2-D ZO CRS stack by considering an acquisition line with smooth topography

2-D ZO CRS stack by considering an acquisition line with smooth topography

Pedro Chira-Oliva^I; João Carlos R. Cruz^II; German Garabito^III; Peter Hubral^IV; Martin Tygel^V

^IUniversidade Federal do Pará, Departamento de Geofísica, Rua Augusto Corrêia 1, Campus Universitário do Guamá, Caixa Postal 1611 - CEP 66017-970; Tel: [phone]; Fax: [phone] - E-mail: [e-mail]

^IIUniversidade Federal do Pará, Departamento de Geofísica, Rua Augusto Corrêia 01, Campus Universitário do Guamá, Caixa Postal 1611 - CEP: 66017-970; Tel: [phone]; Fax: [phone] - E-mail: [e-mail]

^IIIUniversidade Federal do Pará, Departamento de Geofísica, Rua Augusto Corrêia 01, Campus Universitário do Guamá, Caixa Postal 1611 - CEP: 66017-970; Tel: [phone]; Fax: [phone] - E-mail: [e-mail]

^IVUniversidade de Karlsruhe, Instituto de Geofísica, Hertzstr. 16, D-76187 Karlsruhe, Alemanha. Tel: [phone]/4567; Fax: [phone] - E-mail: [e-mail]

^VUNICAMP - IMECC, Praça Sergio Buarque de Holanda 651, Cidade Universitária, Barão Geraldo, Caixa Postal 6065 - 13083-859 Campinas, São Paulo, Brasil. E-mail: [e-mail]

ABSTRACT

The land seismic data suffers from effects due to the near surface irregularities and the existence of topography. For obtaining a high resolution seismic image, these effects should be corrected by using seismic processing techniques, e.g. field and residual static corrections. The Common-Reflection-Surface (CRS) stack method is a new processing technique to simulate zero-offset (ZO) seismic sections from multi-coverage seismic data. It is based on a second-order hyperbolic paraxial traveltime approximation referred to a central normal ray. By considering a planar measurement surface, the CRS stacking operator is defined by means of three parameters, namely the emergence angle of the normal ray, the curvature of the normal incidence point (NIP) wave, and the curvature of the normal (N) wave. In this paper the 2-D ZO CRS stack method is modified in order to consider effects due to the smooth topography. By means of this new CRS formalism, we obtain a high resolution ZO seismic section, without applying static corrections. As by-products the 2-D ZO CRS stack method we estimate at each point of the ZO seismic section the three relevant parameters associated to the CRS stack process.

Keywords: CRS stack, smooth topography, measurement surface curvature, near-surface irregularities.

RESUMO

Os dados sísmicos terrestres são afetados pela existência de irregularidades na superfície de medição, e.g. a topografia. Neste sentido, para obter uma imagem sísmica de alta resolução, faz-se necessário corrigir estas irregularidades usando técnicas de processamento sísmico, e.g. correições estáticas residuais e de campo. O método de empilhamento Superfície de Reflexão Comum, CRS ("Common-Reflection-Surface", em inglês) é uma nova técnica de processamento para simular seções sísmicas com afastamento-nulo, ZO ("Zero-Offset", em inglês) a partir de dados sísmicos de cobertura múltipla. Este método baseia-se na aproximação hiperbólica de tempos de trânsito paraxiais de segunda ordem referido ao raio (central) normal. O operador de empilhamento CRS para uma superfície de medição planar depende de três parâmetros, denominados o ângulo de emergência do raio normal, a curvatura da onda Ponto de Incidência Normal, NIP ("Normal Incidence Point", em inglês) e a curvatura da onda Normal, N. Neste artigo o método de empilhamento CRS ZO 2-D é modificado com a finalidade de considerar uma superfície de medição com topografia suave também dependente desses parâmetros. Com este novo formalismo CRS, obtemos uma seção sísmica ZO de alta resolução, sem aplicar as correições estáticas, onde em cada ponto desta seção são estimados os três parâmetros relevantes do processo de empilhamento CRS.

Palavras-chave: empilhamento CRS, topografia suave, curvatura da superfície de medição, irregularidades próximas da superfície.

INTRODUCTION

In order to obtain a high-resolution image of the earth sub-surface the geophysicists use the multi-coverage seismic data acquisition, that yields to overlap registers of geological targets. In time domain, the ZO section is the seismic image obtained by considering coincident sources and receivers. This is simulated by stacking the amplitudes using a traveltime operator, which is defined by means of stack parameters.

By the conventional seismic processing, the ZO section is simulated using the well-known normal moveout/dip moveout (NMO/DMO) stack method. Mann et al. (1999) presented a new stack method, so-called Common-Reflection-Surface (CRS), based on a hyperbolic second-order paraxial approach. By considering a planar measurement surface, it depends on three parameters, namely, the emergence angle b_o of the normal ray, the curvatures K_NIP and K_N of the two hypothetic wavefront, so-called NIP and N waves, respectively (Hubral, 1983).

Land seismic data are in general affected by the existence of surface topography and irregularities in the near-surface (e.g. weathering base and weathering velocity). In the conventional seismic processing, these effects are interpreted by deviations from hyperbolic NMO correction in the common-midpoint (CMP) gather. The topography effects are corrected by using field and residual static corrections. By applying specifically the field static correction, based on refraction seismic data, we remove the most part of these traveltime anomalies. Nevertheless, this correction usually does not account for rapid changes of the topography, in the weathering base, and of the weathering velocity. It is very sensitive to the choice of parameters involved in the picking phase.

According to Guo & Fagin (2002), land surveys should always be processed considering a floating datum that represents the topography. They showed that velocity analysis from a flat seismic reference datum creates errors to estimate the depth and interval velocities, even in the case of a flat topography, due to deviations of the take-off angles of the seismic ray paths.

Chira-Oliva & Hubral (2003) studied the sensibility of the interval velocity and reflector depth by considering a hypothetical circle measurement surface. They showed the NMO velocity by considering the curvature of the earth surface is more accurate to recover the interval velocities and the depths of the reflectors than the NMO velocities obtained by using a planar measurement surface approach. Chira-Oliva & Hubral (2003) and Zhang et al. (2002), respectively, presented the 2-D ZO CRS formalism for measurement surface with smooth and rugged topography. Chira-Oliva et al. (2001) modified the 2-D ZO CRS operator for including effects of near-surface inhomogeneity. In this paper, the 2-D ZO CRS stack performance is tested by considering a multi-layer model with smooth topography.

THEORY

The 2-D ZO CRS stacking operator depends on three parameters of two hypothetical waves, namely the normal-incidence-point (NIP) and Normal (N) waves (Hubral, 1983). These parameters are the emergence angle of the normal ray, and the radii of curvatures of the NIP and N waves. The emergence point, X₀, of the normal ray is called central point. The NIP wave propagates upwards from a point source located at the normal ray incidence point; and the N wave propagates upwards starting at the reflector, like an exploding reflector source.

Based on the hyperbolic second-order paraxial traveltime approach, the 2-D ZO CRS stacking operator with smooth topography is given by (Chira-Oliva et al., 2001),

Equation (1) describes the reflection time t of the paraxial ray SPG in the vicinity of a normal (ZO) ray X₀NIP X₀ (Figure 1a). The ZO travel-time and the central point coordinate are t₀ and x₀, and v₁ is the near-surface velocity of the P-P wave at the central point X₀. The coordinates x_m and h are, respectively, the midpoint and half-offset referred to the x₁-axis, that is tangent to the topography surface with origin at the central point X₀ (see Figures 1a,b). The emergence angle of the normal ray at the central point is . The parameter K₀ is the local curvature of the earth surface at a point of the acquisition line, that is positive (or negative) if this line falls below (or above) its tangent at X₀. The radii of curvatures of the emergence hypothetical NIP and N wavefronts at X₀ are R_NIP and R_N respectively.

In order to normalize the processing coordinates, we apply a transformation from the local (x₁,x₃) into the global cartesian system (x, z) in Figure 1b. The midpoint and half-offset coordinates, (x_m, h) and (x'_m, h'), in the local and global coordinate cartesian systems, respectively, are related by the expressions

where is the dip angle of the tangent x₁-axis at point X₀. Introducing the relationships (2) into equation (1), we find (e.g. Chira-Oliva & Hubral, 2003; Chira, 2003)

We now consider a pure diffraction, i.e., the situation in which the reflector reduces to a single diffraction point. In this case, the NIP and N waves are coincident, i.e. both propagate from a point source at NIP and have identical radii of curvatures at X₀, R_N º R_NIP. As a consequence, equation (3) becomes

Equation (4) depends on two CRS parameters (R_NIP, ) associated to the NIP wave. This equation will be used at the first step of the CRS strategy. The CRS stacking operator defined by equation (4) is interpreted as an approach of the pre-stack Kirchhoff migration operator with smooth topography.

Setting the condition h' = 0 to the general hyperbolic travel-time equation (3), the CRS stacking operator for reflected events in the ZO configuration becomes

Following Garabito et al. (2001) the three optimal CRS parameters (, R_NIP, R_N) are searched by three steps. At the first step we use formula (4) to determine and R_NIP. At the second step we use formula (5) to determine R_N; and at the third step we use formula (3) to refine the three parameters.

2-D ZO CRS STACK

In the 2-D situation, for each point P₀ (x₀,t₀) at the ZO section to be simulated, the amplitudes in the seismic data will be summed (stacked) along the CRS surface defined by equation (3). The resulting (stacked) amplitude is assigned to the point P₀.

The three CRS stacking parameters are estimated by means of an optimization process, having the semblance as objective function. The CRS stacking optimization problem consists of estimating the parameters that maximize the semblance. In general, the problem requires a combination of multi-dimensional global and local optimization algorithms. The mathematical intervals defined for the parameters are -p/2 < < p/2, -¥ < R_NIP, R_N < ¥. Optimization strategies to estimate these parameters are found in the literature (e.g. Müller (1999); Birgin et al. (1999); Garabito et al. (2001)).

In this paper, we apply the strategy given by Garabito et al. (2001) to estimate the CRS parameters triplet, but using the new equations (3), (4) and (5).

CRS STACK PROCESSING STRATEGY

The proposed strategy to carry out the CRS method involves a combination of global and local search processes and is divided into three steps. The curvature, K₀, of the seismic line at each central point is supposed to be a priori known or calculated by means of some interpolation method by using elevation values. At the first and second steps we used the Simulated Annealing (SA) algorithm (Sen & Stoffa, 1995), and at the third step the Quasi-Newton (QN) algorithm (Bard (1974); Bard (1981)). Each step is performed on each sample point P₀ (x₀,t₀) that pertains to the ZO section to be simulated. The objective function is the semblance calculated for each point in the ZO section.

Step I: Pre-Stack Global Optimization

The multi-coverage pre-stack seismic data is the input. The inverse problem consists of simultaneously estimating the two parameters and R_NIP that provide the maximum semblance value, according equation (4). The results of this step are: 1) maximum coherence section, 2) emergence angle, - section, 3) NIP-wave radius of curvature, R_NIP-section, and 4) simulated (stacked) ZO section.

Step II: Post-Stack Global Optimization

The post-stack seismic data is the input. The inverse problem consists of estimating the single parameter, R_N, that provides the maximum semblance according to equation (5), in which the previously obtained parameter, , is kept fixed. In this step the results are: 1) maximum coherence section, 2) N-wave radius of curvature, R_N-section, and 3) re-stacked simulated ZO (stacked) section.

Step III: Pre-Stack Local Optimization

The multi-coverage pre-stack seismic data from step I is the input. The inverse problem consists of estimating the best parameter triplet (, R_NIP, R_N) that provides the maximum semblance. In this case the CRS stacking operator is equation (3), applied to the full multi-coverage data set with suitable apertures. In this step the results are: 1) maximum coherence section, 2) optimized -section, 3) optimized R_NIP-section, 4) optimized R_N-section, and 5) optimized ZO (stacked) section.

Example

In order to test the CRS stacking algorithm we applied it to a synthetic data set computed for 2-D homogeneous layered model shown in Figure 2. The model is constituted of four layers above a half-space. The acquisition system is lying on a smooth topography line. Based on this model, we generated a synthetic data set of multi-coverage primary reflections, using the ray-tracing algorithm, SEIS88 ( & Psensik, 1988). In order to test the accuracy of the CRS method, it was added random noise with signal-to-noise ratio of S/N = 10. The data set consisted of 201 common-shots (CS) with 72 receivers with interval of 50 meters. The minimum offset was 50 meters. The source signal was a Gabor wavelet with 40 Hz dominant frequency and the time sampling was 4 ms. An example of part of these data is presented in Figure 3, represented by a CS section.

Figure 4 shows the ray-theoretical modelled ZO section with random noise added. Figure 5 shows the simulated ZO section that results from the application of the CRS stack method for a curved measurement surface. Due to the fact that the CRS method involves a larger number of traces during the stacking process, the simulated ZO section presents enhanced primary reflection events, with larger signal-to-noise ratio than the corresponding ones in the modelled ZO section (Figure 4). Figure 6 shows the maximum coherence (semblance) section that corresponds to the best parameters. We note that the coherence values become smaller for larger traveltimes (deeper events). Figures 7, 8 and 9 show the sections of emergence angle and radii of curvature of the NIP and N waves, respectively. These sections correspond to global maxima determined at the third step.

A comparison between the emergence angles, , estimated by the CRS algorithm (curves of red points) and by modelling (curves of blue points), respectively, is shown in Figure 10. We can see the emergence angle has been well estimated along all reflectors. Figures 11 and 12 show the analogous comparison for the other parameters, R_NIP e R_N, respectively. These parameters are also well estimated, with the exception of the locations where abrupt changes of the curvature K₀ are present (Figure 13).

CONCLUSIONS

A new formula for the CRS stack method that considers the smooth topography of the acquisition line has been tested in synthetic data sets with successful results. The parameters were correctly estimated, excepting the regions where there are abrupt changes of the curvature of the topography line. In these regions, the errors of the estimated parameters increase with depth. Besides the simulated ZO sections, we have obtained the coherence section and the sections referred to the attributes of the NIP and N waves.

ACKNOWLEDGMENTS

We thank the financial support in part by the IMAGAM Project of the CTPETRO/FINEP/CNPq/UFPA. The first author also thanks to the National Council of Technology and Development (CNPq), Brazil, for the scholarship, and to the sponsors of the Wave Inversion Technology (WIT) Consortium (Germany) for their support.

Recebido em 12 janeiro, 2005 / Aceito em 6 abril, 2005

Received on January 12, 2005 / Accepted on April 6, 2005

NOTES ABOUT THE AUTHORS

Pedro Chira-Oliva. He received his diploma in Geological Engineering (UNI-Peru/1996). He also received his MSc in 1997 and PhD in 2003, both in Geophysics, from Federal University of Pará (UFPA/Brazil). He took part of the project of scientific research "3D Zero-Offset Common-Reflection-Surface (CRS) stacking" (2000-2002), sponsored by Oil Company ENI (AGIP Division - Italy) and the University of Karlsruhe (Germany). Since 2003 he is researcher of the UFPA, responsible for the scientific project "Generalization of the Common-Reflection-Surface (CRS) stacking applied to real data in the Amazon region", financed by the PROSET/CT-PETRO/CNPq. He is an associate member of the Society of Exploration Geophysicists (SEG), Brazilian Geophysical Society and also is member of the Wave Inversion Technology (WIT) Corsortium (Germany).

João Carlos Ribeiro Cruz. He received a BS (1986) in Geology, MS (1989) and Ph.D. (1994) in Geophysics from the Federal University of Pará, Brazil. From 1991 to 1993 was with the reflection seismic research group of the University of Karlsruhe, Germany, while developing PhD thesis. Since 1996 he has been a full professor at the Geophysical Department of the Federal University of Pará (UFPA). Since 1999 he has been Dean of the Geophysical Graduate Scholl at UFPA. His current research interests include velocity analysis, seismic imaging, and apllication of inverse theory to seismic problems. He is a member of SEG, EAGE and SBGf.

German Garabito. He received his BSc (1986) in Geology from University Tomás Frias (UTF), Bolivia, his MSc in 1997 and PhD in 2001 both in Geophysics from the Federal University of Pará (UFPA), Brazil. Since 2002 he has been full professor at the geophysical department of UFPA. His research interests are data-driven seismic imaging methods such as the Common-Refection-Surface (CRS) method and velocity model inversion.

Peter Hubral. He obtained his MSc in Geophysics (Clausthal/1967) and Ph.D. in Geophysics (Imperial College London/1970). He is Professor at the University of Karlsruhe since 1986, having lectured at PPPG-UFBa (Salvador/1983-1984), Institute of Geophysics at University of Pau (France/1985-1986) and UFPa (2002-2003). He received the awards:Schlumberger (EAGE/1978), Erasmus (EAGE/2003), Reginald Fessenden (SEG/1979), Professor estrangeiro (SBGf/1999). He is a honorary member of SEG (1979) and EAGE (2002). He coordinates the Wave Inversion Technology (WIT) Cosortium and is director of the Institute of Applied Geophysics at Karlsruhe (Germany).

Martin Tygel received his B.Sc in physics from Rio de Janeiro State University in 1969, his MSc in Mathematics at the Catholic University of Rio de Janeiro 1976 and his PhD in 1979 from Stanford University, both in mathematics. He was a visiting professor at the Federal University of Bahia (PPPG/UFBa), Brazil, from 1981 to 1983 and at the Geophysical Institute of Karlsruhe University, Germany, in 1990. In 1984, he joined Campinas State University (UNICAMP) as an associate professor and since 1992 as a full professor in Applied Mathematics. Professor Tygel has been an Alexander von Humboldt fellow from 1985 to 1987. In that period, he conducted research at the German Geological Survey (BGR) in Hannover. From 1995 to 1999, he was the president of the Brazilian Society of Applied Mathematics (SBMAC). In 2002, he received EAGE's Conrad Schlumberger Award and also UNICAMP's Zeferino Vaz Award for academic recognition in 1995 and 2004. Prof. Tygel is the founder in 2001 of the Laboratory of Geophysical Computing at UNICAMP. The Laboratory is a member of the Wave Inversion Technology (WIT) Consortium, with headquarters in Karlsruhe. Prof. Tygel's research interests are in seismic processing, imaging and inversion. Emphasis is aimed on methods and algorithms that have a sound wave-theoretical basis and also find significant practical application.

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Publication Dates

• Publication in this collection
  23 Mar 2007
• Date of issue
  Mar 2005

History

• Received
  12 Jan 2005
• Accepted
  06 Apr 2005