S32A-01 10:20h
Retrieving the Green's Function by Cross-correlation: a Comparison of Approaches
Recently it has been shown by various authors that the Green's function of a random medium can be obtained by cross-correlating the recordings of a diffuse wave field at two receiver positions (Weaver and Lobkis, 2001; van Tiggelen, 2003; Snieder, 2004). The resulting Green's function is the wave field that would be observed at one of these receiver positions if there were an impulsive source at the other. This theoretical result has first been demonstrated with ultrasonic measurements and later with seismic surface waves (Campillo and Paul, 2003). The accuracy of the reconstructed Green's function depends on the amount of disorder of the medium parameters and the duration of the signal. Ideally the cross-correlations should be done in the equipartitioned regime (where the net energy flux is equal to zero), which takes place after sufficiently long multiple scattering of the wave field between the heterogeneities in the disordered medium (Malcolm, Scales and van Tiggelen, 2004). An initially independent line of research, developed by exploration seismologists, deals with the reconstruction of the seismic reflection response of a deterministic medium from (passive) recordings of the transmission response. Already in 1968 Claerbout showed that the autocorrelation of the transmission response of a horizontally layered earth yields the superposition of the reflection response and its time-reversed version. The source in the subsurface may be a transient or a noise signal; in both cases the source signature in the reconstructed reflection response is the autocorrelation of the source signal in the subsurface. Claerbout's derivation was strictly one-dimensional. Later he conjectured for the 3-D situation that `by cross-correlating noise traces recorded at two locations on the surface, we can construct the wave field that would be recorded at one of the locations if there was a source at the other'. Although it was not explicitly stated, this conjecture applies to deterministic media: in exploration seismology the earth is usually considered to be built up of geological layers with smoothly varying properties, separated by well-defined curved interfaces and faults which act as the main reflectors; scattering due to disorder of the parameters within the geological layers is generally considered a second order effect. Numerical modelling studies have been carried out to confirm Claerbout's conjecture (Rickett and Claerbout, 1996). These modelling studies showed that `longer time series, and a white spatial distribution of random noise events would be necessary for the conjecture to work in practice'. The cross-correlation approach has been applied successfully to helioseismic data (Duvall et al., 1993). Recently Claerbout's conjecture has been proven by the authors. The proof also explains the observations of the numerical modelling studies. [Wapenaar, K., J. Thorbecke, and D. Draganov, Relations between reflection and transmission responses of 3-D inhomogeneous media, Geoph. J. Int., 156, 179-194, 2004]. In this paper we compare the `random medium approach' (Weaver etc.) with the `deterministic medium approach' of exploration seismology and discuss the underlying assumptions. Moreover, we discuss applications in passive seismic imaging.
S32A-02 INVITED 10:45h
Virtual Sources, a new Reality for Imaging.
We have developed a technique for sensitive imaging and hydrocarbon reservoir monitoring which we call Virtual Source. Although the methodology is different, our description will be different and the motivation is different, the underlying physics is similar to that employed in Passive Imaging. Our ability to seismically image the subsurface for exploration and to closely repeat surveys for monitoring is compromised by earth heterogeneity. With heterogeneity at all scales we can never make a property model complex and accurate enough to enable a full bandwidth high-resolution image from the energy we put into the ground. To be able to measure small differences in our reservoirs we would like to have multi-path scattering travel times to repeat within 0.1msec or better. This requires impractical positioning repetition of source elements and unexpected stability of naturally changing overburden. We therefore need fixed sources and receivers under the overburden. Our budgets demand the sources, at least, be virtual ! With buried geophone receivers Rj under the troublesome overburden we can shoot over them with conventional sources Si. We record the direct arrivals in traces Tij. This direct arrival energy going down through Rj is the result of a particular source waveform, a particular coupling to the earth, multi-path scattering, reverberations, transmission losses; the whole real heterogeneous earth transmission response from Si to Rj. We may invert this "wavelet" to a pulse by deconvolution. A practical deconvolution is correlation and zero phase spectral shaping to a desired bandwidth to give a custom zero phase pulse W. The deconvolution filter Tij-1 would be of form Tij*.W/[Tij.Tij*]. If we filter the whole shot Si gather into all receivers Rk with Tij-1 then that signal energy from Si which goes through Rj will be received at Rk as Tik.Tij-1 as though it was sourced from Rj at zero time with waveform W. If we now sum over all Si we may simulate a downward radiating source at Rj with known wavelet W being received in receivers Rk. We thus have the ability to recast the seismic experiment as having a controlled downward radiating Virtual Source at each receiver location in turn. What is nice is that we can do this without any knowledge of the overburden, without knowledge of the physical source waveform or exact location. We can get repeatable results even if the overburden should change, as this will be automatically compensated by Tij-1. Another interesting property is that the worse the overburden is for conventional seismic the better it is for Virtual Source work. With a highly scattering overburden we need less physical source locations to give a good Virtual Source radiation pattern. The method has given encouraging results on initial tests. We obtain remarkable repeat of detail from survey to survey and we can see reservoir changes with improved resolution and sensitivity. It is interesting to compare the Virtual Source with passive imaging from the surface. By going underground we can avoid the ground roll and other near surface noise. By using arrays and active sources we can recognize and choose energy for a desired radiation direction. We can use the method with low energy continuous sources such as uncontrolled Vibroseis or natural noise. We can also move our Virtual Source to the surface as in a conventional seismic survey and take advantage of a pulse waveform to select up and down going energy. Being on the surface and using natural noise would perhaps be the most difficult combination.
S32A-03 INVITED 11:10h
Migration, Tomography and Datuming by Seismic Interferometry
Inteferometric seismic data can be defined as the correlation between two selected events, event $P^A_g = e^{i \omega (\tau^{unint.} + \tau^{interest}_g)}$ at trace $g$ and event $P^B_{g'} = e^{i \omega (\tau^{unint.} + \tau^{interest}_{g'})}$ at trace $g'$. The $A$ and $B$ events are similar in the sense that a portion of their raypaths coincide through uninteresting parts of the media (such as overburden or the crust), so that the correlated data $\phi(g,g',s) = P^A_g~{P^B_{g'}}^*$ $ = e^{i \omega (\tau^{interest}_g-\tau^{interest}_{g'})}$ removes the uninteresting kinematic effects (i.e., eliminates $\tau^{unint.}$) as well as source+receiver statics. Applying interferometric imaging formulas to crosscorrelation data allows one to, e.g., automatically migrate multiples in seismic data, redatum sources and receivers below uninteresting parts of the medium (such as the crust or overburden) without needing to know the associated velocity model, migrate converted reflection and transmission waves, and detect the location of hidden sources. Synthetic and field data sets are used to demonstrate the effectiveness of interferometrically converting VSP, CDP or HSP (horizontal seismic profile) data to effective single well imaging data where both sources and receivers are at depth; no velocity model is needed. This could be important for VSP experiments along the San Andreas fault. Other uses of interferometry include imaging the reflectivity structure of the crust by ghost reflections (preliminary results for Utah earthquake data are presented), velocity analysis by interferometric traveltime tomography and generalization of the receiver-function imaging method.
S32A-04 INVITED 11:35h
Generation and Mitigation of Surface-Generated Scattered Phases in Teleseismic Imaging.
Many of the current algorithms that earthquake seismologists use to image the forward scattered teleseismic wavefield have been borrowed from the oil industry. Of course, modifications to the original algorithms allow earthquake seismologists to image forward scattered wavefields rather than reflected, or backscattered, wavefields. A common theme that is represented in these imaging algorithms is the single-scattering, or Born, approximation. The problem is when the data contains arrivals that do not adhere to this approximation (for example, free-surface reverberations, scattered surface waves, etc.). In a typical oil industry type seismic reflection survey, the recording geometry and the density of the geophones allow the application of techniques to "remove" the arrivals that are not considered to arise from primary reflections. Although such techniques are arguably model-based, applying them to the raw data often removed enough of the non-primary arrivals to a degree that allows the application of Born-type imaging algorithms. In teleseismic imaging, we are faced with the same problem of non-primary arrivals "contaminating" our data. Therefore if we wish to use imaging algorithms that are based on the Born Approximation, we must develop techniques to remove non-primary arrivals. In this talk, I will focus on the documentation and mitigation of surface scattered phases that are generated by high magnitude, near-surface, lateral velocity variations. Although most of the work here will be with synthetic data, the generation of surface-scattered phases in synthetic velocity models is consistent with observations of field data. However, because the algorithms presented require data that is not spatially aliased, I will discuss the effects of spatial sampling interval (i.e. station density) on the efficacy of such mitigation techniques.
S32A-05 12:00h
Computational Problem in Three-Dimensional, Plane-wave Imaging with P to S Converted Waves Recorded by Broadband Arrays
We have developed a set of computer programs that implement the three-dimensional imaging methods described by Poppeliers and Pavlis [2003]. The original work developed the theoretical framework for this technique, but did not implement the fully three-dimensional version due to the requirement that efficient computational procedures are necessary to make the algorithm feasible. We implemented the concepts in two programs linked by a relational database. The first program implements a plane wave decomposition of the wavefield using the pseudostation stacking method to interpolate the wavefield onto a regular spatial grid. We then use a relational database to sort data into plane wave components linked to this interpolated, regular grid. Imaging is completed using sums of plane wave components in an integral equation derived from the inverse generalized Radon transform. This allows the image to accumulate incrementally as a weighted sum of plane wave components with the weights varying in space reducing the memory requirements to practical levels on modern, distributed-memory, parallel systems. This imaging algorithm is readily adaptable on massively parallel machines by distributing back propagation computations for plane wave components among processors and devoting one node to summing the components. The algorithm is built around a new object-oriented, C++ library for manipulating seismograms and the two and three-dimensional data objects that are intermediaries in the algorithm.