Full waveform inversion of physical model data Jian Cai*, Jie Zhang, University of Science and Technology of China (USTC) Summary Full waveform inversion (FWI) is a promising technology for next generation model building. However, it faces with many challenges when dealing with real data. To understand some of the issues in real data, we study data collected from ultrasonic lab with a known model by forward simulation and full waveform inversion. We identify the differences in phase, amplitude, and AVO effects between recorded data and synthetics, and explore the impact on FWI results due to these differences. We compare two amplitude scaling methods in FWI to handle AVO effects and find that scaling shot gathers combined with 3D-to-2D conversions producing more accurate result. Introduction Full waveform inversion, based on the finite difference approach for forward modeling and backward propagation of residual wavefield for gradient calculation, was originally proposed in time space domain (Tarantola, 1984; Gauthier et al., 1986), and then successfully implemented in the frequency domain (Pratt et al., 1999). Since full waveform inversion naturally takes into account more general wave propagation effects compared to the high frequency method of traveltime tomography, it should be able to estimate a wide range of slowness wave-numbers. Full waveform inversions attempt to iteratively minimize the misfit between synthetic and input data; therefore, the wavefield calculation must be sufficiently accurate. Physical modeling data provides a useful link between theory and field scale experiments. Using a physical model approach, we are able to measure changes in the acoustic response to simulate real structures in pseudo ideal settings. In this study, we simulate the numerical wavefield with a physical model for Qianshan area (South of China) (Wei and Di, 2006). Then we compare the waveforms between synthetic and real data, and we find that acoustic approximation is sufficient, while AVO effects are very different. Finally, we apply FWI to the data and try different amplitude scaling approach to mitigate the AVO effect. Theory Ultrasonic modeling seismic experiment, as one of important methods in geophysical modeling study, is based on real wave propagation whereas numerical modeling is based on algorithms which are by necessity simplified and discretized versions of the real world. While observed in much smaller scale, physical modeling obeys the same wave propagation rules, which is known as the similarity criterion (Sun et al., 1997).

Consider the wave equation in the regular isotropic media

22 2

2v t∂

∇ Φ = Φ∂

where v is model velocity, Φ is displacement potential, and t is time. Assuming the model velocity in physical model is vm, the displacement potential is Φm, and time is tm, then we obtain the wave equation as

22 2

2m m mm

vt∂

∇ Φ = Φ∂

We set the ratio of parameters in difference scales as

Lm m m m

L x y zRL x y z

= = = =

, ,t vm m m

t vR R Rt v Φ

Φ= =

Φ=

where RL, Rt, Rv, RΦ are the space, time, velocity and displacement potential ratio, respectively. Substituting equation (3) into equation (2) we then obtain

22 2

2( )v t

L

R RvR t

∂∇ Φ = Φ

∂

According to the similarity criterion, equation (1) and (4) are equivalent, thus we derive equation as

1v tL

R RR

=

On the basis of equation (5), we set appropriate ratios for the experiment (Table 1). Table 1: Scaling factors for the physically recorded data.

Scaling description

Variant

Scaling factors

Unit of converted physical data

Space Time

Velocity Frequency

Sampling rate

10000:1 10000:1

1:1 1:10000 10000:1

m s

m/s Hz

KSPS

Laboratory Settings The model used in the experiment is simulated for a fault of Qianshan (South of China) (Wei et al., 2002; Di et al., 2008), with a size of 757.2mm × 756.9mm × 243.5mm (Figure 1a). The properties of the model are set similar to the real geological medium, of which the details are shown

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Full waveform inversion

in Table 2. A 27mm column of water (with P wave velocity of 1480 m/s) is created above the physical model to simulate a marine case. According to description, we build a synthetic model with the same detail properties (Figure 1b). The central frequency of the source generator is 225 KHz and the diameters of the source and receiver are 3 mm and 5 mm, respectively. There are 161 shots in total, with 44 receivers on the west side. The shot and receiver intervals are both set as 2 mm in the experiment. a)

b)

Figure 1: (a) Sketch of the physical model (courtesy of Bangrang Di and Jianxin Wei). (b) Synthetic model built from the sketch, with same detail properties. Table 2: Detail properties of the physical model.

Physical model description

Layer

Designed velocity

(m/s)

Field P wave

velocity (m/s)

Field S wave

velocity (m/s)

Density (g/cm3)

T2

1350

1281

422

1.062 T3 1600 1597 563 1.075 T4 1850 1880 803 1.118 T5 2100 2028 896 1.140 T6 2600 2707 1241 1.194 T7 2800 2933 1479 1.616 T8

3000

3233

1686

1.809

Numerical Simulations The numerical method we use is time domain staggered grid finite difference with a perfectly matched layer (PML) for boundary conditions. Three kinds of comparisons between synthetic and physically recorded data are shown as follow: 1) 3D elastic and physically recorded waveform 2) 3D acoustic and 3D elastic waveform

3) 2D acoustic and 3D acoustic after 3D-to-2D conversion 3D elastic and physically recorded waveform: In the first comparison, we show the waveform, amplitude spectrum, phase spectrum and AVO overlays in Figure 2. From the waveform (Figure 2a) and AVO (Figure 2e) overlays, it can be seen that amplitude of data from the physical model, decreases at the same rate at both near and far offsets, although with different RMS amplitude magnitudes. The input model lack of frequency under 7 Hz, also we do not account for the attenuation effect in the wavefield modeling, hence those might be the reason that the amplitude spectrum does not fit well in lower and higher frequency (Figure 2c). The phase difference around the centre frequency is mostly below 20 (Figure 2d). a) b)

c) b) e)

Figure 2: (a) 3D elastic (Red) and physically recorded (Black) waveform overlay. (b) Waveform difference. (c) Amplitude spectrum overlay. (d) Phase spectrum overlay. (e) AVO overlay. 3D acoustic and 3D elastic waveform: In the second comparison, it is observed from the waveform overlay (Figure 3a) that there is barely converted S wave in the waveform which means the waveform is dominated by the P wave, and it is a useful property for acoustic full waveform inversion. We can see the amplitude spectrum (Figure 3c) fits well. The phase difference is mainly around 20, and it decreases as frequency goes lower. The AVO overlay (Figure 3d) shows similar variations as in the first comparison. a) b)

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Full waveform inversion

c) d) e)

Figure 3: (a) 3D acoustic (Red) and 3D elastic (Black) waveform overlay. (b) Waveform difference. (c) Amplitude spectrum overlay. (d) Phase spectrum overlay. (e) AVO overlay. 2D acoustic and 3D acoustic after 3D-to-2D conversion: Techniques to resolve the 3D-to-2D problem can roughly be classified into three categories (Roberts, 2005): reformulating the problem in cylindrical coordinates and synthesizing line source data by integrating over many point sources in CMP domain (Wapenaar et al., 1992); asymptotic 2.5D filtering procedures (Bleistein, 1986) to convert the 3D data into 2D data; and the true 2.5D modeling approach by using Fourier transform to solve the 2D problem (Song and Williamson, 1995). The 3D-to-2D data conversion using an asymptotic filter is by now the most widespread approach. It has the advantages of both simple implementation and inappreciable computational cost. The asymptotic filter (Bleistein, 1986) for a homogeneous acoustic medium is

2 3 2( ) ( ) exp( )4

D D iG G πσ πω ωω

= ⋅ ⋅

where ω is the angular frequency, the quantity σ is defined as σ= cr, c is the acoustic wave speed, and r is the distance. Applying an inverse Fourier transform to equation (6) yields the filter in time domain (Aki and Richards, 2002):

2 3(t)(t) 2 * (t)D DUG Gt

πσ ⎛ ⎞= ⎜ ⎟⎝ ⎠

We apply the filter function of equation (7) to 3D acoustic waveform, and the comparisons are shown in Figure 4. The waveform (Figure 4a) and AVO overlay (Figure 4e) show that 2D acoustic and 3D acoustic after 3D-to-2D conversion fit very well. The amplitude spectrum (Figure 4c) shows very small difference, and the phase spectrum shows acceptable fitting, except for the low frequency (

Full waveform inversion

a) b)

c) d)

Figure 6: (a) Inverted model. (b) Difference of initial and inverted model. (c) Waveform from initial model (Red) overlaying on input data (Black). (d) Waveform from inverted model (Red) overlaying on input data (Black). Scale input data to synthetic data: In order to make it comparable with synthetic data, we apply a scalar to modify the AVO effect on real data

2

syn obsE d dλ= −

where E is the objective function, dsyn is the synthetic wavefield, dobs is the observed wavefield, andλ is the scalar for each input trace, which is defined as

2 2syn obs

t t

syn obs

d d

N Nλ =

∑ ∑

where Nsyn and Nobs are the sample numbers of each input trace of synthetic and observed data, respectively. After applying the scalar to real data, we then try to minimize the scaled residual energy. a) b)

c) d)

Figure 7: (a) Inverted model. (b) Difference of initial and inverted model. (c) Waveform from initial model (Red)

overlaying on input data (Black). (d) Waveform from inverted model (Red) overlaying on input data (Black). Figure 7a shows the final inverted velocity model, and Figure 7b is the difference of initial and inverted model. Comparing with previous results, we can see that this scaling approach performs better than the one using trace normalization, for which totally disregards the AVO effect. Conclusions We simulate the numerical wavefield with a physical model for Qianshan area (South of China), and find that there are some mismatches on the amplitude, phase and AVO effects, which should make impact on full waveform inversion. During the analysis, we find that the AVO effect affects the FWI results, therefore we apply different amplitude scaling approaches to mitigate the effect, and the results show that scaling shot gathers along with 3D-to-2D conversion can handle AVO effects reasonably well.. For further work, we will try to investigate the approaches that can correct the AVO effect more efficiently. Acknowledgements We would like to thank our colleagues for discussion about the topics. We thank the support from GeoTomo, who offered the software package TomoPlus for this study. We also thank China University of Petroleum (Beijing) CNPC geophysical Key Laboratory for supplying us with the physical model data.

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http://dx.doi.org/10.1190/segam2014-1253.1 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2014 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES

Aki, K., and P. G., Richards, 2002, Quantitative seismology: University Science Books.

Bleistein , N., 1986, Two-and-one-half-dimensional inplane wave propagation: Geophysical Prospecting, 34, no. 5, 686–703, http://dx.doi.org/10.1111/j.1365-2478.1986.tb00488.x.

Di, B., X. Xu, and J. Wei, 2008, A seismic modeling analysis of wide and narrow 3D observation systems for channel sand bodies: Applied Geophysics, 5, no. 4, 294–300, http://dx.doi.org/10.1007/s11770-008-0022-6.

Gauthier, O., J. Virieux, and A. Tarantola , 1986, Two-dimensional nonlinear inversion of seismic waveforms: Numerical results: Geophysics, 51, 1387–1403, http://dx.doi.org/10.1190/1.1442188.

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Tarantola , A., 1987, Inverse problem theory: Methods for data fitting and model parameter estimation: Elsevier.

Wapenaar, C. P. A., D. J. Verschuur, and P. Herrmann, 1992, Amplitude preprocessing of single and multicomponent seismic data: Geophysics, 57, 1178–1188, http://dx.doi.org/10.1190/1.1443331.

Wei, J., and B. Di, 2006, Properties of materials forming the 3D geological model in seismic physical model: Geophysical Prospecting for Petroleum, 45, 586–590.

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Zhang, W., and J. Zhang, 2011, Full-waveform tomography with consideration for large topography variations: 81st Annual International Meeting, SEG, Expanded Abstracts, 30, 2539.

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