TA1]) a covariance matrix Ca, to take algorithmic constraints into account. Stated in this way, this matrix is a generalization of linear damping which is.
Constrained Seismic Well Data Waveform Inversion Marwan Charara 1, Christophe Barnes 1 2 and Albert Tarantola 1 1 2
Institut de Physique du Globe de Paris, France E cole des Mines de Paris, France
1 An underdetermined problem The inversion of seismic data allows one to obtain an image of the underground through the determination of a certain number of physical parameters. Borehole data (PSO) are of special interest because of the geometry of acquisition which provides more informative signals on the medium properties than surface seismic data (high frequency signal, energetic P-S conversions). However, the lack of seismic data redundancy (one shot) renders the inversion problem underdetermined. Classical least squares inversions (without constraints) fail to give a sensible image of the subsurface when the medium has a complex geological structure (not tabular). This arises because of the non-uniqueness related to the degrees of freedom in the inversion problem. One mean to overcome the lack of redundancy in the data is to extract as much information as possible from them. Therefore a ne forward problem such as nite di erence method, when solving the elastic wave equation, is needed. The possibility of explaining various seismic events contained in the data (transmitted, re ected, multiple re ected, converted P and S waves) requires one to nd an Earth model that can explain all these events, thus removing part of the underdetermination of the problem. We will show in this paper that the other means to reduce signi cantly the degrees of freedom in the inverse problem is to incorporate constraints applied to the data and the model space. Using the least squares criterion, these constraints can be expressed simply through covariance matrices.
2 Least squares inversion Using a probabilistic formalism (Tarantola [TA1]), we can rewrite the mis t function as the limit of a series of convergent functions Sn linearized in the vicinity of mn: Sn(mn + m) = (dn ? Gnm)tt C?D?11(dn ? Gnm) +(mn ? m) Cm (mn ? m) + ( m)t C?a 1 ( m) (1)
2
Marwan Charara, Christophe Barnes and Albert Tarantola
where
{ m is the perturbation in the vicinity of the current model mn { dn are the residuals for the model mn { mn is the di erence between the current model mn and the a priori model mprior { CD is the covariance over the data space (which combines experimental and theoretical uncertainties) { Cm The covariance over the model space (de ning the a priori probability density, here gaussian) { Ca is a covariance over the model space damping high perturbations. This
term will disappear at the convergence. Applied to waveform inversion problem of estimating the density and the elastic parameters of the earth, the minimization of the mis t function can be solved by iterative gradient methods (Tarantola [TA2]). Each iteration consisting of the propagation of the actual source in the current medium, and the propagation of residuals as if they were sources acting backward in time. The time correlation of these two elds yields the correction of the medium parameters. In principle, all the constraints listed above in the mis t function can be incorporated. In practice, for reasons of simplicity, covariance matrices are set to identity (Kolb and al. [KOL], Pica and al. [PIC], Crase and al. [CRA]), although if well introduced they can be a powerful tool, as we will show below, to constrain the inverse problem. PSO Geometry 0 Vp=1500 m/s Vs=0 m/s
3
density=1000 kg/m
200 3
Vp=1800 m/s Vs=750 m/s density=1400 kg/m
400
Depth (m)
3
Vp=2300 m/s Vs=1000 m/s density=1650 kg/m
600
Vp=2000 m/s Vs=700 m/s 3 density=1550 kg/m
800
1000
3
Vp=2600 m/s Vs=1300 m/s density=1700 kg/m
1200 0
200
400
600
800
1000
1200
Offset (m)
left: Synthetic experiment con guration where the cross near the surface denotes the source position, whereas the solid line denotes the location of the two-component receivers. Right, the two displacement components seismograms obtained for this experiment (to be inverted).
Fig. 1.
Constrained Seismic Well Data Waveform Inversion
3
3 Constraints on the data space Quanti cation of uncertainties of data is not done systematically, because in many cases it is a dicult task. Although, for instance, data errors generated by receiver response are correlated in time, means to account for them are not straightforward. However, to illustrate the possible contribution of constraints provided by the analysis of uncertainties of data, we have performed two inversions from the synthetic data experiment (Figure 1). The rst inversion is done without any constraints, whereas in the second one we have incorporated polarisation wave analysis in the covariance matrix CD . This matrix corresponds physically to treat uncertainties on displacement according to eigenbasis of the particle motion. The results of these inversions (Figure 2) show that constraints on the wavefront incidence to receivers contribute to recover the sharp model interfaces at the receivers level. Whithout constraint
receivers level
With polarization constraint
receivers level
Fig. 2. Two inversion experiments: On the left without constraints, on the right with polarisation constraints on particle motion. Dashed lines for the initial model, thick solid line for the parameter model at the current iteration (here after 20 iterations) and the thin solid line for the true model. At the level of the receivers, the constrained inversion contributed to the recovery of the sharp interfaces.
4
Marwan Charara, Christophe Barnes and Albert Tarantola
4 Constraints on the model space One often nds in the literature that the covariance on the model space, which should in principle describe the geological information, is used as an algorithmic tool. To avoid this confusion, we have explicitly separated them.
4.1 Algorithmic constraints We have added to the classical formulation of the mis t function (Tarantola [TA1]) a covariance matrix Ca , to take algorithmic constraints into account. Stated in this way, this matrix is a generalization of linear damping which is usually taken as a constant diagonal matrix. This Ca term forces the algorithm to stay in regions where the model are, a priori, sensible and when convergence is reached its e ect disappears.
4.2 Constraints on relationships among parameters Statistical studies of parameters can be obtained from well logs, or in the general case from laboratory studies such as the well known relation of Gardner and al. [GAR], which links P-wave velocity to density. This available information, for the case of statistical study, can be simpli ed to obtain a gaussian probability density or in the case of geological laws, linearized around a mean model. Then, the result will be introduced in the covariance matrix Cm . If we invert the same synthetic data (Figure 1), but this time putting a constraint on the correlation between parameters (here arbitrarily we have taken a 0.5 correlation between the three parameters), we can see on (Figure 2) that compared to non-constrained inversion, a perturbation on one parameter (here on P-velocity) will lead to a perturbation on the other parameters.
4.3 Constraints on spatial distributions of parameters We can distinguish two kinds of geological a priori information that can be incorporated in the inversion process. On one hand information obtained by measurements (well logging), that we consider as \objective" information, allows the de nition of an a priori model at the well vicinity. On the other hand, information coming from geological interpretations such as geological layer dips, that we qualify as \subjective" information, can be simpli ed and introduced into the covariance matrix Cm .
5 Quanti cation of number of degrees of freedom In order to evaluate the importance of constraints on inversion, it is necessary to quantify the number of degrees of freedom. For that purpose, we have stated the problem in the following way.
Constrained Seismic Well Data Waveform Inversion
5
With correlation constraint
Whitout constraint
Fig. 3. Two inversion experiments: on the left without constraints, on the right with correlation between parameters. Dashed lines is the initial model, thick solid line is the parameter model at the current iteration (here rst iteration) and the thin solid line is the true model. A perturbation on a parameter (here P-wave velocity) is propagating on other parameters.
Let us consider a discrete 1D random eld. If the eld is Gaussian with a given variance and without correlation (which corresponds to what we are doing using least square with a constant diagonal covariance matrix). A realization of such a eld is equivalent to independent realizations of the same Gaussian random variable at each point (Gaussian white noise). For each realization k of the random eld, the values for all points are considered independent and sorted in an n-classes histogram. By comparing this histogram with the one given by the theoretical law, through the equation:
Dk =
Xp (ni;k ? Nfth ith) = N Xp (fi;khist ?th fith) 2
2
(2) Nfi fi i=1 (p being the number of classes, ni being the number of values in the ith class of the histogram, fith the theoretical frequency of the ith class, fihist = ni =N the observed frequency of the ith class and N the total number of points) the Dk values follow the 2p?1 distribution. i=1
6
Marwan Charara, Christophe Barnes and Albert Tarantola
If the Gaussian eld is correlated, the realizations at each point are not independent and Dk does not follow the 2p?1 distribution. On the other hand, if we replace N by Nfree (to be determined), in order to turn the Dk into a 2p?1 distribution, we are able to estimate the equivalent number Nfree of independent points which would have given the expected 2p?1. If, in the inversion, we use such a correlation, the number of free parameters is then Nfree (and not N ). This shows that a lled covariance matrix could signi cantly decrease the degrees of freedom of the inversion.
Real data example We have performed two constrained inversions on ltered borehole data obtained from the North Sea. The geometry of acquisition can be seen from (Figure 4). The spatially discretized window of parameters to invert contains 230 horizontally elements by 456 vertical. The choice of a 1D model is motivated from a surface seismic migration image of the region, which shows an earth model almost 1D down to the depth of 3km. For the rst inversion, we have applied a classical 1D inversion which in term of number of parameters to invert, corresponds to a column (456 points) times the three physical parameters to invert (P-wave velocity, S-wave velocity and density), in total 1368 free parameters. The second inversion is a 2D inversion assuming a lateral and vertical stationarity of relation among points that have been described by the exponential model of correlation function, with a horizontal range of 15 points and a vertical range of 1.2 points. An estimation of the number of degrees of freedom (using the above described method) shows that we have reduced the number of parameters by a factor of 3. The 1D inversion converged after 45 iterations and was unable to explain the data, leaving 30 % of coherent residuals (Figure 5). The constrained 2D inversion explains most seismic phases, leaving 10 % of residuals that cannot be explained by our forward problem (3D e ects) and some small re ected phase residuals that could have been explained if we had not stopped the inversion at the 250th iteration due to slowness of convergence. Even though was weak our a priori knowledge of the geological structure of the region, we have been able to impose sensible constraints, reducing by a factor of 3 the number of parameters, and successfully explaining the data with a model (Figure 4) which is sensible from the geological point of view.
6 Conclusions Taking into account the physical phenomena observed in the data, will consequently increase the number of parameters. In our case, we have neglected the e ects of attenuation by ltering the data (thus loosing high frequency information on the properties of the medium); However, should we invert for the attenuation parameters, new constraints on these parameters should be found to slow in ation of number of degree of freedom. The incorporation of all our a priori knowledge of the parameters and all statistical studies on data, allow
Constrained Seismic Well Data Waveform Inversion
7 2D Inversion
0
500
500
1000
1000
1500
1500
2000
2000
Depth (m)
Depth (m)
1D Inversion 0
2500
2500
3000
3000
3500
3500
4000
4000
4500
4500
5000
5000 0
500
1000
1500
2000
2500
0
500
Offset (m) 700
1000
1500
2000
2500
Offset (m) 2700
4700
P Wave Velocity (m/s) Fig. 4. On the left the result of a 1D inversion model, on the right the result of a 2D inversion model with horizontal correlation. Even with horizontal correlation constraints applied to the 2D inversion, the earth model obtained shows dipping layers at the well bottom which explains the associated re ected wave observed in data.
the stabilization of the inversion process: the purpose being not necessarily to converge quickly towards a good model (in term of residuals), but to prospect regions of model that looks for us a priori sensible with the lowest mis t as possible.
References [CRA] Crase, E., Pica, A., Noble, M., McDonald, J., Tarantola, A.: nonlinear inversion: application to real data. Geophysics 55 (1990) 527{538 [GAR] Gardner, G. H. F., Gardner, L. W., Gregory, A. R.: Formation velocity and density{the diagnostic basis of stratigraphic traps. Geophysics 39 (1974) 770{ 780 [KOL] Kolb, P., Collino, F., Lailly, P.: Pre-stack inversion of a 1D medium. Proceedings of the IEEE 74 (1986) 498{508 [PIC] Pica, A., Diet, J. P., Tarantola, A.: Practice of nonlinear inversion of seismic re ection data in a laterally invariant medium. Geophysics 55 (1990) 284{292 [TA1] Tarantola, A.: Inverse Problem Theory: methods for data tting and model parameter estimation. Elsevier Science Publishing Co. (1987) [TA2] Tarantola, A.: Theoratical background for the inversion of seismic waveforms, including elasticity and attenuation. Pure Appl. Geophys. 128 365{399
8
Marwan Charara, Christophe Barnes and Albert Tarantola
Fig. 5. at the top: synthetic on the left and residuals on the right seismograms for 1D inversion; at the center: low pass ltered real data (0-12Hz), with amplitude gain on the left and without on the right; at the bottom, synthetic on the left and on the right residuals seismograms for 2D constrained inversion. The coecients of the gain (used for display purposes) has been computed from real data, and applied to all seismograms excepting real data at the right center. The 1D inversion was unable to explain correctly the data, the residuals (upper right) are still important and coherent. This not the case for the 2D horizontally constrained inversion, the residuals (bottom right) are mainly unstructured and the phases still unexplained are due to 3D e ects and not total convergence of the inversion. The obtained seismograms are amazingly very close to the real data, reproducing its complexity.
Constrained Seismic Well Data Waveform Inversion This article was processed using the LaTEX macro package with LMAMULT style
9
