enders Robinson. He is apioneer who has inspired many. He inspires many to thisday. I am one of them.

INTRODUCTION

A little history first. I was in South Africa in 1964, on apostdoctoral fellowship at the Bernard Price Institute of Geo-physics, University of Witwatersrand, working with one of thefinest of people, Lui Nicolaysen, on a search for the oldestleads in the world. Jack Jacobs, then the head of the Instituteof Earth Sciences in the Department of Physics at Universityof British Columbia (UBC) called me and asked whether Iwanted to come to UBC in 1965. In retrospect, this must havebeen a rhetorical question. What are my teaching responsibil-ities? Fourth-year applied geophysics, mainly. Well, that’s justgreat: Grant and West, some gravity, magnetics, seismology,∇28 = 0, and Pythagoras. I can handle that. Then I lookedat the latest copy of GEOPHYSICS, just to reassure myself thatall was in place—to help the students with the latest literature.Suddenly the shock set in. I was faced with autocorrelation (Iknew something about auto collision, but . . .), minimum phase,deconvolution.

It was time to update, and I was fortunate. Those who havebecome my scientific heroes were writing articles and books onthese subjects. Enders Robinson, Sven Treitel, Jon Claerbout—people whom I am proud to have known as colleagues and

Manuscript received by the Editor March 9, 1998; revised manuscript received November 6, 1998.∗University of British Columbia, Dept. EOS, 2219 Main Mall, Vancouver, British Columbia V6T 1Z5, Canada. E-mail: ulrych@geop.ubc.ca.c© 1999 Society of Exploration Geophysicists. All rights reserved.

friends. Others as well, also my heroes, whom I do not knowpersonally: Ron Bracewell, Gilbert Strang. I began my up-graded education then and continue it to this day. It was, inmy opinion, to a large extent the work of Enders Robinsonthat transformed applied seismology and made the study whatit is today, profound and at the edge. I will describe this workbriefly because it has made such impact and because it illus-trates so clearly the points I want to emphasize.

Before I do this, however, let me talk briefly concerning thetitle of this article. Whiteness because that is at the core, as arereflectivity and inversion. Enders because, to such a great ex-tent, he started it all for me (and for us). Why chaos? Is it herejust to add some pizzazz, a catchy word that attracts? Not atall. I believe very strongly that chaos is a paradigm for a philo-sophical (and mathematical) understanding of our earth (andmusic and architecture). It gives one a manner to rationalizedeterminism with unpredictability. For those who believe in anAlmighty, to put into place the deterministic hand of God withthe stochastic outcome of free will. The philosophical aspectis very important to me. As time passes, one ages. With age,one becomes more consciously philosophical. I believe thatone does so simply because one can no longer cope with thedetail of it all (nor actually desire to). We leave the detail tothose who can cope with it—the young. With the passing oftime, we begin to assimilate and synthesize, to philosophize. Itis not without reason that most of the philosophers whom weadmire and respect are dead.

THE IDEAS OF ENDERS ROBINSON

The model for the seismic trace is ultimately simple (a prop-erty that is encouraging to some and discouraging to others).It is linear and, as such, embodies the principle of superposi-tion. We envisage a pulse of energy that we call the seismicwavelet and designate by w(t), injected in some manner intothe earth. This pulse encounters impedance contrasts on itsjourney through the subsurface and reflects energy back to thesurface, energy that is recorded as x(t), the seismogram. Let usdenote the series that represents all the reflections that wouldarise, primaries as well as all possible multiples, as a result of

1512

The Whiteness Hypothesis 1513

a delta-like pulse traveling through the earth (without atten-uation and dispersion) by r (t). Because of linearity and timeinvariance, the resulting seismic trace is the superposition ofw(t) on every sample of r (t) and is written as

x(t) = w(t)⊗ r (t), (1)

where ⊗ indicates convolution. We ignore noise to make thediscussion even simpler.

It is r (t) (demultipled, of course) that is the goal of the inter-preter. To achieve this goal, w(t) must be removed. And hereis the impasse. We do not, in general, know w(t), and equa-tion (1) is an equation with two unknowns. Enders Robinson(1967) was, I believe, the first person to seriously tackle this ill-posed inverse problem in seismology. He approached it in theonly way that makes sense. By imposing a priori information—information based on physical and geological insight. I stressthis point here. Enders approached the task of finding a plausi-ble solution from the infinity of possible solutions—not by anad hoc regularization such as ridge regression (i.e., a little bitadded to the diagonal of the regressor matrix), but by carefullyreasoned, physically motivated constraints. He required andproposed two solutions.

1) The primary reflectivity of the earth is white.—Whitemeans that all frequencies exist and are represented withequal power. Consequently, the autocorrelation is a deltafunction at zero lag and the process is completely uncorre-lated. This is an absolutely reasonable hypothesis. Surelythe sedimentary generator is not aware on Monday whatlayer it will deposit on Tuesday. The reflectivity seriesmust exhibit no memory and certainly has no predictablefeatures.

2) The seismic wavelet w(t) is minimum phase.—(Or mini-mum delay; it’s the same thing.) What does this actuallymean? I believe, the physical meaning of this constraintis an expression of Fermat’s principle. The elastic energythat travels through the earth as a result of an impulseon the surface arrives at the receiver in the minimumtime possible. The energy of the arriving pulse thereforetends to be front loaded. Robinson expresses this phe-nomenon in his minimum delay theorem (Robinson andTreitel, 1980). This hypothesis feels correct. Given a deltafunction at the source, w(t) would indeed be minimumphase. The problem is that the initial energy input is not animpulse but, in all likelihood, some nonminimum phaseseries of impulses that represents the oscillation of thecavity following the detonation of the physical charge.For this reason, marine sources are much more likely togive rise to minimum-phase wavelets.

I will not discuss assumption two here. Various people havewritten about the minimum-phase nature of the attenuation-dispersion relationship that applies to the earth.

The first assumption is the subject of this article. In fact,this perfectly reasonable assumption has been shown in re-cent times to be incorrect (Hosken, 1980; Walden and Hosken,1985). The reflectivity series appears not to be white but, rather,blue. Blueness implies that the power spectrum of the reflec-tivity is attenuated at low frequencies. Its autocorrelation isnot only a spike at zero lag. Lag 1 is also significant and, in

fact, negative in polarity. The remarkable property of this neg-ative value at lag 1 is that it is scale independent: apparentlyit does not matter whether the sampling rate is 4 or 0.4 ms.For most reflection coefficient series, lag 1 is around−0.4 for anormalized autocorrelation.

What does this mean? Does the generator in fact have amemory? I do not think so. In my opinion, the autocorrelationreflects what is commonly termed the flicker noise characterof the reflectivity, which in turn must be related to the chaoticnature of the generator. The scale independence is the fractalnature of chaotic outputs.

A reminder of how the dual constraints allow the reflectiv-ity and the wavelet to decouple: Taking z transforms in equa-tion (1), we obtain

X(z) = W(z)R(z). (2)

The autocorrelation is now computed as

X(z)X(z−1) = W(z)W(z−1)R(z)R(z−1)

which, because of the whiteness constraint, leads to

X(z)X(z−1) = W(z)W(z−1)σ 2r , (3)

where σ 2r is the variance of the reflectivity. It is of interest to

examine the effect of the blueness of r (t) in a little more de-tail. Let the value of the lag 1 coefficient of the normalizedautocorrelation be r1. Equation (3) may now be written as

X(z)X(z−1) = W(z)W(z−1)σ 2r

(1+ r1z+ r1z−1

)which, designating Pxx(ω) and Pww(ω) to be the power spectraof the trace and wavelet, respectively, we obtain

Pxx(ω) = Pww(ω)σ 2r

(1+ r 2

1 + 2r1 cosω). (4)

For a typical value of r1 = −0.4, we obtain a weighting ofthe desired amplitude spectrum,

√[Pxx(ω)], that varies from

0.6 at ω = 0 to 1.4 at the Nyquist frequency. This weighting isof second-order importance in wavelet deconvolution. Variousmethods, however, have been developed to whiten the reflec-tivity so that equation (3) still applies to the decomposition ofx(t) (Walden and Nunn, 1985; Jensen et al., 1988; Rosa andUlrych, 1991).

Equation (3) shows that now, particularly after applying therequired color correction, we have the autocorrelation of thewavelet in terms of the scaled autocorrelation of the knowntrace. Assumption two now comes into play. Since the waveletis minimum phase, the Kolmogorov spectral factorization(Claerbout, 1976) lets us compute the phase of the waveletfrom its amplitude spectrum:

8(ω)Kolmogorov7→ |W(ω)|,

which lets us compute W(ω) as

W(ω) = |W(ω)|e−i8(ω). (5)

Finally, we can determine the required reflectivity from

R(z) = X(z)W(z)

.

The decomposition is complete. The inversion has been reg-ularized by means of a priori information, physically and geo-logically motivated.

1514 Ulrych

WHITENESS AND CHAOS

Let us now look in more detail at the relationship betweenwhiteness and chaos. I, and many others, have long been fasci-nated by the simplest of chaotic equations, the canonical logis-tic equation. In nondimensional form, it is

xn+1 = λxn(1− xn), (6)

where λ is a constant, now called the Feigenbaum parameter,and the limits relate to the recursion of xn (discussed below).

I first encountered this type of equation in an article byR. M. May (1976). Chaos was not mentioned specifically.Rather, the strange behavior of this equation and possible bio-logical significance were the issues discussed. In particular, themain focus was the modeling of the pseudoperiodic nature ofthe cicada life cycle. The logistic equation models (in a verysimplistic fashion) the birth and death cycle that we all are in-volved in. Perhaps this is the reason for its fascination. Theseductive qualities of the logistic equation are exemplified bythe story concerning Feigenbaum (Gleick, 1987) who studiedthe influence and properties of equation (3) in relationship toλ. Apparently, following months of obsessive lambda isolationin his apartment, his friends had to literally drag him outsideto a little respite. He had not eaten for days and would prob-ably have otherwise expired—productively, happily, but in anuntimely manner.

It is of interest to briefly look at some mathematical high-lights of equation (6) (see Moon, 1987, for a more comprehen-sive treatment).

A brief mathematical look at the logistic equation

Equation (3) is a particular case of the general mappingxn+1 = f (xn). Steady motion, called period 1 motion, has twofixed, or equilibrium, points, xe, which are given by the solu-

a)

b)

FIG. 1. (a) Logistic equation chaotic process, λ = 3.65. (b) Return map for logistic equation, λ = 3.65.

tion of xe = λxe(1 − xe), namely, xe = 0 and xe = (λ − 1)/λ.The stability of the map, i.e., whether the solution moves to-ward or away form xe, depends on f ′(x), the slope of f (x), atxe. The value | f ′(xe)|< 1 implies stability; | f ′(xe)|> 1 impliesinstability. Thus, for 1 < λ < 3, the solution at the origin is un-stable, whereas the solution at (λ− 1)/λ is stable. At λ = 3 thesolution at the second fixed point becomes unstable and bifur-cates at λ = λ1. The motion, called period 2 motion, becomesharmonic.

For 3 < λ < 4, equation (6) exhibits many different mul-tiple periods and chaotic motions where the chaotic behavioris characterized by amplitudes that can adopt any value be-tween±1. In particular, period doubling occurs until the value

FIG. 2. Bifurcation diagram.

The Whiteness Hypothesis 1515

FIG. 3. Return map for lagged logistic equation, λ = 1.98.

a) b)

c) d)

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SPECTRAL PROPERTIES OF THE LOGISTIC EQUATION

But what does all this have to do with whiteness? As I havementioned above and as shown in Figure 2, chaotic regionsexist for certain λ ranges. Let us look at the spectral proper-ties of xt of one of these regions. I chose a value of λ = 3.9, arepresentative value for a chaotic region. A realization of re-sulting process appears in Figure 4a; Figure 4b shows the nor-malized raw periodogram estimate (essentially the square ofthe modulus of the Fourier transform) of the power spectrum.The corresponding autocorrelation is illustrated in Figure 4c.

As is well known, the raw periodogram is a poor spectralestimator. Consequently, I have used an information criterionto estimate the order of the autoregressive representation ofxt that allows not only an optimally smooth spectral estimateto be computed (shown by the thick line in Figure 4b) but

a) b)

c) d)

FIG. 5. (a) Logistic equation chaotic process, λ = 3.9999. (b) Spectra: full line=AR; thin line= FT. (c) Normalized autocorrelation.(d) Scaled AIC.

also characterizes the whiteness of the process. A few remarksconcerning the information criterion are in order.

Akaike information criterion (AIC)

Determining the whiteness of a process is often ofimportance. One can, for example, use the Smirnov—Kolmogorov test (Jenkins and Watts (1968), but this test tellsus whether the process is white—it does not tell us the order ofcorrelation the process exhibits. The Akaike information cri-terion (AIC) does. The literature on the subject is vast. I referthe interested reader to some original Akaike papers (Akaike,1973, 1974), the excellent book on the subject by Sakamotoet al. (1986), and the article by Matsuoka and Ulrych (1986).

The AIC is defined in terms of the logarithm of the likelihoodfunction in the following form:

The Whiteness Hypothesis 1517

AIC(k) = −maxk

2`(kθ̂)+ 2k, (7)

where `(kθ̂) is the log likelihood for the vector kθ̂, which isthe estimate of kθ, the k-component vector of parameters thatdefines the process. The likelihood function is the probabilitythat we would get the particular sample that we got if we knewthat the vector of parameters took on some particular values.

We seek the minimum AIC since AIC(k)|min is the optimalcompromise between errors in parameter estimation and er-rors in fitting the model. For normally distributed errors,

AIC(k) = N log s2k + 2K ,

where s2k is the residual sum of squares given by

s2k =

1N

N∑i=1

q2i

a) b)

c) d)

FIG. 6. (a) Logistic equation chaotic process, λ = 4.0. (b) Spectra: full line = AR; thin line = FT. (c) Normalized autocorrelation.(d) Scaled AIC.

and qi is computed from the model xt = a1 + xt−1 + a2xt−2 +· · · + akxt−k + qk.

We now return to the discussion of the spectral propertiesof the chaotic process. Next to the autocorrelation of xt is plot-ted the variation of the AIC versus the order k. The minimumis obtained at k = 3. In this particular case, the nonwhite-ness of xt is obvious both in the spectrum and the autocor-relation. As we will see later, this is not always the case, andthe usefulness of the AIC will become that much more appa-rent.

The remarkable thing is that a common thread links all es-timates of the AIC order of xt in the chaotic regions. All esti-mates of the orders are greater than zero. In other words, allthe processes are nonwhite; they are all autocorrelated to someextent—all, that is, except the process generated with λ = 4.This value is very special indeed: for values greater than 4, xt inequation (6) diverges. But the value is also special because for

1518 Ulrych

a)b)

FIG. 7. (a) White input process. (b) Return map for a Gaussian, white reflectivity.

a) b)

c) d)

FIG. 8. (a) White reflectivity. (b) Spectra: full line =AR; thin line = FT. (c) Normalized autocorrelation. (d) Scaled AIC.

The Whiteness Hypothesis 1519

this value of λ—and apparently this value only—xt is white. Ihave no formal proof of this supposition. However, some verylong realizations generated with λ = 3.9999, for example, showAIC variations as illustrated in Figure 5d. The minimum alwaysoccurs for orders greater than zero for λ 6= 0. I have never gen-erated a long (5000 points) realization of xt with λ = 4 thathas shown other than an AIC order of zero. A typical panelof spectral properties for this limiting λ value is illustrated inFigure 6.

What does this mean? I do not know, but I think there un-doubtedly is much to learn from this observation.

WHITENESS AND THE REFLECTIVITY

Finally, we consider some aspects of whiteness and the re-flectivity. To do this, I consider the simple yet so instructive

a) b)

c) d)

FIG. 9. (a) Reflection seismogram. (b) Spectra: full line =AR; thin line = FT. (c) Normalized autocorrelation. (d) Scaled AIC.

1-D, zero-offset model of the layered earth—the so-calledGoupillaud earth. Sven Treitel once told me that this modelnever ceases to amaze him. Every time he studies it, he learnssomething new. Basically, given a series of reflection coeffi-cients associated with the layered earth, one generates a pre-diction error operator. (For details, refer to Claerbout, 1976,and Robinson and Treitel, 1978.) The Levinson recursion isnow used in conjunction with this prediction error operator togenerate the zero-offset reflection seismogram that faithfullyreproduces all primary and multiple energy. The first arrivalis the unit pulse at time zero. The important point I want toillustrate concerns the centrality of the whiteness assumptionin the problem of wavelet estimation and deconvolution. To dothis, I generate a reflection seismogram from a white reflectioncoefficient series for a full-band, impulsive input and examinethe statistics of the resulting seismic trace.

1520 Ulrych

a)b)

c) d)

FIG. 10. (a) Chaotic reflection seismogram. (b) Spectra: full line = AR; thin line = FT. (c) Normalized autocorrelation. (d) ScaledAIC.

a)b)

FIG. 11. (a) Return map for a chaotic reflectivity, λ = 4.0. (b) Return map for a chaotic seismogram, λ = 4.0.

The Whiteness Hypothesis 1521

a)b)

c) d)

e)

FIG.12. (a) Real reflectivity. (b) Return map for a real reflectiv-ity. (c) Spectra: full line =AR; thin line = FT. (d) Normalizedautocorrelation. (e) Scaled AIC.

To begin with, for comparison with the chaotic process thatwe have examined, let us look at the return diagram for a whitereflectivity series generated from a normal probability distribu-tion. Figure 7a shows the input series, while Figure 7b illustratesthe return path. This figure demonstrates what I call the scat-tergun effect, an effect that one would predict for a completelyunstructured time series. The chaotic series is a useful inputto the seismic inverse problem for structures such as the oneFigure 11a exhibits. We now examine the effect of the forward-layered earth model that transforms the input reflectivity intothe recorded seismogram. To clarify the effect of the layersalone, the input wavelet is just the full-band impulse at the sur-face. Figure 8 demonstrates the spectral properties of the inputreflectivity series. Specifically, I have taken a 100-point whitereflectivity as input into the layered earth. Figure 9 shows theeffects of this transformation where I have generated a seismo-gram of 500 points. We note, in particular, the AIC in Figure 9dthat clearly demonstrates the nonwhiteness of the ensuing seis-mogram reflected also in the spectra shown in Figure 9b. It isindeed intriguing to observe that the white layered earth de-stroys whiteness in the process of generating the reflection re-sponse. That is because of multiple reflections, the bane of theexploration geophysicist.

1522 Ulrych

What about a chaotic reflection seismogram—one generatedusing a chaotic reflectivity (Figure 5a) as input? The answer isillustrated in Figure 10. The spectrum of the resulting seismictrace is also certainly shaped by the resulting multiple reflec-tions in the modelled chaotic, white earth. The interesting ques-tion here is, does the chaotic signature of the input, so clearlyvisible in the return diagram, manifest itself in any manner inthe chaotic seismogram? The answer is no and is graphicallydepicted in Figure 11. Figure 11a, the return path diagram forthe chaotic reflectivity, is mapped into Figure 11b in the journeythrough the layered earth. All signs of the input chaos are gone.

Finally we look at the behavior of a real reflectivity. This isa well log from the Campos basin in Brazil, one of many an-alyzed by Andre Rosa (Rosa and Ulrych, 1991). Figure 12ashows the reflectivity itself and Figure 12b its return path. Wesee again the scattergun pattern, descriptive of unstructuredrandomness. The particular pattern reflects the underlying dis-tribution of the reflection coefficients. Figures 12c–e illustratethe statistical properties. (I have, for simplicity, considered thewhole 1000-point log as a stationary process, which it clearly isnot.) As expected, the reflectivity is blue.

DISCUSSION

The whiteness assumption is a physically and geologicallyreasonable constraint that is required in the stochastic inver-sion of the linear, nonunique, convolutional model that de-scribes the recorded seismogram. In fact, the primary reflectiv-ity is not white but blue. It also appears to be fractal, as shownby the autocorrelation that is invariant with sampling inter-val. This characteristic probably reflects the chaotic propertiesof the generating mechanism (as far as we can measure it).The full-band seismogram that describes the simple 1-D, zero-offset, layered earth, which results from a purely white primaryreflectivity, not surprisingly is not remotely white and clearlydemonstrates the effect of multiple reflections. The spectralcharacteristic of the seismogram stresses the importance of re-

a) b)

FIG. 13. (a) Pdf(x): full = Gaussian; thin = real. (b) Pdf(x): full = chaos.

moving multiple reflections before inversion using the white-ness constraint.

Real reflectivity series do not demonstrate any obvious struc-ture in the return diagram. This is certainly not surprising. Afterall, velocity and density and the relationship between them thatwe use to define the reflection coefficient are only attributes ofthe real reflectivity. Such factors as porosity, for example, areobviously important, as is the physical law that binds the con-tributing factors together. Another factor concerns the returndiagram itself. What are the forms of such diagrams for chaoticprocesses that may, realistically, generate the sedimentary se-quence? Clearly, logistic chaos is not the description of realreflectivity generators. Apart from being intuitively obvious,this is also quite clear from the probability density functions,which describe the two processes. Figure 13 shows a Gaussian,a real, and a chaoic (λ = 4 in the canonical equation) probabil-ity density function. The reflectivity probability density func-tion exhibits a Laplacian or Lorentzian form that resemblesthe ubiquitous Gaussian shape. The chaotic probability den-sity function on the other hand, at least for the logistic equa-tion, resembles the position-probability curve for the harmonicoscillator.

Many avenues remain to be explored. One concerns all theissues involved in the identification and attenuation of surfaceand internal multiples. Another is the physical and mathemat-ical description of the realistic chaotic signature of the reflec-tivity generator. The latter may prove to be too difficult, tooimpractical; perhaps quantum theory and human flight also ex-hibited such features. One thing is clear: All advances in ourdiscipline are based on the deformation of present boundaries,and this only results through basic, adventurous, and thought-provoking research.

ACKNOWLEDGMENT

This article is meant as a tribute to Enders Robinson. Hisfootprints are discernible on most of the scientific beaches that

The Whiteness Hypothesis 1523

I have visited. He has been and remains my teacher. He hasand continues to inspire my work. Thank you, Enders.

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