CSIRO PUBLISHING
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Exploration Geophysics, 2009, 40, 227–232
Frequency-domain waveform inversion using an l1-norm objective function Sukjoon Pyun1,3 Woohyun Son2 Changsoo Shin2 1
Research Institute of Energy and Resources, Seoul National University, 599 Gwanangno, Gwanak-gu, Seoul 151-742, Korea. 2 Department of Energy Systems Engineering, Seoul National University, 599 Gwanangno, Gwanak-gu, Seoul 151-742, Korea. 3 Corresponding author. Email:
[email protected]
Abstract. In general, seismic waveform inversion adopts an objective function based on the l2-norm. However, waveform inversion using the l2-norm produces distorted results because the l2-norm is sensitive to statistically invalid data such as outliers. As an alternative, there have been several studies applying l1-norm-based objective functions to waveform inversion. Although waveform inversion based on the l1-norm is known to produce robust inversion results against specific outliers in the time domain, its effectiveness and characteristics are yet to be studied in the frequency domain. The present study proposes an algorithm for l1-norm-based waveform inversion in the frequency domain. The proposed algorithm employs a structure identical to those used in conventional frequency-domain waveform inversion algorithms that exploit the back-propagation technique, but displays robustness against outliers, which has been confirmed based on inversion of the synthetic Marmousi model. The characteristics and advantages of the l1-norm were analysed by comparing it with the l2-norm. In addition, inversion was performed on data containing outliers to examine the robustness against outliers. The effectiveness of removing outliers was verified by using the l1-norm to calculate the residual wavefield and its spectrum for the data containing outliers. Key words: back-propagation algorithm, l1-norm, waveform inversion.
Introduction Seismic waveform inversion has received much attention since Tarantola (1984) proposed an algorithm for efficiently calculating the gradient direction of an objective function using the adjoint characteristics of the wave equation operator. Since then, researchers including Gauthier et al. (1986), Mora (1987) and Pica et al. (1990) have published studies regarding the numerical implementation of waveform inversion. During this time, most studies of waveform inversion dealt with application in the time domain. As quantified numerical approaches for modelling of the wave equation in the frequency domain were introduced, researchers began active studies of waveform inversion techniques in the frequency domain (Shin, 1988; Pratt and Worthington, 1990; Song et al., 1995). In addition, with the recognition of the advantages of wave equation modelling in the frequency domain (Pratt, 1999), studies of waveform inversion shifted from the time domain to the frequency domain. Numerous studies have been conducted to improve the waveform inversion technique in the frequency domain, and the results of studies on 2D data can now be applied to real data successfully (Sirgue and Pratt, 2004; Ravaut et al., 2004; Shin and Min, 2006; Operto et al., 2006; Shin et al., 2007). Waveform inversion generally adopts an l2-norm as the objective function in both time domain and the frequency domain. However, it is widely known that the l2-norm is sensitive to statistically invalid data (Claerbout and Muir, 1973). Several other objective functions, including the l1-norm, have been considered as alternatives and applied in waveform inversion (Crase et al., 1990; Djikpesse and Tarantola, 1999). Although these alternative objective functions have been Ó ASEG 2009
applied to waveform inversion in the time domain, their effectiveness and characteristics have not been studied in the frequency domain. Accordingly, the present study aims to perform waveform inversion using an objective function defined with the l1-norm in the frequency domain, and to analyse its behaviour. In the next section, the descent direction is derived for an objective function defined with the l1-norm in the frequency domain and applied to the theory of the back-propagation technique. Inversion was performed on the Marmousi model for verification through numerical examples, and the properties and benefits of the l1-norm were analysed by comparison with the l2-norm. To verify the robustness against outliers, inversion was performed on data containing outliers, and the characteristics and spectrum of the residual wavefield were analysed.
Inversion theory The objective function defined with the l1-norm in the frequency domain is written as EðmÞ ¼
ns X nr X fjRe½uij ðmÞ d ij j þ jIm½uij ðmÞ d ij jg; ð1Þ i¼1 j¼1
where m is the model vector that parameterises the P-wave velocity of the medium, uij (m) is the wavefield acquired from modelling, dij is the measured wavefield, subscript i is the source number, j is the receiver number, ns is the total number of sources, and nr is the total number of receivers for each corresponding source. Differentiating equation (1) with respect to parameter mk 10.1071/EG08103
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to minimise the objective function yields the following directional derivative: " # ns X nr X quij ðmÞ rij ; rmk EðmÞ ¼ Re ð2Þ qmk i¼1 j¼1
operator matrix, and ri indicates the residual wavefield vector. In the proposed algorithm using the l1-norm, ri in equation (4) is defined using rij of equation (3) as follows: ri ¼ ½ri1
ri2
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rinr
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0 L 0T :
where rij ¼ sgnðRe ½uij ðmÞ d ij Þ þ i sgnðIm ½uij ðmÞ d ij Þ;
ð3Þ
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and sgn() is the signum function. Applying the back-propagation algorithm, the partial derivative wavefield of equation (2) can be obtained from the inner product of a virtual source and the Green function, and the steepest descent direction can be established from the directional derivative of every parameter as follows: " # ns X t 1 ð4Þ ðFi Þ S ri ; rE ¼ Re
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Fig. 2. A common-shot gather seismogram generated by a finite-element modelling algorithm.
Fig. 3. Residual seismograms obtained by (a) the l2-norm and (b) the l1-norm.
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The diagonal elements of the pseudo-Hessian matrix proposed by Shin et al. (2001) were used to calibrate the amplitude in the steepest descent direction calculated from equation (4), and a damping term was added to stabilise the solution: rE reg ¼ ðdiagðHp Þ þ lIÞ1 rE;
ð6Þ
where Hp denotes the pseudo-Hessian matrix and l the damping constant. The descent directions acquired for each frequency are added so that they have equal weight factors to establish the final descent direction.
Numerical examples Marmousi model and synthetic data The Marmousi model was used to verify the waveform inversion algorithm proposed in this paper (Versteeg, 1994). The density was assumed to be uniform, and the P-wave velocity was distributed as shown in Figure 1. The model included 288 sources in the oceanic layer, 16 m below the sea surface at 32 m intervals, and 575 geophones at 16 m intervals for each source. The synthetic data were generated with a frequencydomain modelling program using the finite-element method.
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The source waveform used to generate the synthetic data was a first-order derivative of a Gaussian function with a maximum frequency of 18.75 Hz. Figure 2 depicts the common shot gather when the transmitting source was located at 4.8 km. Inversion of data without outliers Inversion was performed using objective functions based on the l2- and l1-norms for verification of the algorithm and comparison with conventional waveform inversion algorithms. Figure 3 illustrates the residual wavefield data between the observed data (Figure 2) and the modelled databased on each norm. (a)
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With some direct waves removed from the observation data, the residual wavefield defined by l2-norm (Figure 3a) displays a structure similar to that of a common shot gather. However, the residual wavefield defined by l1-norm (Figure 3b) displays reinforced high-frequency components and outliers as if the data had been processed with a spiking filter. To perform
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Fig. 5. Inverted velocity models using (a) the l2-norm and (b) the l1-norm objective functions.
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Fig. 6. A common-shot gather seismogram including outliers.
Fig. 7. Residual seismograms obtained by (a) the l2-norm and (b) the l1-norm for data including outliers.
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detailed analysis of the properties of the two residual wavefields, a trace was selected at random, and its spectrum was calculated (Figure 4). Whereas the residual wavefield defined by the l2-norm displays a typical amplitude spectrum (Figure 4b), the residual wavefield defined by the l1-norm shows a uniform amplitude spectrum. However, unlike the amplitude spectra, the phase spectra (Figure 4c) are almost identical. Because of these observations, it is believed that an objective function defined by the l1-norm produces a stable inversion result against outliers that distort the amplitude spectrum. Figure 5 displays the inversion results of the two methods. It can be seen that there
is a velocity distortion at the left 1 km point of the velocity model inverted by the l2-norm (Figure 5a). In comparison, it can be confirmed that inversion by the l1-norm (Figure 5b) yields an accurate distribution of velocity down to the deep layer. Inversion of data containing outliers Several outliers were added to the data used in the previous section to verify the response to outliers (Figure 6). The residual wavefields for the data containing outliers are shown in Figure 7. The qualitative properties of the wavefields are similar to those of
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Acknowledgment
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This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST; No. R0A2006–000–10291–0) and the Brain Korea 21 project of the Ministry of Education.
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References
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expected to reduce pre-processing efforts such as data editing and provide a means of robust inversion.
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Fig. 9. Inverted velocity models using (a) the l2-norm and (b) the l1-norm objective functions.
the example presented in the previous section, the only difference being the presence of outliers. For detailed analysis, traces where the outliers are located were selected and the amplitude and phase spectra were displayed (Figure 8). The traces in Figure 8a indicate that there is an outlier around 1.3 s. Although the amplitude of the outlier is clearly conspicuous in the residual wavefield defined by the l2-norm, it is very small in the residual wavefield defined by the l1-norm. Although outlier removal was not as evident as in the time domain, it was confirmed that the effects of outliers were alleviated. The amplitude spectrum for the l2-norm (Figure 8b) is clearly different from that without the outlier (Figure 4b) but identical for the l1-norm. The phase spectrum (Figure 8c) was similar to that for the data without outliers (Figure 4c). The differences in residual wavefields according to norm definitions are evident from the inversion results. Figure 9 displays the velocity models inverted with each of the two methods. It can be confirmed in the result of l2-norm inversion (Figure 9a) that the velocity structure of the trace containing outliers was distorted below the measurement point. However, the result of l1-norm inversion was identical to that for the data without outliers (Figure 5b). Conclusion This paper proposed an algorithm that adopts the l1-norm for waveform inversion in the frequency domain. As with conventional waveform inversion using the l2-norm, the proposed algorithm uses the adjoint characteristics of wave equations to efficiently obtain the descent direction of an objective function. The difference with conventional approaches is that the back-propagated residual wavefield relies only on the signs of the real and imaginary parts. Unlike applications in the time domain, the l1-norm tends to flatten the amplitude spectrum, compressing the waveforms in the residual wavefield to a spike in the time domain. The effect reinforces the low-frequency region, reducing the non-linearity of the objective function and enhancing the result of inversion. It is believed that flattening of the amplitude spectrum of the data containing outliers minimises the effect caused by the outliers, and the relatively small variation of the phase spectrum allows an accurate inversion of the velocity structure that determines the traveltime of seismic waveforms. The proposed algorithm is
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Manuscript received 4 April 2008; revised manuscript received 20 August 2008.
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