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Internal multiple attenuation using inverse scattering: Results from prestack 1 & 2D acoustic andelastic syntheticsR. T. Coates*, Schlumberger Cambridge Research, A. B. Weglein, Arco Exploration and Production Technology

S u m m a r y

The attenuation of internal multiples in a multidimensionalearth is an important and longstanding problem in explo-ration seismics. In this paper we report the results of ap-plying an attenuation algorithm based on the inverse scat-tering series to synthetic prestack data sets generated in onand two dimensional earth models. The attenuation algo-rithm requires no information about the subsurface structureor the velocity field. However, detailed information aboutthe source wavelet is a prerequisite. An attractive feature of:the attenuation algorithm is the preservation of the amplitude(and phase) of primary events in the data; thus allowing forsubsequent AVO and other true amplitude processing.

Introduction

Seismic processing typically assumes that reflection dataconsists of primaries, i.e., that a single upward reflection hasoccured between source and receiver. Signals which do notconform to this model are usually regarded as noise to beattenuated. Multiples have two or more upward, and oneor more downward reflectons between source and receiver,figure 1, and thus are regarded as noise in seismic data. Mul-tiples may be divided into two groups: surface multipleswhere one or more of the downward reflections occur at thefree surface, and internal multiples where all downward re-flections occur below the free surface. Here we concentratesolely on internal multiples, assuming that all free surfacemultiples have already been removed from the data.

Multiple attenuation is a classic and only partially solvedproblem in exploration seismics. Existing attenuation meth-ods generally make assumptions about the earth, e.g. thatit is flat layered with a white reflection series, or about thecharacter of the primary and multiple signals, e.g. that theyhave significantly different moveouts. In many cases theseassumptions are violated and the effectiveness of the attenu-ation or the preservation of the primary signal are degraded.

In this paper we consider an attenuation method for internalmultiples based on inverse scattering theory. The derivationdoes not assume that the earth is 1D, indeed the theory re-quires no information about the subsurface structure or ve-locity field. It works by predicting and subtracting internalmultiples directly from the data. However, the method doesrequire an accurate knowledge of the source wavelet.

T h e o r y

The internal multiple attenuation algorithm tested here ispresented in detail in Weglein and Araujo (1994), Araujo(1994) and Weglein et al. (1996); here we provide only thebriefest summary. In the forward problem the scattered field,

Figure 1: A schematic illustration of primaries (solid)and multiples (dash).

is given by the Lippmann-Schwinger equation, viz

(1)

where is the Greens function in a homogeneous refer-ence medium, G is the Greens function in the actual mediumand the perturbation V is the difference between the waveoperators in the actual and homogeneous medium. Thisequation may be expanded in powers of the perturbation,

(2)

Similarly, if we define the data, D, as the scattered fieldrecorded at the surface we can write the perturbation, V as aseries in the data, i.e. we write V as

(3)

where Vn is the portion of the perturbation that is orderin the data. Substituting (3) in (2) and equating orders of thedata we obtain

(4) (5)

(6)

Data, D, is input and the model perturbation, V, is output.One of the tasks of inversion is the elimination of multiples.Since is linear in D, and the latter consists of primariesand multiples, the multiple removal must be carried out by

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the higher terms. In fact, the first contribution to the multipleremoval series, fur multiples of a given order, is determinedby the number of changes in direction of vertical propaga-tion of the multiples, e.g. first-order multiples have threereversals of vertical propagation direction and the first con-tribution comes from part of the third term in equation (6).The multiple attenuation series we consider consists of theleading order attenuation term for each order of multiple.

For first-order internal multiples. the portion of the thirdterm chosen is determined by restricting the limits of inte-gration. These restricted limits allow us to focus only on sig-nals which interact with V1 on the first and third occasions atpoints lower in the earth (or later in time) than on the secondoccasion, thus satisfying our definition of an internal multi-ple event of first-order.

For a 2D earth the first attenuation term may be written ex-plicitly in the source and receiver slowness, and do-mains as

is a small time interval which ensure that multiple scat-tering of an event with itself is excluded fmm the multipleattenuation series. For infinite bandwidth data may be asingle time sample, for bandlimited data should be greaterthan the wavelet duration. M1 when added to the data at-tenuates all first order internal multiples at a single step, Al-though, equation (7) may be extended in a straight forwardway to higher-order multiples (see references above), it hasbeen out experience on synthetic data that these higher orderterms are rarely required due to the rapid decay with order inthe amplitude of internal multiples.

In a 1D earth each slowness propagates independently thusthe data in the slowness domain has a delta function depen-dence on p1 p2. i.e. t) and equation (7) simplifies accordingly.

Example: 1D Prestack Acoustic Synthetics

To demonstrate the method in a simple 1D acoustic modelfinite difference synthetics were calculated for a model con-sisting of a 250 m thick layer = 2000 = 2.25g/cc) separating two half-spaces (V, = 1500 m/s, = 1g/cc). The source and receivers were located 125 m abovethe top of the layer with offsets from 0 m. Therewas was free surface. The synthetics are shown in figure 2.

The two primaries and the first order multiple are clearly vis-ible in the data with the second order multiple less so.

Internal multiple attenuation

Figure 2: The ID prestack acoustic synthetics.

Figure 3 shows a detail of the synthetics after the calcula-tion and addition of the first order mulitple attenuation term.The first order multiple at 0.7 s has been significantly attenu-ated. Although not shown on this time window the primariesremain untouched. Note that the second order multiple at0.95 s also experiences a reduction in amplitude; an addi-tional degree of attenuation will be achieved by calculatingthe second order attenuation term.

Example: 2D Acoustic Synthetics

To demonstrate the action of the 2D algorithm. equation (7),we generate data directly in the plane-wave domain. This isdone for a simple wedge model = 4000 m/s. p = 1 g/cc)separating two half-spaces = 1500 = 1 g/cc) byilluminating it with a single plane wavefront with a variety ofdifferent slownesses, see figure 4. Again the synthetics weregenerated using finite differences. A single incident planewave generates two primaries with distinct slownesses anda series of multiples (only one is shown) also with distinctslownesses. figure 5.If the ID single slowness algorithm was applied to eachslowness component independently. then the result would bea zero multiple attenuation term since each slowness traceexhibits only a single event. Figure 6 shows the result of ap-plying the 1D - single slowness algorithm before transforma-tion of the reflected wavefield into the slownesses domain atthe receiver and the result of applying the full 2D algorithm,equation (7). to the data shown in figure 5.The result of applying the single slowness algorithm shows

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Internal multiple attenuation

Figure 3: A detail of the 1D prestack acoustic synthet-ics before (solid line) and after (filled trace) first ordermultiple attenuation.

Figure 4: The 2D acoustic model, showing the incidentand reflected plane waves schematically.

the multiple has been amplified for both incident slownesses.The single slowness multiple attenuation term has incor-rectly predicted the phase of the multiple and hence whenadded to the data it has amplified not attenuated it. In con-trast the 2D algorithm, equation (7), has correctly predictedthe phase of the multiple, and hence, when added to the datahas significantly attenuated the multiple signal.

Example: 1D Elastic Synthetics

Finally we show the results of applying the inverse scatteringmultiple attenuation algorithm to synthetics from an elasticmodel. The model is shown in figure 7 and consists of anacoustic half-space overlying an elastic layer above an elas-tic half-space. Again the model is illuminated by a planewave. The data, figure 8 now consists of four primaries, anevent from the top interface and three events from the bottominterface with different modes of propagation (PP, PS and SPtogether and SS), as well as a variety of multiple events witha mixture of P- and S-wave legs. The central panel show thefirst order multiple attenuation term and the right hand panela comparison of the data before and after multiple attenua-tion.

The multiple events consisting of only P-wave legs are sig-nificantly attenuated; this is not surprising since the formof the inverse scattering algorithm we are using assumesan acoustic reference wave propagation and a P-wave defi-nition of the multiple. More surprising is the fact that theevents with one or more S-waves are also attenuated, if only

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Figure 5: The reflected plane wave primaries and firstorder multiples for two distinct incident plane waves.

slightly. Extending the inverse scattering multiple attenua-tion to an elastic reference medium, which we might expectto attenuate S-wave events more effectively, is the subject offurther research.

Discuss ion

We have presented the results of testing an inverse scatteringseries internal multiple attenuation algorithm on 1- and 2Dacoustic and 1D elastic media. The method attenuates allmultiples of a given order at a single step and does so withoutaffecting primary signals.

The method requires no information about the subsurfacestructure or velocity field. It predicts multiples directly fromthe data. A prerequiste of the method is a detailed knowl-

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Figure 6: (Left and center panels) Applying the 1Dalgorithm to each slowness amplifies the multiple.(Right panel) The 2D algorithm attenuates tbe multiple(solid line) compared to the input data (dashed line).

edge of the wavelet. The results from the acoustic models arevery encouraging showing significant attenuation in both 1Dand ZD. The algorithm is equally effective for elastic modelsin attenuating multiple events of an entirely P-wave history.Events with S-wave legs are not as well attenuated, this mo-tivates our elastic reference medium internal multiple atten-uation research.

References

Arajo, F.V., 1994. Linear and non-linear methods derivedfrom scattering theory: back scattering tomography and inter-nal multiple attenuation: Ph.D Thesis, Universidade Federalda Bahia. Brazil (in Portuguese).

Weglein, A.B., and Arajo, F.V., 1994. Processing reflectiondata, Patent Application No. GB94/O2246.

Weglein, A.B.. Gasparotto. F.V., Carvalo, P.M. and StoltR.H.. 1996. An inverse scattering series method for attenu-ating multiples in seismic reflection data, submitted to Geo-physics.

Figure 7: The 1D elastic model illuminated by a singleplane wave.

Figure 8: Internal multiples in an elastic model illu-minated by a single plane wave. Data and first ordermultiples (left panel), first order multiple attenuationterm (center) and a comparison of the data before andafter attenuation (right) (where tbe later has been siftedslightly for visiblity). Multiple events consisting ofonly P-wave legs have been significantly attenuated.Events with one or more S-wave legs are less well at-tenuated.

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