Interferometry in dissipative media: Addressing the shallow sea problem for Seabed Logging applications

Transcription

1 Evert Slob and Kees Wapenaar, Departent of Geotechnology, Delft University of Technology Roel Snieder, Center for Wave Phenoena and Departent of Geophysics, Colorado School of Mines SUMMARY We derive interferoetric field representations that are valid for diffusive field ethods. The ethod retrieves the reflection response of the earth, as if the doain above the receiver depth level is hoogeneous. It is represented as the flux-noralized up going field deconvolved by the down going field. The deconvolution step can be seen as a weighted crosscorrelation step, which is the usual operation in interferoetric ethods. Because the ethod effectively redatus the source depth level to the receiver depth level and reoves the overburden effects, the shallow sea proble that exists for frequency doain Seabed Logging applications is solved in theory. INTRODUCTION Interferoetry is the branch of science that deals with the creation of new field responses by crosscorrelating observations at different receiver locations. Since its introduction in exploration seisology around the turn of the century, the literature on seisic interferoetry has grown spectacularly. Many interferoetric ethods have been developed for rando fields and for controlled-source data. The underlying theories have in coon that the ediu is assued to be lossless and non-oving, see e.g. the suppleent of the 2006 July- August issue of Geophysics. The ain reason for this underlying assuption is that the wave equation in lossless and non-oving edia is invariant for tie-reversal. Until 2005 it was coonly thought that tie-reversal invariance was a necessary condition for interferoetry, but recent research shows that this assuption can be relaxed. Slob et al. (2006) analyzed the interferoetric ethod for ground penetrating radar data (GPR), in which losses play a proinent role. They showed that losses lead to aplitude errors as well as the occurrence of spurious events. By choosing the recording locations in a specific way, the spurious events arrive before the first desired arrival and can thus be identified (Slob et al., 2007). By choosing one receiver in a lossless ediu, e.g. air, and a configuration with all dissipative paraeters outside the surface distribution of noise or transient sources, crosscorrelation ethods work without spurious events and aplitude errors in the tie window of interest (Slob and Wapenaar, 2007). This approach holds for waves and diffusive fields in dissipative edia. Snieder (2006) showed that a volue distribution of uncorrelated noise sources, with source strengths proportional to the dissipation paraeters of the ediu, precisely copensates for the energy losses (Snieder, R. 2007, Extracting the Greens function of attenuating edia fro uncorrelated waves, JASA, accepted). As a consequence, the responses obtained by interferoetry in such configurations are error free. Also this approach holds for waves in dissipative edia and for pure diffusion processes. Recently we showed that interferoetry, including its extensions for waves and diffusion in dissipative and/or oving edia, can be represented in a unified for (Wapenaar et al., 2006; Snieder et al., 2007). These representations can also be used for ore exotic applications like electroseisic prospecting and quantu echanics. We have loosened the definition of interferoetry to also include crossconvolution ethods. Slob et al. (2007) introduce interferoetry by crossconvolution and show that it is valid for arbitrary dissipative edia. The crossconvolution ethod does not require a volue distribution of sources, but a restriction is that it only works for transient signals in specific configurations with receivers at opposite sides of the source array. Fro these observations we conclude that none of the above described ethods provides a practical approach to controlled source electroagnetic (CSEM) applications. Here we deonstrate that interferoetry-by-deconvolution is applicable in CSEM or in any other exploration ethod eploying diffusion processes. Hence we further loosen the definition of interferoetry to also include deconvolution ethods, which is particularly useful for Seabed Logging ethods. INTERFEROMETRY IN DISSIPATIVE MEDIA The D version of interferoetry-by-deconvolution was introduced by Riley and Claerbout (976). It relies on the decoposition of a field at a particular depth level into flux-noralized down going and up going parts. To facilitate such decoposition we eploy the reciprocity theore for one-way fields and apply it on a doain with two horizontal boundaries, see Figure. The necessity of flat horizontal boundaries can be relaxed under certain conditions and the derived representations also hold for soothly curved boundaries (Frijlink, M. and K. Wapenaar, 2007, Reciprocity theores for one-way wave fields in curvilinear coordinate systes, JASA, subitted). Following Wapenaar and Gribergen (996) we write the frequency doain one-way field reciprocity theore for two independent states A and B as, x 2 x cyl @V 0 ID V n = (0 0 ) n = (0 0 ) n = (n n 2 0) Figure : Configuration for one-way reciprocity theores. D n 3 (ˆp A ) t Nˆp B d 2 x = {(ˆp A ) t Nŝ B +(ŝ A ) t Nˆp B }d 3 x, () D where D denotes the two flat boundaries with outward unit noral n 3 and the superscript t denotes transposition. The 4 electroagnetic field vector p contains flux-noralized down and up going fields ˆp = (ˆp +, ˆp ) t and the 4 electroagnetic source vector contains fluxnoralized down and up going source coponents ŝ = (s +,s ) t, given by ˆp ± = ˆp ± (x,ω) and ŝ ± = ŝ ± (x,ω) (Reid, 972; Ursin, 983). The atrix N is given by ( 0 I N = I 0 ), (2) the atrices I and 0 being the 2 2 identity and null atrices, respectively. To construct the vector ˆp we ust record all horizontal coponents of the electric and agnetic field strengths on a grid and we apply decoposition to these coponents. Note that equation () holds for equal edia in the two states inside D, while outside D the edia in the two states can be different. No derivatives occur because we use flux-noralized field quantities.

2 First we assue that in both states the sources are outside D, this reduces equation () to {(ˆp + A )t ˆp B (ˆp A )t ˆp + B }d2 x = {(ˆp + A )t ˆp B (ˆp A )t ˆp + B }d2 x. D D In the following analysis state B represents the actual state of the easured response of the real earth. Consider the arine CSEM acquisition geoetry with a sea surface at level D 0, see Figure 2, with a source at x S in the water layer and the receivers x at the botto of the sea at level D. Both the water layer and the doain between D and D can be heterogeneous. For each source coponent and after decoposition we have for state B the down going and up going coponents of the recorded earth response, given by { ˆp + x D B (x,ω) = ˆp + (x,x S,ω), ˆp B (x,ω) = (4) ˆp (x,x S,ω). 0 x S (3) State A represents the desired reflection response with a redatued source at the receiver level of state B in an earth with different boundary conditions than the real earth, obtained through interferoetry-bydeconvolution. The difference is that the ediu above the boundary D is hoogeneous and has the sae properties as just below D, see Figure 2. For state A we choose a down going source coponent just above the level D and put receivers at the sae level D. We define the reflection response of the ediu below D as the 2 2 atrix ˆR + 0 (x,x A,ω), where the subscript 0 denotes that no reflectors exist above D and the superscript + indicates the reflection is a response to a down going source field. We therefore find in state A ( ˆP + A (x,ω) = δ(xh x H,A ) 0 0 δ(x x D H x H,A ) ˆP A (x,ω) = ˆR + 0 (x,x A,ω), (6) where the subscript H is used to denote the horizontal coordinates only, hence x H = (x,x 2 ) and x H,A = (x,a,x 2,A ) (the latter denoting the horizontal coordinates of x A ). At x D we have again only down going fields, x D ), { ˆP + A (x,ω) = ˆT + (x,x A,ω), ˆP A (x,ω) = 0, (7) ˆp ˆp where ˆT + (x,x A,ω) is the 2 2 transission response between the levels D and D. Substitution of equations (4)-(7) into equation (3) and using sourcereceiver reciprocity, i.e. ˆR + 0 (x,x A,ω) = ( ˆR + 0 )t (x A,x,ω), yields ˆP (x A,x S,ω) = ˆR + 0 (x A,x,ω) ˆP + (x,x S,ω)d 2 x, x D (8) x State B Rˆ x x, A xa (, ) 0 where the up and down going responses, ˆP, ˆP +, are now 2 2 atrices because we have two source coponents. Equation (8) is a Fredhol integral equation of the first kind in the reflection coefficient atrix ˆR + 0 (x A,x,ω). The reflection coefficient atrix is the retrievable flux-noralized Green s function, representing the ipulse response at a receiver location x A D due to a down going source coponent at position x D. For laterally invariant edia it can easily be solved by siple 2 2 atrix inversion for each wavenuberfrequency coponent separately. Of course this requires two independent source coponents. For general 3D heterogeneous edia it can only be solved when the decoposed data, ˆP (x A,x S,ω) and ˆP + (x,x S,ω) is recorded at a sufficient nuber of receiver positions x A D and for a sufficient nuber of source positions x S. It follows that two horizontal source electric dipole orientations are sufficient to solve equation (8) uniquely, see e.g. Holvik and Aundsen (2005) for an elastic exaple. In atrix notation (Berkhout, 982), equation (8) can be written as ˆP = ˆR + ˆP 0 +. (9) State A Figure 2: State B: the easured response of the real earth. State A: the response of the ediu inside D with a hoogeneous upper half space, above D. Both states have a hoogeneous lower half space, below D. We choose the level D to be below all heterogeneities, hence there are only non-zero down going field coponents at the level D, { ˆp + x D B (x,ω) = ˆp + (x,x S,ω), ˆp B (x,ω) = 0. (5) For exaple, the coluns of atrix ˆP + contain both coponents of ˆp + (x,x S,ω) for fixed x S and variable x at D, whereas the rows of this atrix contain ˆp + (x,x S,ω) for fixed x and variable x S and both source coponents at D S, where D S represents the depth level of the sources. Inversion of equation (9) involves atrix inversion, according to ˆR + 0 = ˆP ( ˆP + ) (0) (Wapenaar K. and D.J. Verschuur, 996, Processing of ocean botto data: The Dolphin Project, Volue I, p ). The atrix inversion in equation (0) can be stabilized by least-squares inversion, according to ˆR + 0 = ˆP ( ˆP + ) [ ˆP + ( ˆP + ) + ε 2 I], () where the superscript denotes transposition and coplex conjugation, I is the identity atrix and ε is a sall constant. Berkhout and

3 Verschuur (2003) used a siilar inversion for transforing surfacerelated ultiples into priaries. Equations (0) and () describe 3D interferoetry-by-deconvolution and is siilar to the least squares redatuing ethod described by Schuster and Zhou (2006). When we ignore the inverse atrix in equation () we arrive at ˆR + 0 ˆP ( ˆP + ). (2) in presence of the reservoir layer in solid red-lines and the responses in absence of the reservoir layer in dashed blue lines. We use a linear offset scale and a ten-base logarithic aplitude scale for all figures showing responses. which is the atrix for expression in the frequency doain of Bakulin and Calvert s virtual source ethod (Bakulin and Calvert, 2006). Of course in our case the atrix ˆP + ( ˆP + ) is not close to I and equation (2) cannot be used for CSEM data. Coparing equation (2) with equation () it can be seen that the here proposed ethod of interferoetry by deconvolution is a weighted for of the usual crosscorrelation ethod, with the inverse atrix in equation () as the weight. NUMERICAL EXAMPLES Electric field aplitude [sv/] 0 2 To illustrate interferoetry-by-deconvolution with a nuerical exaple, we apply it to siulated 2D CSEM data as a siple deonstration of the advantage of this ethod for hydrocarbon exploration with the Seabed Logging ethod. Aundsen et al. (2006) showed already that decoposition of CSEM data into down going and up going fields iproves the detectability of hydrocarbon reservoirs. Here we show that the cobination of decoposition followed by interferoetry-bydeconvolution not only iproves the detectability but also results in iproved quantitative inforation about the reservoir paraeters. h w w =3 S/ 250 =.5 S/ =0.5 S/ 00 3 =20 S/ 2 =0.5 S/ x S =50 Figure 3: Configuration for the 2D exaple for Seabed Logging applications. The odel consists of a plane layered Earth and the TM-ode configuration is used, which is the proper two-diensional approxiation of the CSEM ethod as applied in Seabed Logging applications with an in-line horizontal electric dipole (HED) and in-line electric field receivers. Since this is a 2D scalar exaple, the in-line HED is sufficient and only a single source position is needed because we assue a horizontally shift-invariant ediu. To allow for decoposition into down going and up going field coponents, we record the in-line electric field strength and the cross-line agnetic field strength. The odel is shown in Figure 3, where the seawater layer contains an in-line electric current source at 50 above the sea botto. The receivers are located at the sea botto with a total extent of 40 k. The water layer is odeled with a thickness of 00 as a odel for a shallow sea. The seawater has a conductivity of σ w =3 S/. Below the sea botto there is a layer with a conductivity of σ =.5 S/ with a thickness of 250. This is followed by a half-space with σ 2 =0.5 S/, which is intersected after 250 by a reservoir-type layer with a thickness of 00 and a conductivity of σ 3 =50 S/. Note that the top of this reservoir layer is located at 500 below the sea botto. To copare with a signal strength in the sae background ediu without reservoir layer, we also odel the response of the background ediu. The source frequency is taken at f S = 0.5 Hz. All plots show the responses D Magnetic field aplitude [sa/] Figure 4: In-line electric field response x D in presence (solid red curve) and absence (dashed blue curve) of the reservoir at ; Crossline agnetic field response at x D in presence (solid red curve) and absence (dashed blue curve) of the reservoir layer. Figure 4 shows the recorded in-line coponent of the electric field at the botto of the sea, while Figure 4 shows the recorded crossline coponent of the agnetic field at the botto of the sea. Fro both figures it can be seen that the presence of the reservoir layer is not visible in the plots because the red and blue curves alost copletely overlap. As a first step in our interferoetry-by-deconvolution procedure we carry out the decoposition of the recorded field coponents into down going and up going flux-noralized field coponents with the ediu paraeters of the layer just below the sea botto and hence they correspond to the fields that would have been easured when they were positioned just below the sea botto. They are shown in Figures 5 and 5, respectively. It is clear fro these figures that the up going field is ore than ten ties saller in aplitude than the down going field. In Figure 5 it can be seen that the blue curve asks the red curve for all offsets, indicating that the presence of the reservoir layer is not visible in the down going field part just as in the total field. As can be seen in Figure 5, the up going field shows the

4 presence of the reservoir layer for offsets between approxiately 2.5 k to 7 k. Still the up going field response is strongly influenced by the shallow sea indicating that there is relatively strong interaction with the sea surface and the layer below the sea botto, which is part of the up going field response. Down going field aplitude [sj /2 /] Up going field aplitude [sj /2 /] Figure 5: Flux-noralized down going field response, ˆp + (x,x S,ω), just below D in presence (solid red curve) and absence (dashed blue curve) of the reservoir layer ; Flux-noralized up going field response, ˆp (x A,x S,ω), just below D in presence (solid red curve) and absence (dashed blue curve) of the reservoir layer. Reoving the effect of the water layer fro the up going field by deconvolving it with the down going field results in a uch clearer reflection response as can be seen in Figure 6, where fro offsets of approxiately 2 k onward the presence of the reservoir is clearly visible. An other iportant aspect is the absence of aplitude saturation for large offsets when the water layer has been reoved, as can be seen by coparing the aplitude behavior of the up going field in Figure 5 and the retrieved reflection response in Figure 6 where the aplitude continues to decrease with increasing offset. Obviously, this continuing decrease in aplitude requires high precision data, finite recording precision and noise will prevent practical applications at very large offsets. However, there is clearly a practical offset range where it will work on actual easured data. In our exaple odel the effect of the first layer of 250 thickness still has a ajor effect on the deconvolved reflection response at near offsets because the lower half space in the ebedding has a uch lower electric conductivity Deconvloved field aplitude [ ] 0 5 Figure 6: Subsurface reflection response, ˆR + 0 (x A,x,ω), as if the the air and sea layers are absent, obtained by interferoetry-bydeconvolution, which response is thus independent of the water depth. The red solid line and blue dashed line are for the situation with and without the reservoir layer, respectively. than he first layer. The contrast is a factor 3 at a vertical distance of 250 below the receivers, while the contrast of the reservoir with its surroundings is a factor 0 at 500 below the receivers. It can be understood that for deeper receivers, e.g. placed in a horizontal well, the proposed ethod can result in reoval of these near sea botto, high conductivity layers and produce an even cleaner reflection response of the target. CONCLUSIONS We have forulated an interferoetric ethod using the deconvolution concept to create new responses using recorded responses that are valid and practical for CSEM as applied in the Seabed Logging ethod. This is an extension of known interferoetric ethods that use the lossless ediu assuption. The ethod is developed into an algorith that can be used on data fro Seabed Logging or fro other electroagnetic recordings at the botto of the sea or in a (horizontal) borehole. The algorith requires sufficient nuber of source coponents (for 3D data) and source positions. These source positions can be on the earth surface for land ethods or in the sea for Seabed Logging ethods, either transient or with only a liited nuber of frequencies. The developed algorith not only oves the source to the receiver depth level ( source redatuing ), but also reoves all overburden effects of heterogeneities above the receiver depth level (changed boundary conditions). The result is a reflection response that is obtained fro positions closer to the target and without disturbing overburden effects. As in all interferoetric ethods, no inforation about the ediu properties is required. Reoving the overburden effect effectively solves the shallow sea probles in frequency doain Seabed Logging ethods, while for possible deep receivers our proposed ethod will reove all overburden effects and produce a clean target reflection response. ACKNOWLEDGMENTS This work is supported by the Netherlands Research Centre for Integrated Solid Earth Science (ISES) and the sponsors of the CWP project.

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