3.2 Sensitivity of seismic wide angle wave field and first arrival times to fine scale ... 3.3.2 Wave form modeling of dipping upper mantle structures . ..... Choice of wave equation The level of complexity increases as we move from acoustic ...... Surface waves are computational expensive due to their relative small velocities.
Sensitivity of seismic waveforms to structure and lithology Thomas Mejer Hansen December, 2002
Department of Earth Sciences University of Aarhus, Denmark
Thesis submitted to the Faculty of Natural Sciences, University of Aarhus for the Ph.D. degree
Contents Preface and Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principal aims and structure of this thesis (list of papers) . . . . . . . . . . . . . . . 1 Introduction
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2 General description of research fields 2.1 Wave form modeling . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 The history of seismic wave form modeling . . . . . . . 2.1.2 The finite difference technique . . . . . . . . . . . . . . 2.1.3 Finite difference wave-form modeling . . . . . . . . . . 2.1.4 Efficient finite difference modeling . . . . . . . . . . . 2.1.5 Boundary conditions . . . . . . . . . . . . . . . . . . . 2.2 Sensitivity of the seismic wave field small scale inhomogeneities 2.2.1 Partial derivatives . . . . . . . . . . . . . . . . . . . . . 2.3 Seismic signatures of crustal and upper mantle structures. . . . . 2.3.1 Deep seismic investigations . . . . . . . . . . . . . . . 2.3.2 The seismic signature of the crust and Moho . . . . . . 2.3.3 Upper mantle reflections . . . . . . . . . . . . . . . . . 2.3.4 Upper mantle scattering . . . . . . . . . . . . . . . . . 2.3.5 The seismic lithosphere . . . . . . . . . . . . . . . . . . 2.4 AVO analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Gradient/Intercept cross plots . . . . . . . . . . . . . . 2.4.2 Additional AVO attributes . . . . . . . . . . . . . . . .
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3 Presentation of results 3.1 Main phase modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Removal of surface waves . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Separation of phases . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Efficient modeling of wide angle data . . . . . . . . . . . . . . . . . . 3.1.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Sensitivity of seismic wide angle wave field and first arrival times to fine scale crustal structure and Moho topography . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Upper mantle reflectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Elastic models of upper mantle dipping slabs . . . . . . . . . . . . . . 3.3.2 Wave form modeling of dipping upper mantle structures . . . . . . . . 3.3.3 BABEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Discussion on upper mantle reflectors . . . . . . . . . . . . . . . . . . 3.4 AVO attributes of long offset P wave data and converted P-SV waves . . . . . . i
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Taylor expansions of reflection coefficients . . Cross plots . . . . . . . . . . . . . . . . . . . Identification of fizz gas - a numerical example Discussion . . . . . . . . . . . . . . . . . . .
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4 Conclusions
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5 Dansk sammendrag
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6 English summary
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7 References
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A Papers A1-A5 (Appendix) A1 Numerical Analysis of the 4th order space 2nd order time staggered grid finite difference scheme for the elastic wave-equation . . . . . . . . . . . . . . . . . A2 Efficient finite difference waveform modeling of selected phases using a moving zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A3 Sensitivity of the full wavefield and first arrivals to topography on Moho and small scale variations in the lower crust . . . . . . . . . . . . . . . . . . . . . A4 Upper mantle reflectors: Modelling of seismic wavefield characteristics and tectonic implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A5 AVO attributes of long offset P-waves and P-SV converted waves . . . . . . . .
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Preface and Acknowledgment This thesis has been the result of collaborations with many fine scientists. Niels Balling and Bo Holm Jacobsen have supervised me for the last 5 years, and prepared me for the time after the Ph.D. to stand on my own feet without falling. I hope they succeeded :) The finite difference modeling technique was represented at this department by Egon Nørmark, in his Ph.D. thesis, an he was kind to share his knowledge. From February to May 1997 I visited Dr. Satish C. Singh and the BIRPS group for 3 months. Both Satish C. Singh and Richard Hobbs were very helpful and inspiring. During that time a visco-elastic modeling code was used, kindly provided by Johan Robertsson. From spring to fall in 1998 I visited ARCO, an oil company based in Plano, Texas, USA. I went back in the spring of 2000 for a short period. Here I worked together with primarily Douglas J. Foster and Brian E. Hornby. Many other people were very helpful and made me feel the southern hospitality. Especially the whole Hornby family who made their home my home whenever I felt like it. The Geophysical Laboratory has been a very fine place to study. I will certainly miss both the atmosphere and the people that created it. Morten Wendell Pedersen was kind to share his insights into computers at all levels (and beautiful strange music). Christian Schiøtt kept the spirit high until the end, and was extremely helpful with comments to this thesis. Lone Davidsen and Kim Bak Olsen tried hard to make my written language better. Thank you for the effort. Finally thanks to friends, family and Jannie for keeping up :) Thank you all.
Principal aims and structure of this thesis (list of papers) The present Ph.D. project has been carried out at the Department of Earth Sciences, Geophysical Laboratory, University of Aarhus. The Ph.D. project was initially labeled ”Full waveform modeling of wide angle seismic data”. When the Ph.D. project started in 1995, full waveform modeling was the state of the art modeling tool to model propagation of seismic waves. At that time, 2D acoustic full waveform modeling of crustal size models was the computing limit, using supercomputers. Today 2D visco-elastic finite difference modeling of lithospheric size models is possible using several weeks of computing time on a supercomputer. Ray theory has for many years been the method of choice to resolve velocity structures from seismic wide angle wave fields. Tomographic methods will undoubtedly still be used in the years to come, but the method is a high frequency approximation, and lacks information on e.g. wave forms and scattered energy. Finite difference modeling accurately computes the complete wave field from complex models. This has been used in this Ph.D. to investigate the seismic response from small scale heterogeneities in the crust, and various structures on Moho and in the upper mantle. Finite difference modeling has thus been used to investigate iii
the sensitivity of the seismic method to various models. Wave form modeling has been used to compute large offset P-wave data and converted P-SV waves, and AVO techniques have been deployed on these data. Furthermore additional contributions to the well founded AVO techniques have been established. The need for efficient modeling in settings of crustal to lithospheric size, resulted in development of an efficient elastic finite difference code. This thesis consists of a summary and 5 appendixes. Appendix A1 is a report on numerical analysis. The following four appendixes are research papers, submitted or intended for submission to geological and/or geophysical journals. Paper A3 and A2 are reviewed and published. Paper A5 is submitted, and paper A4 is in ready for submission. Appendix A1 Numerical analysis of the 4th order space 2nd order time staggered grid finite difference scheme for the elastic wave-equation Thomas Mejer Hansen. Paper A2 Efficient finite difference waveform modeling of selected phases using a moving zone Thomas Mejer Hansen and Bo Holm Jacobsen Computers and Geosciences, June 2002, Volume 28(7), pp 819-826. Paper A3 Sensitivity of the full wavefield and first arrivals to topography on Moho and small scale variations in the lower crust Thomas Mejer Hansen, Satish C. Singh and Bo Holm Jacobsen Geophysical Research Letters, Vol. 26(16), pp 2573-2576. Paper A4 Upper mantle reflectors: Modelling of seismic wavefield characteristics and tectonic implications Thomas Mejer Hansen, Niels Balling and Bo Holm Jacobsen Prepared for submition to Geophysical Journal International. Paper A5 AVO attributes of long offset P-waves and P-SV converted waves Thomas Mejer Hansen, Douglas John Foster and Brian E. Hornby Submitted to Geophysics
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1 Introduction The main focus of this Ph.D. has been to investigate the sensitivity of the seismic wave field to different geological structures. Three geological settings have been considered : Crustal models with stochastic contents both in the crust and on Moho topography have been proposed in the recent years. We investigate which parts of the velocity field the wide angle seismic wave field show sensitivity to, and which parts seismic first arrivals shows sensitivity to. We find that the wide angle wave field does not show sensitivity to wavelengths in the velocity model smaller than two times the dominant wavelength of the propagating wave field. Further we find that the wide angle wave field is sensitive to scales in the model space of a factor of 2 to 3 smaller than the first arrivals of the wide angle wave field. Reflected energy from features in the upper mantle has been observed in many deep seismic reflection seismic experiments, and also in some wide angle data sets. The origin of these reflectors is a matter of debate. Several numerical experiments investigate the reflected energy from an upper mantle reflector. A slab of partly converted eclogite shows the best promise of producing the magnitude of reflected waves, that has been observed. Long offset seismic data and converted waves contain information on the sub surface, not present in conventional reflection seismic experiments, that primarily record relatively short offset P-wave data. However longer streamers, and the use of ocean bottom cables to record P to S converted waves, has become normal for commercial seismic surveys. AVO1 analysis of such data has been investigated. AVO analysis is a common technique used to extract information on hydrocarbon content. It is however difficult to extract information on the water saturation using conventional small offset P-wave data, i.e. less than 30 degrees incident angle. Using long offset P-wave data and converted P-S wave data, information on variation in water saturation can be examined. Finite difference modeling of the wave equation has been the primary used tool to investigate the above described geological settings. Finite difference modeling requires large computer power, both CPU and RAM wise. In the beginning of this Ph.D. I had access to a high end super computer. In the last years I have had access to primarily high end Pentium computers. Even with access to supercomputers some problems are either not possible or not feasible to model today. The lack of computer power and the need to model large scale crustal-size models at reasonable frequencies spawned a project, where an efficient implementation of finite difference wave form modeling enabled calculation of high quality seismic wave form data, from lithospheric size models, on a, as of today, conventional 800 MHz Pentium machine with 512 1
Amplitude Versus Offset
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Mb of RAM. The technique also allows efficient and accurate wave form modeling of selected seismic phases. This is valuable as an interpretation tool to e.g. separate seismic phases in a complex wave field. A general description of the research field is presented in section 2. The scientific results obtained through this Ph.D. are presented and discussed with reference to the general research field, in section 3. The main conclusion and considerations are presented in section 4.
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2 General description of research fields Four main research areas, that constitutes the scientific foundation of this thesis, will be described in this section : Finite difference wave form modeling. The sensitivity of the seismic wave field to small scale inhomogeneities. Seismic signatures of crustal and upper mantle structures. AVO Analysis. The subjects are linked to each other in that sensitivity studies of seismic wave propagation is studied in two geological settings of different scale, i.e. crustal to lithosphere scale seismic experiments and AVO experiments. Finite difference wave form modeling provides the most accurate way to model seismic wave propagation, and is thus the tool of choice in seismic sensitivity analysis.
2.1 Wave form modeling The ability to model seismic wave propagation is of great importance. It can be used to justify proposed models of the underground. If a computed seismogram is comparable to an observed data set, the proposed model is consistent with the seismic data. This means that the proposed model is possible, but also just one of many different consistent models. It can also be used with inversion methods to estimate models of the underground, (Tarantola 1986; Zelt 1992). There are several ways of modeling seismic wave propagation. Ray theory is a high frequency approximation that is very efficient. Using inversion methods smooth velocity models of e.g. the crust and upper mantle can be obtained, (Zelt 1992). However only the travel time and amplitude of selected phases can be modeled. Typically only the arrival time is modeled. Features like the wave field around the arrivals and scattered energy are ignored. The most general approach to model seismic wave propagation, is finite difference modeling of the wave equation. The whole wave field, including scattered energy from e.g. small scale heterogeneities, can be accurately modeled. One drawback of finite difference modeling is that it has been and continues to be a large computational task in most cases. 2.1.1 The history of seismic wave form modeling Alterman and Karal (1968) pioneered in the use of the finite difference technique in geophysics, starting with calculations of P- and S-waves in a stratified earth. A similar approach was used by Fuchs and M¨uller (1971). Booth and Crampin (1983) extended this technique to include anisotropy. Boore (1972) used 2nd order finite differences to compute acoustic waves in a heterogeneous model. Kelly et al. (1976) did the same for the elastic wave equation. 3
Kosloff and Baysal (1982) introduced the pseudo spectral schemes, where the modeling is performed in the frequency domain rather than in the space domain. This method has advantages when multiple source models are investigated and when a specific frequency range is of interest. Dablain (1986) investigated the effect of higher order schemes and compares 2nd, 4th and 10th order space, 2nd order time schemes. He found that the 10th order scheme almost eliminates numerical dispersion, and only requires 3 nodes per wavelength, and thus reduces the amount of memory significantly. Strand (1999) showed that some wave phenomena could only be modeled accurately using a high (10th order) spatial finite difference operator. Virieux (1986) proposed a velocity stress finite-difference method, based on Madariaga (1976) for an in-homogeneous elastic media. Levander (1988) evolved this and proposed a 4th order space 2nd order time staggered grid formulation. Robertsson et al. (1994) further evolved the staggered grid formulation to be used on the visco-elastic wave equation. Finite difference modeling of anisotropic wave propagation started in the 90’s, (Igel et al. 1995; Alkhalifak 2000). The computational power of today allows 2D visco-elastic modeling of low frequencies (up to 5 Hz) of lithospheric size models, (Nielsen et al. 2000). A finite difference code specifically designed to run as a parallel job on a super computer with 12 CPUS is under construction at Rice University, (Nielsen, L., pers. comm.). 3D visco-elastic modeling is now being performed on models of size 300*300*300 grid cells (e.g. Olsen, 2000). Facing both the third dimension and the potential to model anisotropic wave propagation, it seems that finite difference modeling of the wave equation will require access to super computers for many years to come. 2.1.2 The finite difference technique The finite difference technique is in short a method to numerically evaluate the solution to a partial differential equation, by substituting the partial derivatives with finite differences. Consider, as an example, a simple hyperbolic differential equation, e.g. Strikwerda (1989) :
Æ Æ (1) Æ Æ where is a function of time and space, is the time, is the position and is a constant. This hyperbolic differential equation can be approximated in a finite grid, by replacing the partial derivatives by finite differences. A point in space and time is described by the discrete point in space, , and the discrete point in time, . The value is thus approximated by the discrete point . There exists many different finite difference approximations to partial derivatives. The set of approximations chosen to substitute the partial derivatives in a partial differential equation is labeled the corresponding finite difference scheme. A simple forward time, forward space finite difference scheme for Equation 1 is given by :
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where and are the sampling intervals in space and time respectively. Typically the evolution of in time is modeled. Equation 2 is trivial to solve for the displacement, , one time step ahead :
Simple looping around this equation will model the evolution of
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Numerical analysis can be used to investigate the consistency, order of accuracy, stability and dispersion of a specific finite difference scheme. Consistency A finite difference scheme must be consistent with the partial differential equation it aims to solve. This means that as the sampling interval in space and time approaches zero, the difference between the finite difference scheme and the differential equation must also approach zero. If this is not the case the chosen finite difference scheme is not valid for the specific differential equation. Order of accuracy Through a consistency check of the finite difference scheme, the order of accuracy of a finite difference scheme can be determined. The wider the differential operator the higher order of accuracy. The width of a finite difference operator, is the number of grid points used to approximate a differential operator. Stability A finite difference scheme must be stable. Some finite difference schemes are unconditionally stable, i.e. the sampling interval can be arbitrarily large. This is most often not the case. Finite difference approximation of the wave equation requires a sufficiently dense sampling of the space domain, defined by the so-called stability criteria. Specifically for the 2D 4th order staggered grid finite difference scheme, the stability criterion is , (Levander 1988). Dispersion Different frequencies travel with different velocity, i.e. propagating waves are dispersive. In addition numerical dispersion is introduced, due the use of a finite grid. It can be minimized by using finite difference schemes of higher order and by choosing a sufficiently dense sampling in the time domain. 2.1.3 Finite difference wave-form modeling The wave equation is a hyperbolic partial differential equation. Three types of the wave equation are used to model the propagation of seismic waves in the earth. They can be either isotropic or anisotropic. Acoustic The acoustic wave equation only considers compressional waves, i.e. P-waves. This can be adequate in some reflection seismic experiments where S-waves can be ignored. In some habitats, e.g. the sun, rotational motions cannot prevail and the acoustic wave equation will accurately describe the wave field (Jensen et al. 2000). For a homogeneous 5
acoustic medium the wave equation in three dimensions is given by e.g. Lines et al. (1999) :
Æ Æ
Æ Æ
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where is the pressure wave field as a function of time and space, and is the acoustic velocity. Elastic Both P- and S-waves are considered by the elastic wave equation. Levander (1988) introduced a 4th order space 2nd order time, hence referred to as , staggered grid formulation that has been widely used in the 90s. Olsen (1994) showed how to implement this scheme in 3D. Visco-elastic The viscoelastic wave equation further includes the absorption, measured by the Quality factor, or Q-factor. The Q-factor is a measure of how much of the seismic energy is absorbed and converted into heat per cycle, and can be defined as, (Sheriff and Geldart 1995) :
(5)
A high Q thus indicates little absorption, which is the case in hard rocks, like seismic basement. Sediments usually have a small Q giving rise to significant absorption. (Robertsson et al. 1994; Sheriff and Geldart 1995) Choice of wave equation The level of complexity increases as we move from acoustic through elastic ending up at visco-elastic modeling. This complexity is reflected in both the seismic wave field as well as in the computational requirements, regarding both memory and CPU time. Therefore it is always desirable to use the simplest modeling scheme that will accurately reproduce the part of the seismic wave field under consideration. Table 1 shows the memory requirements and the number of floating point operations per time step when using the staggered grid formulation for the three wave equations, in 2D and 3D , (Levander 1988). Choosing a visco-elastic modeling code over an elastic will almost double the memory requirements, and will increase the computation time by a factor of 3. Furthermore it can be seen that shifting from 2D to 3D is computationally expensive. The number of grid points increases due to the 3rd dimension, and additionally the number of parameter grids increases by approximately 50 %, and the required computation per grid points increases by a factor of 2. Changing from a 20*20 2D model to a 20*20*20 3D grid, using the elastic wave equation, will thus increase the memory requirements by a factor of
and the computation time will increase by a factor of . The choice of finite difference scheme can also greatly affect the efficiency of the code. To suppress numerical dispersion for the staggered grid scheme, 5 grid points per wavelength is required, (Levander 1988). Moczo et al. (1999) claims 6 grid points per wavelength is required. This is an improvement of a factor of 2 over the 10 grid points per wavelength required by the 2nd order space 2nd order time, , schemes, e.g. Virieux (1986) and Kelly 6
Wave equation Acoustic Elastic Viscoelastic
2D Ngrids 6 8 14
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TABLE 1: Number of parameter grids that needs to be loaded into physical memory, Ngrids, and the number of computations per grid point per time step, CPT, for three wave equations in 2D and 3D. Partly from Robertsson (1994).
et al. (1976). This reduces the required memory demands by a factor of four in two dimensions and computation time consequently decreases. In general the required number of grid points per wavelength will be reduced as the order of accuracy goes up, (Dablain 1986), lowering the demands for physical memory. All in all the computation time will increase. The stability criteria is the same for all three wave equations when using the staggered grid formulation, (Dablain 1986; Levander 1988; Robertsson et al. 1994). For a comprehensive description and numerical analysis of the staggered grid scheme proposed by Levander (1988), see Appendix A1.
2.1.4 Efficient finite difference modeling Probably due to the high demands for computer power, the idea to optimize finite difference calculations is old. Boore (1972) introduced the ’expanding grid’ concept : Only a steadily increasing part of the grid needs to be evaluated, since there is no need to compute the wave field ahead of the first arrival. Vidale (1988), who elaborated on the expanding grid concept to focus only on the first arrival, presented a finite-difference scheme that provided the timing of the first arrival at all grid points at all times. This can be used to define a time window around the first arrival, in which finite difference calculations are performed. Outside this region no calculations are performed. Kvasnicka and Zahradnik (1996) implemented the method described by Vidale (1988). They used ray based theory to obtain the first arrival time at any grid point, and showed encouraging results for geological models with complexity similar to a smooth velocity model obtained from ray theory inversion. Falk et. al. (1996,1998), showed how the finite-difference schemes could be implemented with both varying time steps and grid spacing. Robertsson and Chapman (2000) describe an efficient way to calculate finite difference seismograms, when a model is repeatedly locally changed. The wave field is propagated down to the local region of interest, where the wave field is saved on disk. For each model alteration, propagation of the wave field down to the local region of interest can be saved, as the saved wave field is around the region of interest. 7
2.1.5 Boundary conditions Boundary conditions for finite difference approximations of the wave equations have been heavily studied, to minimize reflections and mode conversion caused by the boundary. One of the simplest boundary conditions is the Dirichlet boundary condition, where the displacement components are forced to be zero at the boundary. This however produces significant reflected energy from the boundaries, e.g. Strikwerda (1989). Clayton and Engquist (1977) proposed absorbing boundary conditions (ABC) based on the par-axial wave equation. This par-axial scheme completely absorbs waves with an incident angle of . However, the amplitude of reflections increases as the angle of incidence increases. Reynolds (1978) used a similar approach. Cerjan et al. (1985) and Sochacki et al. (1987) enlarged the grid and used damping in a buffer zone along the boundaries. Randall (1988,1989) proposed a boundary condition based on Lindman (1975), where P- and S-waves were treated separately. In theory reflected amplitudes should decrease down to 1 % or less for incident angles below
. The method requires Fourier transformation to the wave number domain. Kosloff and Kosloff (1986) suggested a method of viscous damping, also requiring a transformation to the wave number domain. Long and Liow (1990) decompose the wave field into dilatational and rotational strains, and use gradients to identify the directions of propagation. They apply a one-way wave equation to each strain. This technique requires that the medium is homogeneous at the boundary. Higdon (1991) promises perfect absorption of all outgoing waves that have a specific value of the ray parameter. Peng and Toksoz (1995) introduced a fast boundary conditions that performs equally well as those of Higdon at large angle of incidence but much better at small angles of incidence. However two ’optimal’ angles of incidence have to be specified to produce optimal transparency. Berenger (1994) introduced the concept of perfectly matched layers (PML) as an efficient absorbing boundary for electromagnetic waves. Hastings et al. (1995) implemented PML for the elastic wave equation, and showed that PML outperforms the ’optimal’ absorbing boundary of Peng and Toksoz (1995) in homogeneous media. Collino and Tsogka (2001) indicated the PML performs well in slightly heterogeneous media, though PML in theory only applies to homogeneous media. Zhu (1999), suggests an implementation called ’A Transparent Boundary Technique’. He separate the P- and S-waves at the points closest to the boundary and use the P- and S-wave velocity to calculate the radial and tangential displacements. This approach requires the knowledge of the angle of incidence at the boundary. Still today one of the most used boundary conditions for the elastic wave equation is the one proposed by Clayton and Engquist (1977), often together with the damping zone of Cerjan et al. (1985). This is probably due to several factors. 1) It is very easy to implement 2) It does not require any additional input 3) The imperfection of the method is well known, and thus quite easy to identify. Some of the above described ABC’s require Fourier transforms, (Randall 1988; Randall 1989; Kosloff and Kosloff 1986; Long and Liow 1990), while others require some extra knowledge of the propagating direction of the wave field, (Higdon 1991; Peng and Toksoz 1995; Zhu 1999). The PML boundary condition, (Hastings et al. 1995), shows promise in conjunction with e.g. Cerjan type damping, to be very efficient. So far PML has only been used on simple models with little complexity. 8
Most of the described absorbing boundaries apply to the elastic wave equation. When using the visco-elastic wave equation an efficient absorbing boundary is ’built’ into the wave equation itself. Simply decreasing the Q factor in a zone about two wavelengths thickness (approximately 20 grid cells) along the borders, i.e. increasing the absorption, provides efficient absorbing boundaries, (Robertsson et al. 1994).
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2.2 Sensitivity of the seismic wave field small scale inhomogeneities In forward modeling there is no point in including details in the model space for which the modeled data shows no sensitivity. Likewise, using inversion theory one can only hope to obtain a model, for which the data shows sensitivity. Therefore sensitivity analysis should be a natural part of forward/inverse modeling of geophysical data. Seismic data are band limited both in space and time due to the discrete distribution of receivers and the digitized recording procedure. Resolution of intermediate wavelengths have been a matter of dispute. Claerbout (1985) argued that there exists an information gap for wavelengths in the model space from 200 to 1000 m. In a numerical demonstration Jannane et al. (1989) found similar results that an information gap existed, in the range from 60m to 300 m, and supported the thesis that the seismic wave field is sensitive to the short wavelengths of the impedances and the large wavelengths of the velocity field, (Tarantola 1994). Mittet and Houlder (1993) showed numerical results, almost opposite to that of Jannane et al. (1989). This spawned an intense discussion and reply, (Tarantola 1994). Mittet and Houlder (1993) used a method where much weight was given to the first arrivals, where Jannane et al. (1989) gave almost zero weight. As Mittet and Houlder performed their experiment without the first arrivals, i.e. using only the reflected wave field, they found a decrease in sensitivity in the range from 150m to 300m. Neves and Singh (1996) elaborated on this discussion and performed yet another numerical experiment, this time including post critical reflection data. They showed that it is possible to extract information from the intermediate range of wavelengths, especially using critical angle data. The conclusion to all these studies is that a reflection seismic experiment is likely to have drop in sensitivity for an intermediate range of wavelengths from approximately 100m to 300m. When including longer offset data, the drop in sensitivity is less prominent. A ’complete’ seismic dataset contains information about all length scales, limited primarily by the geometrical setup of the seismic experiment. 2.2.1 Partial derivatives Tarantola (1986) showed analytically the 2D P- and S- wave field generated by point perturbations of the elastic parameters. This was illustrated using numerical finite difference modeling, (Hansen and Jacobsen 1996). Figure 1 is a graphical illustration of the amplitude of the diffracted wave field from point perturbations of the elastic parameters, i.e. the partial derivatives of the wave field, for three different parameterizations. Changes in P-wave velocity, P-impedance and give the same result, that energy is scattered out in all directions. Perturbations of density give very little diffracted energy perpendicular to the direction of motion (left-right) for all parameterizations, but the diffracted energy in the direction of motion differs significantly. Density variation for the velocity parameterization (Figure 1a) causes much reflected energy, but almost no transmitted energy, which is the opposite effect of density variations for an impedance parameterization (Figure 1b) where mostly transmitted energy is observed. Density perturbations in a Lam´e parameterization gives both reflected and transmitted P-wave energy (Figure 1b). Variations of S-velocity or S-impedance gives similar results, with 10
energy perpendicular to the direction of motion. Finally a Lame´e parameterization shows almost the same diffracted energy for all three parameters, only and density changes do not cause reflections perpendicular to the direction of motion.
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F IGURE 1: Amplitude of the diffracted P-wave from point perturbations of elastic parameters for three different parameterizations of the wave equation. Each sub-plot is generated by introducing a small perturbation of one elastic parameter at (x,z)=(1000,1000), while the other two elastic parameters for that parameterization are held constant. A plane P-wave traveling from the top to the bottom passing the perturbation creates the observed diffracted wave field. Three parameterizations are considered : A) Velocity (the top 3 plots), B) Impedance(IP and IS) and C) Lam´e parameters From Hansen and Jacobsen (1996)
11
2.3 Seismic signatures of crustal and upper mantle structures. Wide angle seismic data has been extensively used to study the composition of the crust. Mohoroviˇci´c (1909) discovered the crust-mantle boundary by investigating seismic data from earthquakes. As recording equipment got more sensitive controlled sources were used. In the 60’s digital recording enabled the collection and manipulation of normal incidence reflection seismic data. Reflection seismic experiments typically give detailed information on the structures of the underground, while wide angle experiments yield information of the velocity field, typically in form of a smooth velocity model. This is the reason why it is desirable to collect both normal incidence and wide angle data in a seismic experiment with the crust and upper mantle as target. This can and has been done, using OBHs2 and land stations to record wide angle data, at the same time as a reflection seismic experiment is carried out, e.g. Abramovitz and Thybo (2000). 2.3.1 Deep seismic investigations Seismic investigations of the crust and upper mantle are very expensive and both time and man power consuming. Typically a ship has to be hired. Several people have to be present on the ship to properly record data, and land stations must be cared for. After the data are collected a lot of work remains processing and interpreting the data. Several corporations have emerged which had the adequate size to investigate the seismic properties of the crust and upper mantle. Since COCORP3 was started in 1976, several corporations followed, e.g. LITHOPROBE4 , BIRPS5 , ECORS6 , ANCORP7 and DEKORP8 . Working groups like BABEL, MONA LISA also contributed. They provided spectacular images of the crust and upper mantle. The focus of deep seismic experiments has shifted from the lower crust, to include the upper mantle and even the base of the lithosphere. The Russian PNE9 project is especially intriguing, since high frequency nuclear explosions were used as sources. The wide angle data profiles cover several thousands of kilometers across Russia and seismic phases like the tele-seismic phase can be followed all along the profiles, (Nielsen et al. 2000). 2.3.2 The seismic signature of the crust and Moho Figures 2 and 3 illustrate many of the puzzles that geophysicists have been working with for the four decades. First, the crust can not be adequately described by a smooth model, like the ones obtained using ray theory. Plenty of scattered energy is visible especially in the lower 2
Ocean Bottom Hydrophone Consortium for Continental Reflection Profiling 4 Canadian integrated lithosphere program 5 British Institution of Reflection Profiling Syndicate 6 French seismic reflection program 7 Andean Continental Research Project 8 German seismic reflection program 9 Peaceful Nuclear Explosion 3
12
F IGURE 2: Unmigrated line drawing of the DRUM deep reflection line. From Warner and McGeary (1987)
F IGURE 3: Line 1 from the MONA LISA data set
crust. It is usual to see normal incident data showing a crust separated typically into a partly transparent upper crust, on top of reflective lower crust, (Mooney and Meissner 1992). Second, the character of the Moho on both wide angle and normal incidence data does in most cases not imply a simple discontinuity. Sometimes Moho can be described as a clear reflector, sometimes as the base of a reflective zone. Quantification of reflectivity Lamellae’s was the preferred description of many coherent, mostly horizontal, reflections from the lower crust, (Glocke and Meissner, 1976). The identification of such in-situ structures, i.e. direct observations in data, gave rise to a speculation of the origin of the observed crustal reflectivity. One way to clarify important seismic features found on in-situ observations in the seismic data has been to use line drawings, emphasizing special features. A line drawing is however highly subjective, in that a human being typically picks the coherent energy. A new approach for elimination of the subjective input into line drawings is 13
described in the following. Seismic skeletonization - quantifying line drawings Seismic skeletonization is a pattern recognition technique developed to obtain statistical information of a reflection seismic data set, (Li. et al. 1997; Cook et al. 1997). Seismic properties such as coherence, dip and amplitude are gathered in a database, that serves as a base for statistical analysis. A simple application of the technique is to produce independent reproducible line drawings. Line drawings are in general an interpreters view of what reflectors are prominent, produced mainly to highlight certain events in the full data set. If a conventional line drawing cannot be replicated reasonably using seismic skeletonization techniques, one can argue that such a line drawing is too much an interpretation rather than a simplified version of plotting a complex data set. Thus seismic skeletonization should be used on a regular basis in interpretation of deep seismic reflection seismic data. The reason that the technique is not widely used today, can be that seismic skeletonization is not yet a part of the seismic processing tools, like e.g. ProMAX or Seismic Unix, (Cohen and Stockwell 2000). Vasudevan and Cook (1998,2000) show several examples of applying seismic skeletonization to deep crustal seismic data. Wave form modeling of crustal reflectivity Wave form modeling has been the preferred tool to test the seismic reflectivity of models of the crust. Until the 90’s the reflectivity method of Fuchs and M¨uller (1971) found wide spread use, due to it’s efficiency. As computer power has increased, finite difference wave form modeling of large crustal and lithospheric-size models has become feasible. Today it is the modeling technique of choice. Sandmeier and Wenzel (1986) used the reflectivity method to model the seismic response from a crustal-size model with a laminated lower crust. They showed good agreement with a data set from the Black Forest (Germany). Investigating the same data and by using finite difference modeling Gibson and Levander (1988) showed that petrophysical 2D model realizations of stochastic processes with an exponential auto-correlation function produced synthetic data consistent with the recorded data. Andrier (1991) showed, using finite difference wave form modeling, that small lateral variations in a laminated lower crust can produce normal incidence reflections that resembles some of the observed lammellae’s. Andrier (1991) also showed that random anomalies in the lower crust could produce normal incidence reflected wave fields that resembled real data reasonably well. The fact that Sandmeier and Wenzel (1986) and Gibson and Levander (1988) could show quite different models, both consistent with the observed data, shows that some kind of uniform description of inhomogeneities causing the observed scattered wave field was needed. In the late 80’s work intensified to quantify how different kinds of reflectivity patterns could be generated from models containing small scale variations. This spawned great interest in the effect of small scale heterogeneities on the seismic wave field. Stochastic properties of the crust A breakthrough in the description of the small scale variations of the crust, came as Holliger and Levander (1992) found a mathematical description of 14
a section of exposed lower crust found in Northern Italy, the Ivrea Zone. They digitized the exposure on the site in Italy, and found that the digitized maps of the exposure showed trends, that could be well described by a 2D von Karman correlation function, see Figure 4(a). Heterogeneities smaller than are self similar, where K is the wave number and A the correlation length. Larger heterogeneities exist but are not self similar in nature. Holliger et al. (1993) showed the first forward finite difference modeling in a model consisting of a transparent crust and Ivrea type lower crust model. The Lewisian complex in Scotland contain exposed segments of upper to middle crustal rocks. They were digitized the same way as the Ivrea zone exposures was, by Levander et al. (1994). Like Holliger et al. (1993) they showed forward finite difference modeling of the obtained models. Percival et al. (1992) gives a general description of exposed crustal sections. Levander et al. (1994) summarizes the work on both the Ivrea Zone and the Lewisian complex. A number of models are proposed using different Lewisian type models as upper crust and Ivrea type 2 lower crust. It turned out that using Lewisian type 3, see Figure 4(b), as upper crust produced a normal incidence migrated section very similar to the Winch Profile. Goff and Levander (1996) showed how to incorporate ’sinous connectivity’, adding even further complexity the random velocity models of the crust. Larkin et al. (1997) published a model consisting of a crust divided into four different stochastic realizations, on top of a Moho, with a von Karman distributed variation of depth to Moho. Forward modelings of both normal incidence and wide angle data was used to justify the model.
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(a) 1-dimensional Fourier transforms of the 1D von Karman auto correlation. A is the correlation length and K is the 1-dimensional wave number. From Holliger and Levander (1992).
(b) Lewisian 3 upper crustal and Ivrea type 2 lower crust. From Levander et. al. 1994.
F IGURE 4: Realizations of a 2D random model with a von Karman auto correlation function.
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Estimating random media properties from seismic data Pullammanappallil et al. (1997) showed how statistical measures such as the Fractal Dimension and horizontal correlation length could be obtained from seismic reflection data, simply by analyzing autocorrelation functions of parts of a normal incidence seismic section. Huric (1996) used a similar approach to obtain estimates of horizontal correlation length and fractal dimension directly from seismic data. Resolution of small scale heterogeneity Brittan and Warner (1996) showed how ignoring the existence of small scale heterogeneities can lead to significant biases in the velocity field. Heterogeneities with correlation lengths longer than the dominant wavelength of the traveling wave, will produce the largest bias in seismic velocities. Certain geometric arrangements of small scale heterogeneities can lead to an under-estimation of the P-wave velocity in a wide angle seismic data set. High frequency models, suggesting small scale heterogeneity, has been accepted as valid models of the crust. Ignoring the small scale heterogeneities will cause errors in the magnitude of the velocity field. However the information of scales on the crust and upper mantle is primarily obtained using seismic surveys. The seismic method is band limited both by the sampling of the data (time and space) and by the limited frequency of the source wavelet. From a typical wide angle seismic experiment, Hansen et al. (1999) show that for a dominant source frequency of 10 Hz, refracted data from the Moho showed significant sensitivity to scales down to 1 km in the crust, and down to 10 km for random variation in topography on Moho. Bean et al. (1999) investigated the role of seismic bandwidth in a reflection seismic experiment with respect to crustal heterogeneity. They show that band limited reflection seismic data cannot give reliable estimates of correlation length that are larger than the seismic wavelength. Using the technique proposed by Pullammanappallil et al. (1997), Bean et al. (1999) show that the estimated correlation length depends almost entirely on the frequency of the source wavelet, and not the stochastic properties of the underground. Thus as extraction of stochastic properties from seismic data is becoming more widely used, one should remember that the information of the length scales of the underground is limited by the seismic signal itself. 2.3.3 Upper mantle reflections Until the first stunning reflection seismic sections were published only a minority of the seismic community expected to be able to record seismic reflections from the upper mantle. However some of the deep seismic corporations and working groups began to image clear reflected energy from the upper mantle in the late 80’s. Dipping structures in the upper mantle In 1984 BIRPS acquired the now famous DRUM line. Two sub crustal dipping events, the Flannan and W reflectors, could be clearly identified, see Figure 2, (Warner and McGeary 1987). They were first identified on the MOIST line, (Smythe et al. 1982). These reflectors are probably the most extensively studied upper mantle 16
reflectors, (Warner and McGeary 1987; Morgan et al. 1994; Jones et al. 1996; Price et al. 1996; Mosegaard et al. 1997; Steer et al. 1998). In 1989 the BABEL working group measured normal incidence data down to 25 s in and around the Baltic Sea. Line 4 showed a classical image of a dipping structure in the upper mantle from Moho and down to bottom of the section at 80 km depth. Wide angle data were also collected, BABEL Working Group (1993a,1993b). An apparent upper mantle reflection, , was observed in both the normal incidence data and on a wide angle dataset (Landstation 10). Nørmark et. al. (1992) ray traced the position of as a dipping slab in the upper mantle. Krishna et al. (1996) modeled , and indicated that extreme variations in elastic parameters might be needed to obtain the observed amplitudes of . Nørmark (1993) used random anomalies with very high velocities around both Moho and to produce a model consistent with the seismic data. However both proposed models contains close to unrealistic seismic velocities, and are thus hard to interpret as geologically valid. In 1993 The MONA LISA project collected 1112 km of normal incidence data, recorded to 26 s, in the North Sea. Prominent reflectors are visible in all 4 reflection seismic sections, and one sub Moho feature on Line 1, see Figure 3, can be followed from the Moho, dipping down to a depth of 90 km, were it flattens out, close the base of the lithosphere, (Balling 2001). Due to primarily such seismic experiments the existence of large geological structures in the upper mantle is widely accepted. The origin of the observed reflected energy is however still a matter of debate, e.g. Steer et al. (1998) Geological models for upper mantle reflectivity The most widely accepted explanations for upper mantle reflectivity are relict subduction zones and mantle shear zones. In conjunction with these two models seismic anisotropy and fluids can additionally cause reflections. Relict subduction and collision : Warner et. al. (1996) concludes that the only geological valid model that explains all the observed features of upper mantle reflectivity in the DRUM profile, is a remnant slab of mafic high velocity eclogite. Balling (2001) investigated reflection and wide angle data from several deep seismic experiment in the Baltic Shield area, and concluded through studies of seismic data and geological evidence, that for many dipping reflectors a remnant slab of eclogite related to relict subduction and collision dynamics could explain the observed features. Furthermore Balling (2001) suggests that simple geometric shapes like a Moho offset zone, where a subducted slab detaches from the Moho, is likely to be found in regions associated with relict subduction and collision. Shear zones : Warner and McGeary (1987) argued that the bright reflectors in the DRUM profile could be explained by the existence of hydrated minerals and/or mafic rocks in a mantle shear zone. Balling (2001) notes that mantle shear zones might exist where slabs of remnant eclogite exists, since this may be a weak zone that can lead to a shear zone in a subsequent compression or extension. From a purely mechanical point of view, Frederiksen and Braun (2001) show that a mantle shear zone can be generated purely by strain softening, i.e. decreasing viscosity in a region, without changing the elastic parameters. Still on some seismic lines reflected energy can be observed in areas where shear zones are likely to occur, (Frederiksen et al. 2001). Thus, a shear zone in itself does not necessarily cause reflections from the upper 17
mantle, but in conjunction with fluids and/or anisotropy it is more likely to happen, (Frederiksen and Braun 2001). If there is a change in petrophysical parameters in a shear zone, it is most likely to be a decrease in both velocity and density, (Warner et al. 1996) Anisotropy : Upper mantle minerals have very little anisotropy, except for olivine, with an anisotropy of up to 10%, e.g. Anderson (1989). In assemblages of minerals, anisotropy larger than 5% is uncommon. Warner and McGeary (1987) found that anisotropy in upper mantle peridotite, with up to 50% of olivine, will lead to reflection coefficients of less than 0.4 %, which is too small to account for much of the observed upper mantle reflectivity. Eclogite has little or no anisotropy, e.g. Anderson (1985). Thus with respect to the observed zones of upper mantle reflectivity, the literature contains little evidence that anisotropy is a significant factor. However ignoring anisotropy can lead to significant errors in the positioning of Moho, and thereby the position of upper mantle reflectors, e.g. Price et al. (1996). Fluids : Fluids have high effect on seismic waves. The contrast in shear modulus to solid rocks is huge, and causes much reflected energy. Water is likely to exist along an active subducting slab. As a slab subducts the water can migrate to the top of the slab, causing a high reflective zone (ANCORP Working Group, 1999). For an old slab it is likely that water has migrated away from the slab. In an extensional young regime it is likewise reasonable to assume that fluids can have some effect in younger regimes, but less effect in older regimes. 2.3.4 Upper mantle scattering From 1971 to 1990 Russian scientists recorded wide angle seismograms across Russia up to offsets of more than 4000 km, using nuclear explosions as source. The data are referred to as the Peaceful Nuclear Explosion data, or PNE. The PNE data provide detailed information on the upper mantle, that is not available from any other types of seismic data. In the PNE data sets a high frequency teleseismic (5-10 Hz) can be observed out to several thousands of kilometers. The teleseismic is a seismic phase arriving slightly later than the refracted wave from Moho, caused by the existence of small scale variations in the upper mantle. Tittgemeyer et. al. (1996,2000) use reflectivity wave form modeling to show that an upper mantle with randomly distributed velocity fluctuation of only can generate a teleseismic with the characteristics of the observed teleseismic in e.g. the Quartz profile of the PNE data. This is the case even when the velocity gradient in the upper mantle is negative. In this case the refracted wave from the Moho is missing, but the teleseismic present. Nielsen et al. (2000) investigated seismic scattering observed at offsets of approximately 800 to 1400 km in the PNE dataset. They show that small scale heterogeneities with correlation lengths of about 3 to 5 km embedded in an upper mantle zone between 100 and 185 km depth is consistent with the observed scattering. 2.3.5 The seismic lithosphere Reflectivity can be identified anywhere from the crust to the base of the lithosphere. The origin of the reflected energy is still a matter of debate. The tool one can use to investigate the recorded 18
seismic energy is currently wave form modeling. The goal is to produce a model that satisfies both the seismic data and has a valid geological explanation. The latter is necessary due to the non-linearity of the seismic problem, where many very different models will produce similar seismic data. The use of random media, and von Karman functions in particular, has proven an important step forward in characterizing the scattered energy in seismic sections. The base of the theory was founded on direct geological measurements, and is partly why the it has become widely accepted in the geological/geophysical society. A next step is to obtain the statistical information directly from seismic data. A goal in the future is to use wave form inversion techniques on lithospheric-size models including data from normal incidence to post critical recorded energy, possibly combined with extraction of stochastic properties.
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2.4 AVO analysis Koefoed (1955) investigated the variation of the reflection coefficient of plane waves with offset, for variation in Poisson’s ratio. He came up with several rules for this relation, known as Koefoeds rules. He suggested that someday one could be able to extract information about lithology directly, by analyzing amplitude versus offset of seismic data. Some 30 years later Koefoeds rules had become the foundation of a geophysical discipline of its own : AVO or Amplitude Versus Offset analysis. Usually the amplitude of a reflection decreases with offset, but for some lithologies containing hydrocarbons the amplitude increases with offset. This observation has led to a significant interest in AVO analysis, mainly driven by exploration geophysicists. The first commercial use of AVO was as a Direct Hydrocarbon Indicator, DHI. AVO attributes As measures of AVO effects, several so-called AVO attributes have been defined. An AVO attribute can be any indicator of a property linked to AVO. Shuey (1985) showed that for angles smaller than about 30 degrees, a good a linear relation between the reflection coefficient and the square of sine of the angle of incidence is a good approximation :
(6)
is the reflection coefficient at an interface with an incident P wave reflected as a P wave. is the intercept and the gradient. Eqn.6 is often referred to as the gradient intercept relation. By far the most used AVO attributes have been and . For larger angles of incidence a linear relation is not adequate. Using a second order polynomium approximations, can be approximated for larger angles on incidence, introducing a C attribute defined by :
!
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Using the assumption that changes in rock properties are small, Aki and Richards (1980) obtained simple and widely used links between these AVO attributes and the elastic rock properties at a plane interface between two elastic solids :
"" ## (8) " $ # $ " " # $ (9) " ! (10) " " . " is the P-wave velocity above the interface, and where " " " and " "
" is the P-wave velocity below the interface. The other parameters follow the same convention. The reflection coefficient at zero offset, , depends on variation in the seismic impedance.
The slope of the linear relation, , is the function of a much more complex set of elastic properties. ! depends only on the variation in P wave velocity. 20
2.4.1 Gradient/Intercept cross plots Simple plots of a seismic section of ’s and ’s have helped to indicate presence of hydrocarbons. Looking at sections of AVO attributes can however be a daunting task. Another way to look at AVO attributes is by cross plotting seismic attributes. There exists a linear relation between compressional and shear wave velocity in non hydrocarbon bearing rock samples, (Castagna et al. 1985). Thus a cross plot of versus from a dry well, shows a linear relationship. Likewise, cross plotting of A versus B for a non gas bearing formation, usually forms a well defined linear trend, known as the background trend, e.g. Foster et al. (1997). Hydrocarbon bearing rocks plot anomalous to these linear trends. Foster et al. (1997) investigated the background trend, and showed that the background trend is closely linked to the ratio. Specifically they found that background trend will tend to fall along a line through the origin with a slope of : . A well log from the North Sea confirmed their result. They labeled this background trend the ’fluid line’ in an A-B cross plot. An example of background trends or fluid lines i shown in Figure 5(a)
(a) Background trend, i.e. the fluid line. A versus B, assuming constant from Castagna et. al. (1998)
(b) Indication of fluid line and Rutherford and Williams (1989) AVO classification
F IGURE 5: Background trend and AVO classes, from Castagna et. al. (1998)
DHI’s in AVO cross plots Points plotting off trend to the fluid line are anomalous. This is indicative of a change in the variation of rock properties, i.e. a possible DHI. Gas sands can be classified into 4 classes, each giving a specific AVO signature, e.g. Rutherford and Williams (1989), Hilterman et al. (2000) : Class I No bright spot. Little amplitude brightening. Class II Maybe bright spot. AVO analysis must be performed on pre-stack data. 21
Class III Produce bright spot. Reflection magnitude increase (amplitude decreases) and brightens with offset. Class IV Produce bright spot. Reflection magnitude decrease (amplitude increases) and brightens with offset. Figure 5(b) shows the position of the 4 classes of gas sands in an A-B cross plot. Class I gas sands are hard to identify, while Class III and IV are best suited for recognition in cross plots. Gas has a high fluid compressibility, and heavy oil has a low compressibility. Figure 6 shows that gas produces the greatest deviation from the fluid line, while heavy oil is close to the fluid line, i.e. giving no anomaly. A change in porosity can be seen as a shift parallel to the background. Thus the position of a Class III gas sand is determined by the fluid compressibility, giving the distance to the background trend, and the porosity giving the position parallel to the background trend.
F IGURE 6: Effect of porosity and fluid compressibility of a Class III gas sand in an A-B cross plot, from Foster and Keys (1999)
2.4.2 Additional AVO attributes Class III and IV gas sands can usually be identified in an A-B cross-plot, Class II is harder to identify, and usually requires AVO interpretation of pre stack data, and Class I is almost invisible. In the recent years longer offset seismic data and P to S converted data have been recorded by many oil companies. These types of data are relatively new in AVO analysis, and are currently the subjects of intense study. Hilterman et al. (2000) show that using angle stacks from long offset P-wave data, Class I AVO anomalies can be converted into Class II AVO anomalies. This enhances the possibility of detecting hydrocarbons. Figure 7 shows that the P-wave velocity from a Class III gas sand with 10% water saturation is very similar to gas sands with higher water saturation. A completely water saturated sand 22
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F IGURE 7: Variation in elastic properties for a Class III gas sand as a function of water saturation ( ). From Enclosure A5
section has a distinctive higher P-wave velocity. Thus it is quite easy to determine that a Class III type gas sand contains hydrocarbons, but not the amount. This is often referred to as the fizz gas problem. Drilling of a well is extremely expensive, and water saturation around 80% and higher is not commercially viable. From a commercial point of view this is a problem of great importance. Fuping et al. (2000) investigated the effect of using P to S converted waves with respect to fizz gas, and showed encouraging results when using both the reflection coefficient of reflected P and reflected S waves. Jin et al. (2000) shows an example of how S-wave velocity and density was estimated using AVO analysis and inversion on an ocean bottom seismic dataset. An application of their technique could very well be to discriminate fizz gas from commercial gas, since a good estimate of density was found. As seen on Figure 7 density has an almost linear relation to water saturation. AVO attributes are but one set of many seismic attributes like velocity, instantaneous phase, energy etc. For list of the most used seismic attributes today see Brown (2001). Castagna (2001) illustrates how seismic data, using seismic attributes, has been used to do lithological analysis. This is exemplified by e.g. Bach et al. (1999), who show how cross plotting of Poisson’s ratio versus the acoustic impedance can help to discriminate between wet sand, shale and gas (Figure 8). As indicated in section 2.2 the resolution increases as more parts of the seismic wave field are considered. With access to long offset P-wave data and P to S converted data there is hope that the resolution in reflection seismic experiments will increase. AVO analysis is already used to extract information from these high resolution dataset, and is likely to continue to be an important and powerful tool as a DHI.
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F IGURE 8: Cross plot of acoustic impedance versus Poisson’s ratio used as a lithology indicator. Crossplot of acoustic impedance versus Poisson’s ratios, gray toned with the gamma-ray values. Large circles indicate gas sands, medium circles indicate oil sands and water sands, small circles indicate shaly sands, triangles indicate sandy shales (large) and shales (small). Plot and data are from Bach et. al. (1999)
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3 Presentation of results The results derived in this Ph.D. dissertation have fallen in four areas, described in detail in enclosures 2-5. This section contains the principal results from these enclosures, as well as results for numerical experiments of efficient finite difference modeling, not presented elsewhere.
3.1 Main phase modeling Finite difference simulations of even simple models can lead to seismic synthetics that are complex and difficult to interpret. This is an advantage of finite difference modeling, but also a reason to choose less detailed modeling techniques in some cases. For an interpreter it can be very hard to correlate observed seismic energy with a specific phase or part of a model. The timing of e.g a reflection from Moho, , can be quickly obtained using ray tracing, but the wave field currently cannot. To model the wave field of one usually models all other seismic phases. Another problem with finite difference modeling is that it is computationally demanding. Nielsen et al. (2000) showed wide angle data computed on a lithospheric size model where a visco-elastic finite difference code was used on a model with 8192*1024 grid points (2000 by 250 km). Data were computed to 300 seconds using a Ricker wavelet with a peak frequency of 2 Hz as source. Such models took more than one week to model on a super computer of the late 90’es. In this case the observed seismic data, from the PNE dataset, contained frequencies of 5-10 Hz, (Tittgemeyer et al. 1996), but such high frequencies were not feasible to compute. Furthermore most part of the computed wave field was never investigated since it was the wide angle data that was of interest. We have developed an efficient way to model seismic main phases using wave form modeling, i.e. Main Phase Modeling or MPM, that deals with both of the problems described above. It is both computationally efficient and allows modeling of selected seismic phases. As an example we have implemented the concept using a 2D 4th order space 2nd order time staggered finite difference scheme, (Levander 1988). The technique could however be used on any space-time finite difference approximation of the wave equation. Paper A2 emphasizes the implementation of MPM, while this section has emphasis on the practical uses and performance of MPM. 3.1.1 Implementation The basic idea is that the wave field does not have to be computed in the whole model space at all times, to model a specific phase. Figure 9 illustrates this efficient modeling of the wave field representing the reflection from a complex interface. The wave field is only modelled in a moving zone around the wave field that generates the reflected wave field, i.e. the down going wave field and the reflected wave field. In this case the moving region is a small rectangular grid of constant size, hereafter the small moving grid. 25
STEADY LARGE GRID MOVING SMALL GRID
REFLECTIVE LAYER
F IGURE 9: MPM: Main Phase Modeling. A box is moved around in a larger model, following the ray path of one selected phase.
Boundary conditions Because the zone of computation follows the wave field of interest the quality of the absorbing boundary conditions is very critical. Several implementations of elastic finite difference codes use a combination of the Clayton and Engquist (1977) absorbing boundary condition and the multiplicative absorbing boundary zone procedure of Cerjan et al. (1985). This procedure is used in the current implementation. Figure 10 shows the effect of the boundary in the direction of movement, for a box moving approximately 0.66 gridcells per timestep (tracking a P-wave). A free surface is used at the top, and Clayton and Engquist absorbing boundaries are used elsewhere. The boundary to the left shows almost no reflected energy compared to the amplitude of the reflections from the bottom and top boundary. In effect the boundary in the opposite direction of movement, where the wave field is moving out of the box, is almost completely transparent. 0
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F IGURE 10: Effect of disappearing grid-space, opposite the direction of movement. Snapshot of horizontal displacement at time 120(a), 360(b), and 600(c) ms. Free surface at the top, 1st order Clayton and Engquist borders at the bottom, and no damping along borders. The gain is the same for all plots, about a factor of 100 larger than the peak amplitude of (a).
Figure 11 compares the boundary reflections for stationary boundaries to reflections from moving boundaries with wave field zero padding. Three cases are examined, a fixed box, a box moving with the speed of the down going S-wave and finally a box moving with the speed of the P-wave. 26
The Clayton and Engquist condition is not effective when the main phase moves almost parallel to the boundary, and also the multiplicative absorption condition may cause problems. Figure 11.a.2 and 11.a.3 illustrate reflections from the boundary due to the high angle of incidence in a non-moving box. Figure 11.c4 illustrates the problem along the steady boundary when the box is moving. Therefore, the multiplicative absorbing zone must be wider and more gentle at boundaries which are almost parallel to the main phase propagation direction. In the direction of main phase propagation the boundary moves away from the energy entering the multiplicative absorption zone. Hence the efficiency of the zone is improved by the zone movement.
In Figure 11.b.2 a P-wave, %& , is moving out of the box in the direction of movement, %& . Reflected energy traveling along the bottom boundary is visible, but all in all artificial reflections are less prominent for the moving box compared to the conventional fixed box.
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F IGURE 11: Absorbing boundaries, using a exponential damping zone of 30 grid cells (3.75km) and a damping factor at the edge of 0.95 . a) No moving box, b) box moving down at = =3.461 km/s, c) box moving down at = =6 km/s. The wave fields are strongly amplified to visualize the weak absorbing boundary effects. Maximum amplitude of the main phase is eight times the saturation color.
There are two main applications of MPM : 1) MPM can lead to efficient modeling of some types of dataset, e.g. wide angle seismic data, and in general large models where only a part of the wave field is of interest. 2) MPM can help separate seismic complex phases from each other. In the following three sections these applications will be addressed. 27
3.1.2 Removal of surface waves Surface waves are computational expensive due to their relative small velocities. To minimize numerical dispersion a small velocity implies a decrease in spatial sampling interval leading to larger models, increasing both memory and CPU demands. Furthermore, in many situations geophysicists are not interested in surface waves at all. MPM can easily be used to overcome both problems. Figure 12a shows the recorded wave field, from a stochastic zone embedded in a homogeneous crustal type model. It is clear that numerical dispersion from the surface waves heavily distort the reflected wave field from the stochastic zone. Figure 12b shows the modelled reflected wave field using MPM, and Figure 12c shows the difference to Figure 12a. A box with a vertical size some samples smaller than the original is used. At approximately 2.2 s of modeling, the box is moved downwards for 0.2 s and then upwards 0.2 s again to be positioned at the top. This causes the moving box to be positioned below the surface waves, which removes them from the modelled wave field. It is clear that most of the surface waves have disappeared from the wave field measured at the top of the model. 0
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3.1.3 Separation of phases Even for rather simple models the modelled full wave field can be very complex. And for models with stochastic content, it can be hard to determine which part of the model causes the observed wave field. MPM can be used as an analytical tool to investigate individual phases. Figure 13 shows a simple crustal model, with two anomalies. Such a crustal model, will typically cause the three main crustal phases, , and . Figure 14a shows the wide angle wave field modelled using a steady box. The three main phases arrive almost at the same time and location. Even for such a simple model, it is hard to identify the three main phases. 28
The individual phases are modelled using a box with a movement path defined from the ray paths illustrated on Figure 13. Figure 14b and 14c show the computed seismic data following and / respectively. Figure 15 shows snapshots as the moving box follows the ray path of . Some effects of non absorbing boundaries are evident, but in general one can follow a specific phase.
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3.1.4 Efficient modeling of wide angle data Modeling of wide angle seismic data is a special case of wave form modeling, since most of the modelled wave field typically is not used. The usual way to interpret wide angle data is by studying reduced time plots of the seismic wave fields, i.e. the time axis is defined by . e.g. in the MONA LISA survey, wide angle dataset were continuously recorded, but reduced travel time plot typically only included approximately 16 s TWT, focusing around the refracted Moho phase . MPM can dramatically reduce computer requirements for such models, and 29
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produce reduced time wave field plots indistinguishable from the wave field computed using conventional finite difference full wave form modeling. Figure 6 in Paper A2 illustrates a model of the lithosphere, based on examinations of the PNE data sets, (Tittgemeyer et al. 1999). The size of the model is km. A peak frequency of 2-3 Hz leads to a spatial sampling interval of km, which defines a model grid of points, when using the 4th order space 2nd order time staggered grid scheme, (Levander 1988). To model Pn out to the maximum offset, 192 s must be recorded. This is equivalent to 12000 timesteps as the required time-sampling rate is seconds. Figure 16a shows the reduced time plot coming from a wave form modeling using a steady box. Figure 16b shows the corresponding wave field, when a moving box with width=24 km or 1000 grid points is used. Down to 26 s it is virtually impossible to see any difference. Figure 16c shows the difference between the two modelings amplified by a factor of 100. In this example the center frequency is deliberately chosen very small. This was done in order to run the full model. If one wants to model with higher frequencies, the model gets bigger, and it can come to the point where MPM modeling is the only solution, even using near future super computers. MPM increases the size of the model, or the frequency range, that can be computed on any computer at any time. Even as computers gets bigger and faster all the time, MPM will always enable one to model wide angle models faster, and/or with higher frequencies. Thus the MPM technique will remain useful for many years to come. 3.1.5 Discussion Comparison to other methods Compared to other types of efficient time domain finite difference modeling techniques MPM is different in a number of ways. The expanding grid intro30
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31
duced by Boore (1972), where the wave field ahead of the first arrival is not considered, is to some degree a simplified version of MPM. In MPM only the part of the wave field of interest is modelled. Compared to the speed up method, MPM can seem rather simple, (Vidale 1988; Kvasnicka and Zahradnik 1996). This is its strength. The present implementation of MPM has the advantage that other phases than the first arrival can be modelled. One does not need to know the exact position of the wave front. MPM works equally well in very simple and very complex models. The speed up method models only the first arrival, and needs the exact position of the wave front. For the very specific case when only the wave field around the first arrival is of interest, the speed up method will outperform MPM, in pure CPU time, but not in memory requirements. Two modeling techniques has to be combined to use the speed up technique, which makes it less accessible for the end user. MPM can be implemented in any existing finite difference time domain scheme with only little change to the original code, see paper A2. Limitations and problems When following a specific phase the box should be as narrow as possible. This calls for good absorbing boundaries, since the wave field will be relatively close to the edges of the model at all time. Recently Collino and Tsogka (2001) showed an implementation of the PML absorbing boundary, in a formulation of the elastic wave equation, as used here. The presented results were quite convincing, though only slightly complex velocity fields were considered. It would be interesting to consider an implementation of the perfectly matched layer technique (PML, Hastings et al. (1995)), in conjunction with Cerjan type damping, (Cerjan et al. 1985). When using a visco-elastic formulation, absorbing boundaries are naturally imposed simply by decreasing the Q-factor in the boundary regions. Robertsson et al. (1994) showed a viscoelastic implementation of the same finite difference scheme used for the modeling presented here. Such an implementation would be desirable both for the enhanced absorbing boundaries, and for inclusion of the Q-factor in the modeling scheme. It would be straightforward to implement a varying size of moving zone with time. The main reason not to, is for simplicity. As the code is presented here, the user only needs to give the approximate position of the moving zone for some timesteps, or simply a constant speed and direction of the zone. This means the code can be used with only little preliminary computation before the modeling can begin. A 3D implementation of MPM could have interest to a number of scientists. E.g. in the numerical 2D modelings presented later in this thesis, section 3.3.3, it cannot consider whether the observed reflectivity is actually positioned where it is imaged in the normal incidence section, and further the studied effects of random velcoity fields, might show different results in a 3D model. Conclusions Main Phase Modeling has two major applications : It can be used as an analytical tool to separate complex wave fields of mixed phases. Today no other tool exists that can model the wave field of a specific phase. 32
For wide angle wave fields, the use of a moving zone can dramatically decrease the need for both CPU time and memory requirements, while at the same time produce wave field modeling of a quality virtually indistinguishable from conventional finite difference modeling. The reduced calculation time and memory requirements advocates the use of MPM as the tool of choice, especially when considering wide angle simulations. Furthermore MPM is easy to implement in existing time domain finite difference codes, and easy to use. Once implemented the code can behave exactly like a conventional finite difference wave form modeling program.
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3.2 Sensitivity of seismic wide angle wave field and first arrival times to fine scale crustal structure and Moho topography Deep seismic wide angle data are commonly used to obtain velocity models of the crust and upper mantle, (Christensen and Mooney 1995). In general, main phase arrivals are picked, and a large-scale velocity model is obtained using forward modeling, (Cerveny 1985), or inversion methods based on ray theory, e.g. Zelt (1992) . Only relatively long wavelength structures can be resolved using a ray theoretical approach. The fine scale structural information is contained in seismic wave forms. Based on the qualitative fit between the observed seismic data and modeling response from geologically derived models, Holliger et al. (1993) and Levander et al. (1994) have proposed very detailed models containing fine scale structure. These models are of stochastic nature and show fine scale variability both in velocity and in Moho topography, (Dainty and Schultz 1995; Larkin et al. 1997). Seismic data contain information about the sub surface structures, which is limited by the bandwidth of the source wavelet and recording offset, Neves and Singh (1996). The study of Neves and Singh (1996) was limited to a deterministic model and did not include turning rays and refracted arrivals, commonly used in crustal studies. Appendix A3 is an investigation on which spatial wavelengths in the velocity field are sensed by travel time picks and the full wave field of wide angle seismic data, respectively. This quantitative study is performed by comparing the modelled output from progressively filtered reference models of the crust, using an -norm. Reference model A model closely related to the stochastic Ivrea model proposed by Levander et. al. (1994) was chosen for the reference velocity structure of the crust. Variation on topography on Moho was inspired by Larkin et al. (1997). Two test reference models were considered : (a) A Fully Stochastic model (FS) where the stochastic crust and depth to Moho are superimposed on the deterministic model and (b) a Deterministic Crust model (DC) with a purely deterministic crust and a stochastic depth to Moho. A part of the reference model is shown in Figure 17(a). Synthetic Models In order to study the sensitivity of the wave field and first arrivals to the random part of the velocity field, the reference model was low pass filtered with varying cut off wave numbers. A spatial low pass filter was designed to produce velocity models with a good compromise between wave number definition and ringing in the space domain (Gibbs phenomenon, e.g. Bracewell (1978)). The filtering was performed in the wave number domain and characterized by the critical wave number, , where is the corresponding critical wavelength in the space domain. is the first wave number that will be affected by the filtering. The filter was 1.0 from 0 to and decreased as a cosine taper from to . Eight filtered models with =0.3, 1, 3, 10, 33, 100, 333, 1000 km were considered for the DC model, and ten models with =0.3, 1, 2, 3, 6, 10, 33, 100, 333, 1000 km were considered for the FS model. 34
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(b) Computed seismograms for the non filtered fully stochastic model. The solid box denotes the region in which the first arrival times and the wave fields are expanded in Figure 18(a) and 18(b). The dashed box denotes the region in which the misfit calculations have been done.
F IGURE 17: Reference model, and computed seismic wave field.
Modeling Synthetic seismograms for a shallow explosive source at x=0 km were computed using a visco elastic finite difference code (Robertsson, Blanch, and Symes 1994). The model was discretized with a spatial distance of 50 m in a 3000x720 grid. The source pulse was a Ricker wavelet with a dominant frequency of 10 Hz, thus with the dominant wavelength in the lower crust being . Part of the resulting data from modeling of the FS reference model is shown in Figure 17(b). The part of the section used for the sensitivity analysis is indicated in Figure 17(b) as a dashed rectangle. For detailed qualitative comparison the first arrivals from the solid rectangle in Figure 17(b) are expanded in Figure 18(a) and the wave fields are expanded in Figure 18(b) for =(0, 1,3,10) km, since this is the interval where major changes happen. 3.2.1 Sensitivity analysis A quantitative comparison of the computed seismograms was performed using the ' norm for both the travel time and the wave field with respect to the data resulting from the unfiltered model. The misfit calculations, were performed for offsets between 110 and 150 km. Figure 18(a) shows a close-up of the first arrival times for the wide angle data, for offsets from 110 to 120 km for 4 different models. The misfit for the full wave field was computed for 35
the reduced travel time interval 8.3 s
9.0 s. Figure 18(b) shows the full wave field for the wide angle data at the same offsets and as in Figure 18(a). The misfit functions, scaled by the misfit for a completely random model, are shown in Figure 19. Seismograms for the unfiltered models can be considered as real data without any distortion, whereas seismograms for the filtered models can be regarded as observed data. We assume pragmatically that a misfit of less than 5% would be due to noise, and that one of 20% is significant.
Fully Stochastic Model Considering the first arrival times of the FS model, it is evident, from Figure 18(a), that only minor differences are apparent between the reference model and the model at . For , the misfit between the first arrival times becomes significant, which can be observed in the full wave field seismograms, Figure 18(b). The wave form changes significantly for . Figure 19a shows the corresponding RMS misfit. For the RMS misfit of the wave field misfit is about 20% , thus we do not expect to find sensitivity for wavelengths smaller than . For the first arrival times, the 20% RMS misfit level is reached at , indicating that no sensitivity should be found for wavelengths smaller than . Thus for the FS model, there seems to be a factor of about 3 between the sensitivity obtained from the full wave field, as compared to the first arrival times only. Constant Crust The effects of topography on Moho can be investigated from the results of the modeling on the DC model. Figure 18(a)b shows that no apparent change is noticeable for the first arrivals until . The corresponding wave field shows similar features. For , the misfit is below 5%. For the misfit is close to 20% for the first arrival times and higher for the wave field. This is just the level at which the qualitative comparison showed difference, indicating that the RMS misfit of 20% is a good choice for ’minimum acceptable noise’. RMS misfit reaches the 20% level at and for the first arrival times and wave field, respectively. Thus the first arrival times are sensitive down to wavelengths of 11 km, and the full wave field is sensitive down to wavelengths of 5.5 km for the lateral variations on the topography on Moho, indicating a factor of 2 in difference in sensitivity. 3.2.2 Discussion It is now widely recognized that the stochastic content in velocity models accounts for much of the energy in the recorded seismic wave field. The first stochastic models were produced by digitization of exposed crustal structures, e.g. the Ivrea Zone, (Levander and Holliger 1992). Forward modeling of these models produced synthetic seismic data that resembles the observed data. Even though seismic data does not show sensitivity to the smaller wavelengths of the velocity model, the velocity model is still well constrained since digitized maps were used. 36
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F IGURE 18: First arrival times and wave field around the first arrival time, for progressively filtered models
However when no digitized maps are available, the comparison of synthetic forward modelled seismic data with real data is not enough. When only seismic data are investigated, the corresponding velocity model should contain only those wavelengths that can actually be resolved. The work presented in this section investigated the sensitivity of the full wave field to small scale variations in the crust and on Moho topography. This clearly illustrated that including small scale variations below a critical wavelength does not affect the wide angle wave field at all. Therefore there is no point in presenting velocity models containing small scale variations with wavelengths smaller than the critical wavelength. Of course the inclusion of other seismic data, or geological data, like digitized maps, can justify velocity models with smaller wavelengths. When trying to obtain stochastic properties directly from seismic data one should also be careful. Seismic data are band limited. Due to finite frequency there will be an upper limit, and due to the finite spacing in the geometric setup, there will be a lower limit to what correlation length can be extracted from the seismic data, (Bean et al. 1999). Correlation lengths outside this range does not describe the model, but the limits in the seismic experiment setup. In the last 10 years stochastic modeling has changed our understanding of the composition of the crust. Stochastic fields have been highly successful in accounting for the observed scattered energy in crustal seismic data. However in some situations small scale heterogeneities has been introduced that has no resolution. This should be avoided.
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3.3 Upper mantle reflectors Deep seismic investigations often image spectacular dipping upper mantle reflectors, as discussed in section 2.3.3. Reflection from these dipping structures are sometimes found in coincident wide angle data, but not always. The most widely addressed models to explain the observed reflectivity are relict subduction zones (Calvert et. al, 1995; Warner et. al., 1996; Balling 2001) and mantle shear zones (Reston, 1990; Braun and Shaw, 1998; Frederiksen et. al., 2001). Paper A4 investigates the reflected wave field from many kinds of perturbation of a typical upper mantle model, using Zoeppritz equations and full wave form modeling. This section will focus on the part of the paper that investigates the observed reflectivity from dipping structures in the upper mantle, as seen in e.g. the BABEL and the MONA LISA datasets. 3.3.1 Elastic models of upper mantle dipping slabs Reflected seismic energy is caused by a change in the elastic parameters, P- and S-wave velocity and density. Thus to be able to model the seismic wave field from the two considered geological models, the elastic parameters within the dipping slabs, the shear or subduction zone, and the surrounding material must be known. Typical upper mantle Based on among others, Christensen and Mooney (1995, Fig. 4), we use a reference model for the upper mantle consisting mainly of peridotite, with %& and density % & . S-wave velocity is chosen as , following Sobolev et. al. (1994). Shear zone A shear zone can take many forms, seen from a petrophysical point of view. Rocks can be squeezed, crunched and metamorphosed and fluids can be present. All of these factors can cause a change in elastic parameters, and may generate reflectivity. Density, as well as P-wave velocity, is most likely to decrease in a mantle shear zone (Warner et. al., 1996). We have investigates models for variation in Vp, Vs, density and impedance. Relict Subduction Zones Oceanic crust consists largely of rocks of basaltic composition. When subducted to great depths it transforms into eclogite. This will cause the petrophysical parameters to change. Depending on the degree of transformation from basalt to eclogite the density will increase. Anderson (1989, Table 3-8) lists measured densities and seismic velocities for different eclogites. Densities are generally large, % & . Christensen and Mooney (1995) report densities of eclogite as % & . Thus eclogite is consistently found to have a density larger than that of normal peridotite. Vp velocity of eclogites shows a much larger range. According to Mooney and Christensen (1995) most eclogites have a Vp of around 7.9 km/s. Anderson (1989) lists velocities from 8.2 to 8.6 km/s and Warner et. al. (1996) discus velocities up to 8.6 km/s as reasonable if a large amount of garnet is present. Rudnick and Fountain (1995) shows that eclogites have a range of Vp from about 7.7 km/s to 8.6 km/s. 39
Measurements of S-wave velocity are less common, but in general the Vp/Vs ratio is larger in eclogites, than other types of upper mantle rocks. Table 3-8 from Anderson (1989) lists Vp/Vs ratios as high as 1.85, substantially larger than a typical upper mantle Vp/Vs ratios of 1.65-1.8. In general the density of a slab of mafic eclogite in the mantle is larger than that of it’s surroundings. We use . The P-wave velocity has a wide range of velocities. We investigate P-wave velocities from 7.6 km/s to 8.6 km/s. For S-waves we use a Vp/Vs ratio of 1.8. The reference models we use for numerical studies is shown in Figure 20a and 20b. Variations of these models are considered in the following. 3.3.2 Wave form modeling of dipping upper mantle structures Both the elastic parameters and the geometry of the geological structures affect the seismic wave field. In the following the wide angle seismic wave field from simplified models of subduction zones and shear zones are investigated. Fig. 20a shows a simplified model of a relict subduction and collision model. Balling (2001) suggests two main characteristic features for such regimes. A Moho offset and a remnant slab of eclogite. These 2D features are imposed on a simplistic 1D reference model of the crust and upper mantle. Fig. 20b shows a simple model of an upper mantle shear zone. The computed wave field from a variety of models based on the two models in Fig. 20 is shown in Fig. 21. Specifically Fig. 21(a) show the computed wave field from a model with no moho offset and no slab in the mantle. Both models were inspired by discussions about, and data from the BABEL and MONA LISA projct and the MOIST line (Balling, 2001; Smythe et. al., 1982) 0
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A relict subduction slab Fig. 21(b) shows the modeled wide angle synthetics from a model with only a Moho offset imposed on the reference model. can be identified by a series of distinct phases compared to the one distinct arrival from the reference model, Fig. 21(a). Fig. 22(b) shows the difference to the reference model for the computed wave field. When adding a slab of high velocity eclogite, Vp=8.6 km/s, a high amplitude phase following can be identified, Fig. 21(c). The amplitude of the reflected wave from the slab is of the same magnitude as and , and a factor of 10 larger than the first arriving . Not all the subducted material needs to transform into eclogite. In particular, if sufficient amounts of fluids are not available, total transformation into eclogite may not occur, (e.g. Austerheim et. al., 1997). Fig. 21(d) shows the computed response, from a model with random distribution of 50% eclogite and 50% oceanic crust. is followed by an up to three second tail of reflected energy. The peak amplitude is smaller than that of a homogenous slab of high velocity eclogite, and at the same level as the first arriving . Both high, medium and low velocity eclogite models has been investigated, and they all give rise to a reflected wave field with significant magnitude of amplitude, an order of magnitude larger than that from a homogeneous slab of eclogite. A shear zone If shear zones can be identified in reflection seismic section, then a change in velocity and density must occur. A decrease in velocity and density is the most likely scenario in a shear zone, (Warner ,1996). Fig. 21(2) shows the computed wide angle wave field from a shear zone, (Fig. 20b), where the velocity and density are decreased 6% relative to the reference upper mantle model. The negative reflection coefficient across the boundary to the shear zone apparently causes an extinction of and the resulting wide angle wave field shows an apparent arriving approximately 0.4 seconds later than the from the reference model. This will obviously cause depth errors if one determines a velocity model, using the apparent as the real . When applying a random distribution of 50% reference model and 50% of the shear model from the previous exampled, almost no scattered wave field can be identified. The computed wave field is very close to the wave field computed for the reference model. A Moho offset alone gives rise to a complex arrival. The addition of a slab of high velocity eclogite causes a strong phase following the phase, with amplitudes comparable to that of and . A partially converted slab gives rise to a zone of diffracted energy following comparable in amplitude to . A homogenous shear zone, with a drop in velocity and density, causes a delay in the apparent but otherwise no sign of the shear zone. A shear zone with randomly distributed properties, causes almost no wide angle seismic energy. 3.3.3 BABEL The BABEL project (Baltic and Bothnian Echoes from the Lithosphere) is a marine deep seismic experiment on the Baltic Shield and its southwestern margin, with data acquisition in 1989. 41
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F IGURE 21: Wide Angle wave field for crustal models based on the models in Fig. 20. a) The reference model, i.e. a flat Moho and no slab. b) 12 km Moho offset at 100 km offset, beginning at 60 km offset. c) Like b) including a homogeneous high velocity slab (Vp=8.5 km/s). d) Like b) including a slab with a random velocity distribution, of 50 % eclogite (Vp=8.5 km/s and 50 % lower crustal material (Vp=7.0 km/s)). e) Homogeneous Low velocity shear zone (Vp and density are down 6%). f) Like e) but with a random velocity distribution in the shear zone consisting of 50 % mantle material. The gain is the same for all plots.
Both near normal incidence reflection and wide angle data (using land stations) were recorded, (e.g. BABEL Working Group, 1993). Unusually strong energy following the prominent crustal phases can be observed at station 10, a land station positioned on the island of Bornholm in the southern part of the Baltic Sea, (Fig. 23). The phase has been labeled , Upper Mantle Reflection. In the same area dipping reflectors are observed beneath Moho, (Balling, 2001). The recorded wide angle energy, Rum, was unusual for two reasons: The very high amplitude and the relative short offset at which the phase could be identified. Nørmark et. al. (1992) used ray tracing to position a dipping reflector in the upper mantle consistent with the arrival time of Rum, and consistent with a less prominent dipping feature in the reflection seismic data set. There has been efforts to determine which models can explain the existence of Rum. Krishna et. al. (1996) suggests that Rum is a wide angle reflection from the bottom of a low velocity zone (LVZ), just above a high velocity slab. A large contrast between the LVZ and the high velocity slab is efficient to produce the relative high amplitude of Rum. Nørmark (1993) indicated that a region of randomly distributed velocity anomalies in a band around Moho and the upper dipping reflector, would create phases comparable to the observed data. Balling (2001) argues that Rum is associated with a compressional tectonic regime, where a remnant slab of subducted eclogitized crust is a likely cause of the observed reflectivity. We 42
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investigate the seismic response from a model consisting of the raytraced velocity model and position of the source of reflectivity of Rum of Nørmark et. al. (1992). Our model includes a 7 km thick dipping layer of various elastic properties resembling a slab of eclogite. Two different models are investigated. A homogenous (Fig. 24(a)) and a randomly distributed velocity field (24(b)) in the slab. The randomly distributed velocity field is a bimodal von Karman field with fractal dimension 2.75, horizontal and vertical correlation lengths, km. A suite of models with different velocities are investigated. Eclogite with both high (Vp=8.6 km/s), medium (Vp=8.3 km/s) and low (Vp=7.9 km/s) P-wave velocity are modeled. For the bimodal field both velocities typical in the uppermost mantle (Vp=8.1km/s) and untransformed material (Vp=7.0 km/s) was considered as ’background’ material. The upper mantle structure is according to Nørmark et. al. (1992) dipping app. 15 . Considering the station position this will give an incidence angle of app. 45 . Investigations using Zoeppritz equations in paper A4, indicate that an increase in density alone will give no reflectivity at an incidence angle of 45 . With a significant velocity contrast reflective energy should be visible. The computed wave field from the original raytraced model of Nørmark (1992) is shown in Fig. 25a. The effect of a homogeneous slab of high and low velocity eclogite respectively, is shown in Fig. 25b-c. At offset larger than 150 km both these models produce reflected energy of the same amplitude. At smaller offsets the high velocity eclogite tends to show more reflected energy. A slab of intermediate velocity (not shown here) shows as expected almost no reflected energy. Thus both a high and low velocity eclogite shows reflected energy with about the same amplitude. However analyzing the full wave field plots, it is evident that the amplitude of the reflected 43
F IGURE 23: Vertical component seismogram, BABEL Station 10, Bornholm. A few seconds after the arrival of a strong phase is identified as Rum. It has a remarkable high amplitude at offsets larger than 100 km. (From Nørmark et. al., 1992)
waves from any type of a homogeneous slab of eclogite is not comparable to e.g. and as seen in data. Additionally no energy is observed at offsets around 90 km, as it is observed in data. Modeling results using the same model as Fig. 25b, but with the homogeneous distribution of eclogite replaced by a binomal velocity distribution of 50 % high velocity eclogite embedded in typical upper mantle material, is shown in Fig. 25d. This has a rather dramatic effect. We observe that seismic reflected energy is apparent with approximately the same amplitude from offset well under 100 km. The character of the reflected wave field from around x=400 km (offset of 110 km) is that of a band of reflected energy. This is similar to observations from the BABEL data (Fig. 8). However the amplitude of the reflected wave field from the upper mantle reflector is not sufficiently high to be comparable to e.g. or . In a subducted slab a full transformation from crustal material (oceanic or continental) to eclogite does not necessarily happen. Only some parts of the slab may have transformed. This implies a slab, with partly crustal type of rock velocities and partly eclogite velocities. The modeled result from such a semi transformed slab, using the same von Karman distribution as in Fig. 25d, is shown in Fig. 25e. The modeled amplitudes are now comparable to that of and . Reflected energy is evident from very short offsets, less than 100 km. The larger amplitudes are caused by the velocity contrasts within the slab. Modelings (not shown here) confirms that different kinds of eclogite with high or low velocity will produce similar wave fields resembling the observed wide angle data, recorded at land station 10. Density variation alone cannot explain the observed Rum phase. A homogeneous slab of eclogite will not produce neither the amplitude, reflectivity pattern nor explain the short offset at which Rum is observed. A random distribution of the velocity field in the slab, will produce the reflectivity pattern and explain the energy observed at small offsets. However, the contrast in elastic parameters must be relatively large to explain the amplitudes, best explained by a randoamlly distributed velocity field, consisting of partly eclogitized crustal material. 44
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F IGURE 24: Velocity model from a section of BABEL Line A, with the addition of slab at the position of the raytraced reflector, Rum (BABEL Working Group, 1993 and Nørmark et. al., 1992).
3.3.4 Discussion on upper mantle reflectors In addition to the results presented above, paper A4 contains an investigation of some fundamental patterns of reflectivity from anomalous zones in the upper mantle. Zoeppritz equations is used to examine the reflectivity coefficient with offset for variation of the elastic parameters in the upper mantle. We find that both a relatively high (Vp=8.6 km/s) and a relatively low (Vp=7.8 km/s) velocity eclogite will produce reflected energy with comparable order of magnitude. This is further supported by numerical experiments shows above (Figure 25). Furthermore we find that it is possible to generate critically reflected energy from high and low velocity eclogites, i.e. from the top of a high velocity eclogite, and from the base of a low velocity eclogite. This expands the conclusion made by Warner et. al. (1996), who found that only a dense garnet rich high velocity eclogite can explain the high reflection coefficients observed. We find that both a high and low velocity eclogite will have a magnitude of wide angle reflected wave field of about the same order. However, the magnitude of simulated wide angle reflected wave field from a homogeneous slab of eclogite is not comparable to some crustal phases, as have been observed in e.g. BABEL landstation 10. If the slab consists of partly eclogitized crustal material, simulated by a bimodal random velocity field, the magnitude of the reflected wide angle wave field becomes comparable to some crustal phases, and the wide angle wave field becomes visible at relatively short offset, as seen in BABEL landstation 10. This model is radically different from the one presented by Krishna et. al. (1996), who introduced a model with variations in the elastic parameters, that was hard to justify using a valid geological model. The normal-incidence data from MONA LISA line 2 and the wide angle data from OBH 28, can be explained by the existence of a Moho offset zone associated with a subduction zone. A dipping slab of partly eclogitized oceanic crust attached to the subduction zone, is consistent 45
with the seismic data. This is illustrated through wave form modeling of OBH 28, presented paper A4. This is different from Abramovitz and Thybo (2000) who argued that the observed reflectivity is likely to be due to a mantle shear zone. In general we have shown that a mantle shear zone might be visible on normal incidence data, but most likely not in wide angle data. A relict subduction zone, is likely to be visible both in normal incidence and wide angle data-sets. The existence of a Moho offset zone, is likely to produce distinct features observable both in normal incidence data-sets, and on wide angle data. The fact that a shear zone and a homogeneous slab of eclogite will cause reflections from the uppermost mantle in normal-incidence seismic data, but not likely in wide angle data, can be the reason why little reflected energy is observed in some wide angle datasets, where normalincidence data on the same region shows upper mantle reflected energy. If seismic energy from dipping upper mantle structures is visible in both normal-incidence and wide angle data, we conclude that a remnant slab subducted oceanic or continental crust is a valid model. The subducted slab should consist of partially transformed crustal material into a higher velocity material, most likely eclogite. A Moho offset zone is likely to be present too. Such a model is consistent with normal-incidence and wide angle data from both the MONA LISA and the BABEL data sets.
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F IGURE 25: Modeled wide angle wave field. a) Reference model without Rum. b) Homogenous slab of high velocity eclogite (Vp=8.6 km/s) c) Homogenous slab of low velocity eclogite (Vp=7.9 km/s) d) Slab with random velocity distribution of high velocity eclogite (Vp=8.6 km/s) e) Random slab of partly converted high velocity eclogite (Vp=8.6 km/s embedded in lower crustal material with Vp=7.0 km/s). The source was positioned at x=288.5 km.
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3.4 AVO attributes of long offset P wave data and converted P-SV waves The gradient intercept method, see Eqn. 6 and section 2.4.1, is based on two assumptions : The existence of a linear relation between the reflection coefficient and the squared sine of the incidence angle, ( , defined by the AVO attributes A and B, (Shuey, 1985), and the assumption that rock property contrasts are relatively small (Aki and Richards, 1980). When hydrocarbons are present the contrast in rock properties can be very large. Thus the small contrast approximation becomes invalid when zones of hydrocarbon is investigated. Ursin and Dahl (1992) and Foster et al. (1997) expanded the Knott-Zoeppritz equation with respect to ( , obtaining exact expressions of and , thereby avoiding the small contrast approximation. This section describes an extension of their work including exact descriptions of the third term in the expansion of the reflection coefficient, , around ( , labeled ! , and the first two terms of the expansion of the P-SV converted wave reflection coefficient, , labeled and . These AVO attributes are investigated for their sensitivity to water saturation and porosity variation in gas bearing sands. Using gradient intercept data it is typically only possible to detect the existence of hydrocarbons, but not the amount, which is crucial from a commercial point of view. This problem, often referred to as the fizz-gas problem, is addressed in the following. The contents in this section is elaborated in paper A5.
3.4.1 Taylor expansions of reflection coefficients Ursin and Dahl (1992) and Foster et al. (1997) used Taylor expansions of Knott/Zoeppritz equations, e.g. Achenbach (1973), around ( , to obtain exact estimates of A and B. Extension to the third term of the approximation of the P-wave reflection coefficient, , and the first two terms of the approximation of the P-SV converted reflection coefficient, is given by
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3.4.2 Cross plots Non-hydrocarbon bearing lithologies tend to plot along a linear trend in a gradient intercept cross plot. Hydrocarbon bearing lithologies tend to plot off trend to the linear backgrounds trend. Gradient intercept cross plots are well investigated, (e.g. Foster et. al., 1997; Castagna et. al., 1998). This is not the case for cross plots involving ! , and . Here the background trends for three new AVO crossplots are investigated in non hydrocarbon bearing rocks, and for hydrocarbon bearing rocks with variation in water saturation and porosity. Background trends - non hydrocarbon For non-hydrocarbon rocks Foster et. al. (1997) derived the background trend for a / crossplots based on two assumptions. First, when no hydrocarbons are present, the change in $ " ratio is small () ), and the variation in density is small (# ). Using these simplifications, the background trends for ! - , - , and - cross plots from Eqn. 11 are determined assuming changes in P-wave velocity tends to 0 (" ), (B-A from Foster et al., 1997b) :
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where ) $ ". Figure 2 in paper A5 show that the background trends in ! - , - and - cross plots are less dependent on the $ "-ratio compared to the background trend for the cross plot. Thus an estimate of the $ "-ratio is easiest to extract from a conventional cross plot. However deciding whether a data point is anomalous or not might be easier on the newer cross plots. These background trends are only approximately linear. The higher order terms can be significant, and it is only around the origin of the cross plot, where changes in rock properties are small, that they apply. An advantage of using exact attributes compared to first order attributes (Aki and Richards, 1980) is that a linear relation between " and $ can only lead to a linear relationship between the attributes. Although first order attributes may accurately model the reflection coefficients as a function of offset, they cannot provide insight into the relationship between the higher order terms and rock properties. Effect of hydrocarbons A well log from offshore Myanmar shows evidence of noncommercial gas, so called fizz-gas, in an unconsolidated gas bearing sand, (Hornby and Pasternack 1998). Elastic properties from this zone, and just above it, are used to generate 11 models with variation in water saturation, using fluid substitution, (Gassman 1951). The rock properties above and below the reservoir zone is for simplicity considered the same. Using the attributes defined by Eqn. 11, AVO attributes are computed for all 11 models. The , ! , 49
and cross plots of the eleven models are plotted in Figure 26a-d. Both first order and the exact attributes are plotted. The AVO-attributes obtained from the original log is plotted as small black dots to show the background trend in the data. The distribution of the dots show clear correlation to the background trend for $ " , the solid black line. The cross plot, Figure 26a, is a good example of the fizz-gas problem. The wet model plots close to the background trend, whereas models with * are almost at the same position, clearly separated from the background trend. In the ! ! cross plot, Figure 26b, decreasing water saturation is seen as a trend along the background with a close to linear increase in the distance to the background trend. In the cross plot, Figure 26c, the models show an almost perfect linear relationship between water saturation and the distance to the background trend. From the top of the reservoir, seems to be insensitive to any variation in * , and is thus alone responsible for detecting the variation in * . From the bottom of the reservoir seems to have a little sensitivity to changes in * . The cross plot, Figure 26a, could be interesting since the term shows the presence of some hydrocarbons and the term shows a gradual variation with respect to water saturation. However, the background trend seems to be parallel to the trend of the variation in * . This could make it difficult to distinguish models with different * . The first order attributes gives perfectly symmetrical response from the top and the base of the reservoirs.All attributes but , which is the same and exact in both the considered formulations, shows non symmetrical response using the exact attributes. Especially the ! ! and cross plots clearly illustrate that the first order attributes deviates significantly from what should be expected, i.e. the exact attributes, from a model with variation in water saturation. Effect of porosity Murphy et al., (1993), proposed an empirical relation between porosity and bulk- and shear- frame moduli, based on laboratory data for a wide range of pure sand stones with different porosity. Their relation are used to create a series of models with varying porosity, +, between 5% and 34%, with a gas saturation of . The empirical value obtained for porosity of 34 %, fits very well the one measured on the well offshore Myanmar. Figure 27 shows the variation of cross plots of the derived exact attributes to porosity. Variations of and , shown in Foster and Keys (1999), are shown here for comparison, Figure 27a. Porosity effects for the other AVO attributes are described here. The variation as the porosity changes can generally be described as a somewhat linear trend subparallel to the background trend. Specifically for the ! cross plot a strong asymmetric response from the top and base of the reservoir is visible, Figure 27b. With decreasing porosity, the C-term gets close to zero from the top of the reservoir, causing both the response from the top and base of the reservoir to plotted at the same side of the background trend. In conventional cross plots of first order attributes such behavior is very unlikely and could be interpreted as the result of bad processing of the seismic data. In this case we use it as an indicator of high versus low porosity. The change in water saturation affects ) , which moves the points away from the background trend, see Figure 26, while porosity variations (which have little affect on ) ) move points subparallel to the trend, Figure 27. These effects are somewhat orthogonal to each other, and 50
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F IGURE 26: Cross plots of AVO attributes from the top and base of gas sand with varying water saturation, x. Above and below the reservoir the parameters are : , , . The values in the reservoir are plotted in Figure 7. Circles(o) are Aki and Richards AVO attributes, and crosses(+) are the exact AVO attributes listed in Appendix A1 and A2 of paper A5. The solid line denotes the background trend for . The small dots around the background trend are the calculated exact AVO attributes derived from the well log.
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