Characteristics of Reflectivity Strength on a Thin Bed

From the point of view of resolution, an existing thin layer may not be detected by seismic wavelets. By numerical experiment and using the reflectivity strength, it is illustrated that the existence of a thin layer with a thickness of less than one-eighth of the dominant '\Tavelength of the propa­ gating seismic wavelet can be detected. An observed seismic wavelet consists of sub surf ace reflectivity, i.e. the composite wavelets are a function of the separations of individual reflectivity alone. For a thin layer, the shape of a composite seismic wavelet is a func­ tion of layer thickness. Using plane wave theory and assuming no energy is dispersed, the authors calculate a synthetic seismogram for a geologically pinchout model based on an input Ricker wavelet. The calculated compos­ ite wavelets are then cross-correlated with the derivative of the input wave­ let. From Widess's (1973) studies, the resolvable ability of a seismic wave­ let is clearly defined and understood by the correlation. To examine the effects of the thickness of a thin layer on reflectivity strength, the Hilbert transform is then used to transfer synthetic wavelets. By destructive inter­ ference, reflectivity strength shows a minimum when the layer thickness is less than one-eighth of the dominant wavelength of the wavelet. The mini­ mum no longer occurs as the layer thickness exceeds the above criterion. This phenomenon of reflectivity strength on the layer thickness of a ''real'' thin layer can be considered as an indication of its existence.


INTRODUCTION
Seismic signals recorded at the surface carry sub surf ace geological information as the wa,1e propagates. With the continuity/discontinuity of the observed signals in a reflection seismogram, the features of this subsurface structure are reconstructed and interpreted. Con seque. n tly� on the base of the attributes of the recorded wavelet, the potential of hydrocarbon resources can be evaluated.
From the perspective of sedimentary and tectonic processes, most geological structures ' TAO, \;'ol. 7, N(J. 3, Septe1n. lJer 1996 have a smaller vertica] dimension than a horizontal one. Seismically, for a geological reser \1oir the ratio of' the \1ertical dimension to the horizontal plays a ve1-y impc)rtant role in reflec tion exploration. As this dimensional ratio of a geological event decreases, the dit' ficulty of identif' ying the e\'ent increase. s. The structure of an "unobvious'' dimensional ratio is treated as a thin layer in i·etlection seismology. The existence of� thin layers themselves exhibits a special meaning to an e . xplora tion ge. ophysicist. In the histc)r)' of petroleum exploration, structures of an ''ob\1ious" dimen sional ratio (e.g. anticlines, faults, ()r domes etc.) have been widely explored and becc)me depleted and exhausted. Therefore, locating and detecting the existence. of hydrocarbon struc tures of an ''unob,1ious"ddimensiona ratio (e.g. pinchout, on-Jap, off-lap, lenticular sand etc.) are gaining mo1·e and more importance. However, to resolve and detect a thin la)1er are not only difficult but also challenging.
To date, most of the research done on a thin layer concentrated on analyzing how an observed wa\1elet is distorted by the ef' f' ects of' the boundaries of a thin layer. This has been studied in de. tail with the conclusion that the limitation of the vertically separated reflectivity to be resolved is dominated by the wa\1elength of the propagating wav·elet by which the com posite wavelet is observed. Widess ( 1973) pointed out that the power of reS()lv'ability ot· a thin ]ayer is frequency depe. ndent, i.e. w·avelength dependent. B)' convoluting a ze. ro-phase wa,1e let 'h1ith t\�to spikes of equal amplitude but opposite polarity, Widess observed that as the separations between spikes decrease to one-eighth of the dominant wa\relength of the propa gating wa\1elet, a "stable" composite \\i'avelet occurs. This stable composite wav'elet has its semblance similar tc) the derivative of the vvavelet CC. )nvoluted. The separation of one-eighth of a wav·elength is thus defined b)' Widess as a criterion by· which the. adjace. nt interfaces, top and bottom boundaries, of' a thin layer could be resolved. He also concluded the magnitude of a composite w·avelet is approximately proportional to the thickness of a thin layer. IV1eanwhile, constructing different combinations of top and bottom reflectivity· of� a thin la)'er, Iv1e. issner and Meixner ( 1973) investigated the shape of the wavelets \Vhich have been distorted and derived the simila1� results.
Thus, the shape of a composite. wavelet has been thoroughly analyzed and has provided information 1� or ide11tit' ying the retlections of the upper and the lo�'er bc)undaries of thin layers. Studying the shape. ot' a composite wavelet dc)es address lots C)f remarkable contributions in thin layer resolution. Instead of dealing with resolv·ing reflections from thin layers, the present authors concentrate on detecting the existence of the la ye. rs. The re. sponse of reflectivity strength and instantaneous amplitude obtained b)' transf' orming a seismic trace using Hilberts tech nique on a composite 'A'avelet are studied.

NUlVIERICAL MODEL
To generate a synthetic se.ismogram of a geological pinchout, a zero-phase Ricker ( 1940) \\iavelet . is calculated to convoluted with spikes of' eqttal amplitude and opposite polarity. The mathematical expression of the wavelet is \\t·here A i is the peak amplitude, and \J1\,1 is the peak frequency of the amplitude spectr· um ot' the \\i'avelet. The calculated zero-phase Ricke. r wavelet and its deri\1ative are shown in Figure  1. The l/J\1 of the Ricker \vavelet is 50 Hz. The peak amplitude, Ai, is arbitrarily set at 100. Assuming no absorption and no transmission loss, composite wa\1elets, with a zero sepa ration bet\\'een spikes to one \\i'avelength (period), are calculated. The mathematical form of the composite wa\1elet, with amplitude A and a separation increment of /l /16 (or T/ 16 ), can be (2) A synthetic seismogram is calculated at a 1-ms sampling interval and is displayed in Figure 2. There are eighteen S)'nthetic traces in Figure 2. Trace 1 is the ·composite wavelet of zero separation (i.e. diffraction). The trace on the left hand side of Trace 1 is its derivative form ( Figure I for comparison). The separation between spikes in Trace 3 is 2 A /16; 3 A. /16; for Trace 4; and 4 A /16 for Trace 5. The coherence (similarity) of the derivative. trace and Trace 3, 4, or 5 are 'risible. The symbol, A, stands for the dominant wavelength of a propa gating seismic \\1ave of the analyzed wa\le]et. Figure 3 shO\\lS the cross-correlation of the derivati\re wavelet and the other composite wa,1elets (Traces 1-17) in the synthetic se. ismogram (Figure 2). According to constructive interference, it is expected that the maximum coefficient of the correlation should occur at the separation of A 11. 6 (Trace 5) between spikes; however, it. seems this is not the case in the present computation. On the contrary, instead of Trace 5, a maximum appears at Trace 4.

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This means the separation between spikes is 3 .A /16. However, this may be easily explained. The expected result would occur only \vhen two . sinusoidal wavelets of equal amplitude and opposite polarity interfere, but here the source '"'avelet by which the seismogram is derived and processed is a zero-pha.f)e R i cker \ti'avelet. From Figure 3, it can also be seen that the normalized magnitude of the cross-correlation is about 0.8 at a separation of' A 18 (Trace 3). When the results are compared to similar research done by Ricker ( 1953 ), Widess ( 1980) and Kall w·eit ( 1982), a coincidence is found. The reflections of a composite wa\le}et are resolvable only it .. their separation exceeds one e : ighth of the wavelength of the analyzed wav·elet. Ho,:v·ever, due to inherent complexities, if the wavelet is deformed, this criterion might change: in other �1ords� the mini1num resolvable separation might increase . .

REFLECTl\l'ITY STRENGTH ANALYSIS
To discuss and v·erify the criterion ot' the resolvable limitations is not the purpc)se of' this study. The objectives here are to i n\1estigate the et' t' ects of layer thickness on reflectivity strength and to detect the existence ot· a thin laye. r. Among the attributes of' seismic data that are considered for stratigraphic interpretation, amplitude is the. most frequently adopted. Bright spot has been considered an indication of hydrocarbon resources; an1plitude versus ot' t"' set (A VO) has been \videly anal)1zed to study the variation of' subsurt·ace lithology. To access 1nore int' ormation from the ampl.itude of a seismic �1avelet, a seis1nic trace is Hilbert tr an sf ormed and the relationship between reflectivity strength and thickness of· a thin layer is investigated.  (2)). At zero spike separa tion the correlation coefficient is zero. The maximum appears at the separation of 3\ A I 16 (Trace 4) for an input Ricker wavelet.

Theory Review
The Hilbert transform of a function f(t) is defined as: by Brace\vell (1986). Mathematically, Eq. (3) can be considered as a convolution. The equiva lent expression of Eq. (3) in convoluted form is The application of two Hilbert transf onns in succession reverses the phases of all compo nents; If the Hilbert transform pair, H[f(t)} andf(t), is itself Hilbert transformed, the resulting pair becomes -f (t) and H[f(t)]. This polarity reversal is simply a result of n/2 phase advances. Hence, a seismic trace f(t) can be defined as the real part of an analytic trace F(t), and thef*(t) is the quadrature series.
A(t) is amplitude spectrum. (I.
F ,11 2 is the Nyquist t' requency of f(t). Once the data is transformed, some instantaneous seismic attributes can be readily obtained. The instantaneous attributes associated w·ith the transf' ormed wavelet include: instantaneous amplitude, instantane(lUS phase� and instantaneous t' requency. \\tTith f(t)and f * (t) calculated, the reflectivity strength of the correspondent instanta neous amplitude, A(t), and the instantaneous phase, 8(t) are directly obtaine. d by performing the t' ollo"\IV·ing manipulations: and i nstanta11eous phase 8(t) == t a n -1 ima g( H a (t )) = t a n _1 .f "' (t ) . 1� eal( Ha(t ) ) f ( t ) Among all of the attributes derived, instantaneous amplitude me. asures the reflectivity strength , which is proportional to the square root of the total energy of the wavelet at an instant time. lnstantane. ous phase measures the c. ontinuity of the e\1ents on a seismic section. Instantaneous frequenC)' is computed from the temporal rate of change of instantaneous phase (Taner, et al., 1979).
Studying the instantaneous attributes obtained by Hilbert transformation he. lps in inter preting seismic data from different points of vie\\l perspective. To obtain more detailed infor mation about how the reflectivity st1�ength ot· composite wa\'elets vary with layer thickness, the increme. nt of laye1· thickness f<lr the successive composite is found and adjusted to A 132. The c. urve of the variation ot· 1·eflectivity strength with layer thickness is shov\ln in Figure 4.
In Figure 4, it should be noted that the magnitude of reflectivity strength decreases right after ze. ro-separati()n and jumps to a 1ninimum. The minimum sho\�/s up at the layer thickness of 0.0313 /l (A /32), where the maximum destructive interference occurs. It can also be seen that as the separation of the laye14 boundaries increases and exceeds 0. J 25 .A (.A /8)'; the effects of destructive interference may no longer be. that ob\1ious. The response of reflectivity strength of composite seismic wavelet which is exhibited on the thickness of' a thin layer is now clearl)' de1nonstrated and understood. The occurrence of the minimum in reflectivity st1·ength for a thin layer thickness less than its resolvable thickness (criterion) can be considered an indication for its existence . . A thin layer, formed by the in.trusion or sedimentary process, commonly exhibits opposite reflectivity. If the separation of the layer boundaries is large enough, say exceeding one eighth of the dominant \\,. avelength of the propagating wavelet, and if it can be resolvable, the volume of the layer can be estimated by analyzing the configuration of composite wavelets. Neverthele. ss, a thin layer of unresolvable thickness will very possible be ignored and become in\'isible on the seismic section.
In this research, a synthetic seismogram is calculated using a Ricker \VTavelet as a source wavelet to convolute with a geological pinc. hout model of equal reflectivity but opposite po larities. Based on the physical properties of wavelet interference, a composite wavelet is studied both to understand the resolvability of the wavelet for an existing thin layer and to in\1estigate the reliability of the reflectivity strength of the wavelet in detecting the existence of a layer.
To see the resolution of a composite wavelet, which is formed by the interfere. nee from reflections from the top and bottom of a thin layer, a Ricker wavelet is generated and differen tiated. For interfered ref1ections of equal amplitude and opposite polarity, the shape of the composite wave. let conv·erges into the derivative shape of an input wavelet at the layer thick ness of one-eighth of the dominant wavelength of the propagating wavelets (Widess, 1973 ) . The similarity between the derivative of input (source) wavelet and the composite wavelet provides a crite. rion for the resolvability of the reflections. Thus, to resolve a composite wavelet in accordance with ·the notion of Widess idea, the correlation te. chnique works rather successful 1 y.

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TA 0. V(J/. 7, ;vcJ. 3, Septemhe1--J 996 Ho\\'ever, to detect the existence of� a "real" thin layer with a layer thickness of' less than A /8, the co1,. relation technique can be no longer be applied. On the other hand, reflectivity strength analysis \\'()fks more efficient])'. For a "real'' thin layer, reflections of opposite polarization occur and mingle clt its boL1ndaries. These results show the existence of· the "real'' thin lay·er can be re\lealed by· 1,. etlectivity strength analy·sis. In short., the thinner the layer is, the more ex.agge1·ated is the sensiti\1ity. Hence, the sensiti\1it)' of reflectivity strength responds t< . ) layer thickness ot· tl1e la)1er can be adopted with cont' idence to locate a ''real'' thin layer.