Computers in Physics Phase shift plus interpolation; A scheme for high-performance echo-reconstructive imaging Ernesto Bonomi, Leesa Brieger, Carlo Nardone, Enrico Pieroni, Ralf Gruber Editor, and Jacques Rappaz Editor Citation: Computers in Physics 12, 126 (1998); doi: 10.1063/1.168623 View online: http://dx.doi.org/10.1063/1.168623 View Table of Contents: http://scitation.aip.org/content/aip/journal/cip/12/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Using of Bezier Interpolation in 3D Reconstruction of Human Femur Bone AIP Conf. Proc. 1389, 1309 (2011); 10.1063/1.3637859 Effective Algorithm for Detection and Correction of the Wave Reconstruction Errors Caused by the Tilt of Reference Wave in Phaseshifting Interferometry AIP Conf. Proc. 1236, 91 (2010); 10.1063/1.3426174 Fast Reference Phase Extraction and Wavefront Reconstruction for Generalized Phaseshifting Interferometry AIP Conf. Proc. 1236, 339 (2010); 10.1063/1.3426137 A temporal interpolation approach for dynamic reconstruction in perfusion CT Med. Phys. 34, 3077 (2007); 10.1118/1.2746486 Comparison of local and global angular interpolation applied to spectral-spatial EPR image reconstruction Med. Phys. 34, 1047 (2007); 10.1118/1.2514090
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ALGORITHMS & APPLICATIONS
PHASE SHIFT PLUS INTERPOLATION: ASCHEME FOR HIGH-PERFORMANCE ECHO-RECONSTRUCTIVE IMAGING Ernesto Bonomi, Leesa Brieger, Carlo Nardone, & Enrico Pieroni
Department Editors:
RalfGruber
[email protected]
Jacques Rappaz rappaz@dma. epfl.ch
cho-reconstruction techniques for nonintrusive imaging have wide application, from subsurface and underwater imaging to medical and industrial diagnostics. The techniques are based on experiments in which a collection of short acoustic or electromagnetic impulses, emitted at the surface, illuminate a certain volume and are backscattered by inhomogeneities of the medium. The inhomogeneities act as reflecting surfaces or interfaces that cause signal echoing; the echoes are then recorded at the surface and processed through a "computational lens" defined by a propagation model to yield an image of the same inhomogeneities. The most sophisticated of these processing techniques involve simple acoustic imaging in seismic exploration, for which the huge data sets and stringent performance requirements make high-performance computing essential. Migration, based on the scalar wave equation, is the 1 standard imaging technique for seismic applications. In the migration process, the recorded pressure waves are used as initial conditions for a wave field governed by the scalar wave equation in an inhomogeneous medium. Any migration technique begins with an a priori estimate of the ve-
E
Ernesto Bonomi is head of geophysics at Centro di Ricerca, Sviluppo e Studi Superiori in Sardegna (CRS4) , Via Nazario Sauro 10, 1-09123 Cagliari, Italy. E-mail:
[email protected] Leesa Brieger is the senior researcher at CRS4 Area Geofisica. E-mail:
[email protected] Carlo Nardone is an expert researcher at CRS4 Area Geofisica. E-mail:
[email protected] Enrico Pieroni is an expert researcher at CRS4 Area Geofisica. E-mail:
[email protected]
locity field obtained from well logs and an empirical analysis of seismic traces. By interpreting migrated data, comparing the imaged interfaces with the discontinuities of the estimated velocity model, insufficiencies of the velocity 2 field can be detected and the estimate improved, allowing the next migration step to image more accurately. The iterative process (turnaround) of correcting to a velocity model consistent with the migrated data, which can last several computing weeks, is crucial for imaging complex geological structures, including those that are interesting for hydrocarbon prospecting. Because subsurface depth imaging is the outcome of repeated steps of three-dimensional seismic data migration, the process requires gigabytes of data. The data must be reduced, transformed, visualized, and interpreted to obtain meaningful information. Severe performance requirements have led in the direction of high-performance computing hardware and techniques. In addition, an enormous effort has historically gone into simplifying the migration model so as to reduce the cost of the operation while retaining the essential features of the wave propagation. The phase-shiftplus-interpolation (PSPI) algorithm can be an effective method for seismic migration using the "one-way" scalar wave equation. This algorithm is particularly well suited to data parallelism because of, among other things, its decoupling of the problem in the frequency domain.
Exploding reflector model The PSPI method will be discussed in the context of coincident source-receiver experiments. With the seismic data-compression technique known as stacking, signals corresponding to all source-receiver pairs having the common midpoint ( x ,y ,0) are collected into a single zero-offset trace that simulates a coincident source-receiver experiment. In such an experiment, the downward ray path and travel time t/2 from the source to a point of reflection (reflector) are identical to the upward ray path and travel time from the reflector to the receiver (see Fig. 1). As a consequence, an equivalent trace would result from a source of appropriate intensity R initiated at the reflector and traversing the medium at half the original velocity. This is the so-called exploding reflector model. 3 The field R(x,y,z) of signal intensity that reproduces the ensemble of seismic traces (the seismic section) gives an acoustic image of the volume:
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large values of R correspond to sharp contrasts in the velocity field. Using the zero-offset seismic section P(x,y,O,t) as a boundary condition and solving the scalar wave equation,
a2 P
a2P
a2 P
ax
ay
az
-+ -2+ -2 2
a2P =0 v 2 (x,y,z) at 2 ' 1
P(x,y,z+ Llz,t=O)= L P(x,y,z+ Llz,w). (1)
in reverse time with zero initial conditions, the exploding reflector model allows us to interpret the migrated section P(x,y,z,t=O) as a map of the local reflectivity and yields an acoustic "picture" of the reflectors: R(x,y,z)
=P(x,y,z,t=O).
Phase-shift formula The original phase-shift migration method was formulated by Gazdag4 as a fast and simple implementation of zero-offset data migration. Assume for the moment that v, the halved velocity of the medium, is constant. A way of solving Eq. (1) and thus arriving at our acoustic image R(x,y,z)=P(x,y,z,O) is to use the depth z as the advancing variable along which to propagate the seismic section P(x,y,O,t); this is known as depth extrapolation or continuation. Equation (1) written in the wave number-frequency domain ( kx ,ky, w) is the second-order ordinary differential equation
in which
kz=~~1- ( ;) (k;+k~).
If we use an inverse transform to map Eq. (5) to the space-frequency domain, we can then evaluate the field R and do the imaging of migrated data at level z + Llz by noting that
(7)
w
Equations (3), (5), and (7) form the basis for the phase-shift migration algorithm. They allow the exact inverse extrapolation of seismic data inside a homogeneous layer [z,z+ Llz] with constant velocity. Since the power spectrum of the seismic source is band-limited with a cutoff frequency far below the temporal Nyquist frequency, mapping data into the space-frequency domain allows significant data compression. To avoid data aliasing in the wave-number domain and thus improve the accuracy of the extrapolated wave, it is often preferable to replace the ideal exponential operator with a "designed" filter, to provide a highest attenuation between the passband region and the spatial Nyquist frequencies.
The phase-shift formulation of migration leads to an elegant parallel implementation.
2
(3)
Equation (2) has two characteristic solutions of the form
F(kx ,ky ,z+ Llz,w) =F(kx ,ky ,z,w )e±ikzAz,
(4)
relating the field at level z with that at level z + Llz by a phase shift. Since agreement in sign between phase and frequency corresponds to a positive displacement of a wave in reverse time, and since by convention the z axis points downward, for depth extrapolation we are interested only in the characteristic solution
F(knky ,z+ Llz,w) = F(kx ,ky ,z,w )eik.fJ.z,
(5)
which back-propagates in time. Notice that this is also a solution of
the paraxial or one-way wave equation. This equation provides a hierarchy of migration methods5 in the spacefrequency domain ( x ,y, w) based on different approximations of kz.
The phase-shift formulation of migration leads to an elegant parallel implementation. In a preprocessing phase, the seismic traces P(x,y,t) are transformed to the spacefrequency domain (x,y,w) by concurrent one-dimensional fast Fourier transforms (FFTs). P(x,y,w) is loaded from disk and distributed among processors by blocks of frequencies. All blocks are then concurrently transformed to the (kx ,ky ,w) domain by two-dimensional (2D) FFTs. In addition, during downward continuation, three highly parallel steps are carried out to image the reflectors for each value of z: (1) for each complex array, a parallel operator masks all spurious frequencies and shifts the phase of each entry, (2) partial sums over w are performed, followed by the execution of 2D FFTs, one per processor, (3) the transformed fields are added and the resulting migrated section R(x,y,z+Llz)=P(x,y,z+Llz,t= O) is discharged onto a disk.
Phase shift plus interpolation Whereas seismic imaging in a stratified medium can be done with Eqs. (3), (5), and (7), the case with lateral veloc-
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A&A
S1
S2
S3
S=R
R2
R1
reference velocity v;">, which
R3
v(l)
