PARTIAL DERIVATIVES AND ITERATIVE INVERSION OF SEISMIC WAVEFIELDS Thomas Mejer Hansen and Bo Holm Jacobsen Departement of Earth Sciences, Geophysical Laboratory, Aarhus University, Denmark (
[email protected],
[email protected])
Introduction Modelling of seismic waves has been done for as many years as seismics have been considered. Nowadays Full Waveform Modelling of the wave equation is one of the most accurate ways to model the wave equation. Full waveform modelling started in the late 60's, Alterman & Karal (1968). Not until the mid 80's was Wave Form modelling widely used. In 1986 Virieux proposed a staggered grid scheme to solve the elastic wave-equation. Through the 80's Albert Tarantola (1984a, 1984b, 1986) proposed ideas on how to use inversion on the whole wavefield : Full Waveform Inversion. The theory he used was rather complex, but the conclusion he made was easy to use : Full waveform inversion could be seen as 'iterative migration'. The problem at the time was that the amount of computer power needed were tremendous. Thus to perform Full Waveform Inversion on small synthetic models required supercomputers. Still today the demand on computer power is great, but now small models can be inverted on PC based platforms on a reasonable timescale. The code used was based on the staggered grid method of solving the full elastic wave-equation using a 4th order space and 2nd order time staggered grid (Levander 1986). The inversion routine is based on Tarantolas work from the mid 80's (e.g. Tarantola, 1986)
The Problem The routine should be valid for all seismic experiments, but here we will mainly discuss the routine in context to reflection seismics : The model is described by m=(λ,µ,ρ), defined in a staggered grid (Levander, 1986). The pulse is assumed to be known. The data is the observed horizontal, U, and vertical, V, displacements at geophone points, (U(r,ttwt);V(r,ttwt)).
The Inversion Routine We have followed the inversion technique as described by Tarantola (1986) : 1 2 3 4
Obtain the data, Uobs Choose a starting model Mcurrent=M0. Model a forward response of the current model Mcurrent, the ‘downgoing’ field. Calculate the residuals : dU = Uobs - Ucurrent .
5 Correlation of the 'downgoing' field with the backward propagating residuals, the ‘residual’ field. 6 Calculate steepest descent direction for the updates of the model parameters, dM. 7 Update the model parameters Mcurrent = Mcurrent+ k*dM (k is a constant) 7a If the updated model is below a certain acceptance level the inversion is ended (8). 7b If not, then the iteration continues with the updated model parameters as the current model (3). Below we consider some of the considerations that have to be done implementing this routine.
Parameterization and the Fréchet Derivatives There are different ways to parameterize the model space. In the elastic formulation of the waveequation, 3 possible parameterizations is considered (following Tarantola, 1986): a. b. c.
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The choice of parameterization is important for how efficiently the inversion will converge. Examination of the Fréchet derivatives can lead to an understanding of the correlation of the model parameters. The parameterization chosen should be the one with the most uncorrelated model parameters. This section is an elaboration of figure 2-4 in Tarantola 1986. The Fréchet derivatives of the 3 model parameters, of the 3 different parameterizations have been approximated numerically by subtracting the result of two forward modellings with a perturbation in the model parameter of + 0.1 0/00. Fig.1 shows the emitted pulse as it is measured 0.3 s after its emittion, and Fig. 2 shows the pertubation. Emitted pulse used in obtaining partial derivatives
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The pertubation is chosen as a Gaussian peak. The half width is roughly 50 m, and the emitted wavelet has a wavelength of 200 m. The Gaussian peak should act approximately as a point source. The pertubation is at (1000m, 1000m). The explosion (P-wave) source is emitted at gridpoint (1000m, 500m). Thus with respect to the pertubation, the incident P-wave is travelling downwards! The Frechèt derivatives gives the diffracted energy as a result of an incoming P-wave Fig. 3-4 shows the divergence an the rotation of the Frechèt derivatives for the 3 parameterizations. The plot of the divergens of the Frechét derivatives is an expression of the amount of energy transferred from the incident P-wave to the generated P-wave. (P-P) (Fig. 3) The plot of rotation of the Frechét derivative is an expression of the amount of energy transferred from the incident P-wave to the generated S-wave (P-SV) (Fig. 4) Incident P → P ( divergence ) (Fig. 3): For near-vertical seismics, the P-P conversion is the most important since S-wave energy is almost absent at small offsets. For all 3 parameterizations we see that one parameter (Fα,FIP,Fλ) leads to an diffracted P-wave in all directions (Fα corresponds to the Frechet derivative of α when β and ρ-velocity is constant) Using velocity as parameterization both Fα and Fρ-v gives a reflected P-wave, though Fβ only shows horizontally diffracted P-wave energy. Using the Lamé parameters as parameterization, all 3 model parameters (Fλ,Fµ,Fρ-l) shows both reflected and transmitted energy. When using impedance as parameterization only FIP shows reflected energy. Fρ-I shows transmitted and FIS shows only horizontal diffracted energy. This means that any reflected P-wave energy, using this parameterization, can be correlated with the P-wave impedance, IP, only. Consequently the impedance should be chosen as parameterization in reflection seismic experiments, when only P→transmitted P energy is considered. However in f.ex. a cross-borehole experiment this will not be the case. Here will the better parameterization be the one that gives the most uncorrealated information on transmitted energy. It's seen that using velocities as parameterization it's only a change in α-velocity, that results in a transmitted P-wave. Thus. seimic velocities would be the best parameterization in a cross borehole seiscmic excperiment, when only P→transmitted P energy is considered.
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Fig. 3 . Plot of the divergens of the Frechét derivatives. Indicates the amount of energy transferred from the incident P-wave, emitted at (x,z)=(1000,500), to the diffrcated P-wave caused by the gaissuian pertubation at (x,z)=(1000,1000).
Incident P → S ( rotation ) (Fig. 4) : The same parameters that led to diffracted P-wave energy in all directions, (Fα,FIP,Fλ), leads to no diffraction of S-wave energy. Fβ, FIS and Fµ leads to similar diffraction patterns. S-wave energy is diffracted in 45O angles to horizontal, but no energy is diffracted in neither horizontal nor vertical direction.
The last 3 Frechét derivatives (Fρ−v,Fρ-I,Fρ-l) shows mutually different patterns. Fρ-l shows diffracted S-wave energy everywhere but vertical, Fρ-I shows reflected non-vertical S-wave energy, Fρ-v transmitted non-vertical S-wave energy.
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Fig. 4 . Plot of the rotation of the Frechét derivatives. Indicates the amount of energy transferred from the incident P-wave, emitted at (x,z)=(1000,500), to the diffracted S-wave caused by the gaissuian pertubation at (x,z)=(1000,1000).
We note that for the impedance parameterization, FIS has amplitudes at 45o reflected energy, but Fρ-impedance only shows up in the transmitted cone. This might be advantegous too.
Since it is the P-energy that is by far the most prominent in seismic sections, we choose the paramterization with respect to P->P wave energy. We choose (as the example shown later) seismic impedances as parameterization for reflection seismics.
Disk Storage A practical problem that has to be considered is how to handle large amounts of data. The problem arises when the 'residual' field is correllated with the 'downgoing field'. Since the two fields, in a modelling sense, run in opposite directions in time, we should store the displacement field at all timesteps, when we model the 'forward' field. This could be stored in RAM, but at present time this is not possible, even for small scale problems. Next, it could be stored on disk. This is however often not possible. So, in order to perform the inversion we must find a way to lower the demands on storage. This could be done by calculating the 'forward' field forward in time from time t=0, to the present time for the residual field. This would make the storage problem vanish, but will be too time consuming to handle computationally. We have chosen a way somewhere in between of these two cases. We have chosen to save the borders of the displacement field at each timestep, and the full wavefield at the last timestep. The final timestep for the 'forward' field is now propagated backwards in time, simultanously with the backwards propagation of the 'residual' field, with the saved borders added at every timestep. Thus we calculate the 'forward' field two times, but the demands on disk storage is lowered with great effect. An example : We save the border just outside of the absorbing boundary, in a band of 4 cells, because the code is accurate of 4th order. In 1000*1000 grid modelled in 1000 timesteps, 2 component, the storage needed, when saving the whole wavefield as 4 byte reals, is : 2 * 1000*1000 * 4 bytes * 1000 timesteps = 7629 Mb. When we save the borders, in a band of 4 cells at 3 borders (At the surface the calculated seismogram acts as the saved 'border') the storage needed is : 2 * (4*3)*1000 * 4 bytes * 1000 timesteps = 91.6 Mb. The required storage capacity is reduced by a factor of 83. The unreduced storage of 7.6 Gb, per shot, is hard to handle on todays workstations. This is not the case with 0.1 Gb. Thus by saving the borders we can model and invert several shots, with the price paid that, instead of 2 modellings, the 'forward' field forward and the 'residual' field backwards, we must perform 3 modellings, foward and backward of the 'forward' field, and backward for the 'residual field'. This means that the time of the inversion is increased with 50 %, but at much less disk communication.
Inversion We will show an example of the implementation here. We have chosen a 3 layer model as shown in Fig. 5. The synthetic 'true' data set has been obtained by a forward modelling of this model. The 'observed' data is seen in Fig. 6 . The first step in the inversion is to set up an initial model. We have chosen a homogenous halfspace with a Vα=3000 m/s, Vβ=1780 m/s, ρ=3000 kg/cm3. This is the conditions in the first layer.
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The next step is to obtain the residuals : The difference between the 'observed' and the calculated seismogram from the present model. The residual seimogram is shown in Fig. 7, and we see that the direct P- and S-wave is gone, because we had chosen the right model parameters as our first guess. 0
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The next step in the iteration is the most important one. Here the 'downgoing' field is correalated with the 'residual' field, the backward propagation of the residulas. When the two fields correalate, it should be interpreted as a spot of mising energy. The correllated field is proportional to the update of a certain model parameter. For an update of IP, dIP, the correalation must be between the dillatation of the two fields (Tarantola, 1986). In Fig. 8 the correalation of the dillatation of the fields is imaged, for a single shot at (x,z)=(600,70)
If there are more shots in the same model, then one just have to add the time-correalated field from all the shots together, to obtain the final update. The sum of the correllation of 8 shots at (x,z)=(400, 600, 700, 1000,1100 ; 70 70 70 70 70), is shown in Fig. 9. Note that mainly those parts of the model are imaged, that are hit by reflected rays (Fig. 10-11). The update of P-impedance, dIP, provides good imaging for these synthetic studies. 500
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Fig. 9. dIP after 1 iteration on 8 shots
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Fig. 10. Reflection ray-patterns from one shot. The bold line indicates where reflected waves hit the reflector. (see Fig. 8)
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Fig. 11. Reflection ray-patterns from five shot. The bold line indicates where reflected waves hit the reflector. (see Fig. 9)
Conclusions A rather complex theory, which leads to computationally heavy calculations, have been implemented to work on an ordinairy 10 Mflop workstation with 32 Mb of RAM and a diskcapacity of 1 Gb. This is the typical capacity of a new Pentium PC date 1996. The code is written as a TOOLBOX for MATLAB, which provides great opportunities for examining Full Waveform inversion. Primarily the graphical interface in MATLAB makes it an easy task to actually get a notion of what this inversion routine does. The MATLAB code can be transformed to Ansi-C if a proper converter is available. This speeds the code up by a factor of about 4. Both the MATLAB-TOOLBOX, MATLAB to C-converted files and a compiled version of the code to run on any SGI machine is free to download via ftp from the URL : ftp://ftp.geofysik.aau.dk/usr/thomas/MATLAB/
References Alterman, Z., and Karal, F. C., 1968, Propagation of elastic waves in layered media by finite difference methods, Bull. Seis. Soc. Am., 58, 367-398. Levander, A. R., 1986, Fourth-order finite-difference P-SV seismograms, Geophysics, 53, 14251436. Tarantola, A., 1984a, Inversion of seismic reflection data in the acoustic approximation, Geophysics, 49, 1259-1266. Tarantola, A., 1984b, The seismic reflection inverse proplem, in Santosa, F., Pao, Y.-H., Symes, W. W., and Holland, C., Eds., Inverse problems of acoustic and elastic waves : Soc. Industr. Appl. Math., 104-181. Tarantola, A., 1986, A strategy for nonlinear elatsic inversion of seismic reflection data inversion, Geophysics, 51, 1893-1903.
