A comparison between time domain and depth domain inversion to acoustic impedance Laurence Letki*, Kevin Darke, and Yan Araujo Borges, Schlumberger

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1 Laurence Letki*, Kevin Darke, and Yan Araujo Borges, Schlumberger Summary Geophysical reservoir characterization in a complex geologic environment remains a challenge. Conventional amplitude inversion assumes reliable seismic amplitudes. In a complex environment, inadequate illumination of the subsurface due to complex geology or the acquisition geometry has detrimental effects on the amplitudes and phase of the migrated image. Such effects are not compensated for in conventional seismic inversion techniques. Consequently, an imprint of various nongeological effects will manifest themselves in the results of seismic inversion, leading to a less reliable estimation of the resultant acoustic and elastic parameters. The depth domain inversion workflow uses point spread functions to capture the dip-dependent illumination effects due to acquisition geometry and complex geology. The amplitude inversion is performed in the depth domain and the output is an acoustic impedance volume corrected for illumination effects. This paper presents the results of a field data example with the objective of comparing the results of the time domain inversion and the depth domain inversion, identifying and explaining both differences and similarities. This leads to an assessment of what should be expected from the depth domain inversion approach, including key advantages and limitations. Introduction Conventional amplitude inversion assumes that the seismic amplitudes are correctly located and can be inverted to elastic parameters from which a true representation of the rock properties can be derived. However, inadequate subsurface illumination due to complex geology or the acquisition geometry can have detrimental effects on the amplitudes and phase of the migrated image. Conventional amplitude inversion techniques do not compensate for these amplitude and phase variations. Consequently an imprint of various non geological effects, including illumination, will manifest themselves in the results of seismic inversion, leading to a less reliable estimation of the resultant elastic and rock properties. Additionally, in a complex geologic environment, depth imaging is required to obtain a reliable image of the subsurface, but current amplitude inversion techniques are usually implemented in the time domain. This creates a disconnect between the imaging and inversion steps which can compromise the fidelity of the attributes derived from seismic inversion. To address this simplification, we propose a technique to perform amplitude inversion directly in the depth domain. Correcting for the dip-dependent illumination effects caused by the acquisition geometry and complex geology, and thus creating consistent and more reliable imaging products and seismic inversion attributes from depth migrated data. Using the results of a real data case study, the objective of this paper is to compare the results of the time domain and depth domain inversion techniques, to identify and explain both the differences and similarities. Method The key input to the workflow is a grid of point spread functions (PSF). These are the impulse response of the modelling and imaging step. In mathematical terms, the migrated image m is related to the true reflectivity r by m = M*Mr = Hr where M is a modelling operator, M* is the migration operator and H=M*M is a Hessian operator, a measure of the illumination effects due to velocity variations and acquisition geometry, which blurs the true reflectivity to give the migrated image. The grid of PSF is an approximation of the Hessian operator. In other terms, it is a representation of the laterally and depth variant 3D wavelet embedded in the migrated image and it captures the dip dependent illumination effects due to acquisition geometry and complex geology. The depth domain inversion workflow then finds the best acoustic impedance model by minimizing the least squares objective function m - Hr 2 (Fletcher et al., 2012), where r is the reflectivity model corresponding to the acoustic impedance model. Additional constraints which can be included in the objective function relate to: Sparsity of the reflectivity model. Lateral continuity of the output along the geological structure. Deviation from a prior low-frequency model. This process can be seen as a least squares migration in the image domain and it is entirely performed in the depth domain. The output is an acoustic impedance image corrected for the dip-dependent illumination effects, as well as the associated reflectivity volume SEG SEG New Orleans Annual Meeting

2 and lower amplitudes associated with high structural dip (Figure 2). Analysis of the grid of RTM point spread functions across the area support this observation. Overall, the PSF show variations with depth but little lateral variations. In the presalt areas studied, the PSF are capturing clear dip dependent variations in the wavelet as illustrated in Figure 2. Depth domain inversion uses the grid of PSF to represent the 3D wavelet embedded in the RTM image and the inversion is performed directly in the depth domain. Conventional time domain inversion is using a 1D wavelet extracted at the well location. The inversion is performed in the time domain and the output is stretched back to depth for comparison purposes. All other parameters of the inversion have been kept as similar as possible. Figure 1: location. Field data example RTM image and available acoustic impedance log This case study is located in a complex salt environment. The target is located in the pre-salt area of the Brazil Santos Basin below a thick salt body consisting of an upper layered salt on top of a consolidated salt layer. The structure of the layered salt is complex with structural dips up to 40. The analysis is located around a well location where an acoustic impedance log is available for a portion of the salt and presalt layers (see Figure 1). An image of the subsurface has been obtained by reverse time migration (RTM) of full azimuth data (Figure 1). Amplitude variations are observed along seismic events with stronger amplitudes associated with low structural dip Figure 3 presents a comparison of the acoustic impedance outputs at the well location. In both cases, a good well tie is observed in the window used to extract the time domain 1D wavelet. At this location, the time domain wavelet is appropriately representing the wavelet embedded in the image and therefore we should not expect any significant difference between the two inversion methods. However, depth domain inversion leads to a better match in the shallower part of the well log interval, with more balanced variations across the layered salt. The depthdependent variations of the wavelet have been captured by the PSF and handled by the depth domain inversion. The benefits of the depth domain inversion over time domain inversion become even clearer away from the well location. In this case, it is particularly visible in areas of significant structural dip variations, such as in the layered salt. The amplitude variations observed in the acoustic Figure 2: Inline through RTM image highliting amplitude differences depending on the local strutural dip (left) and PSF exracted at both highlighted location showing little lateral variations (right). The FK spectra illustrates the dip dependent effects.

3 stable, this case study shows that the depth domain inversion results are comparable to the time domain results, with no degradation of the well tie, admittedly at the significantly increased cost of generating the grid of PSF. The time domain inversion must be constrained to an area small enough to justify using a spatially invariant wavelet. On the other hand, since the dip dependent spatial wavelet variations are captured by the grid of PSF, the depth domain inversion can in principle be run on a much larger volume of interest. Note that the PSF will not only capture dip dependent wavelet effects but also spatially variant illumination effects due to the complex geology and acquisition geometry. Although the spatially variant effects were limited in the present case study, depth domain inversion has proven reliable in correcting such effects in field examples (Letki et al., 2015). Figure 3: Comparison of time domain inversion results (stretched back to depth) and depth domain inversion results at well location. In the highlighted area and above, the amplitude variations are more balanced in the depth domain inversion results, with a better well match in the highlighted area. impedance volume from the conventional time domain inversion are strongly correlated to the dip dependent amplitude variations of the RTM image. The depth domain inversion output shows an overall better balanced acoustic impedance and a reduced imprint of the RTM image amplitudes. This is illustrated in Figure 4. In the pre-salt area, with no significant structural dip variations, the results of the time and depth domain inversion are more comparable. The 1D wavelet remains appropriate across the survey area in this particular layer. Finally, the reflectivity outputs show an increased bandwidth compared with the original image, with the flatter spectra coming from the depth domain inversion results (see Figure 5). Advantages and limitations With the depth domain inversion workflow, the grid of PSF is capturing the dip dependent variations of the 3D wavelet across the target area. This leads to a more balanced acoustic impedance volume compared to the time domain inversion results which shows a strong imprint of the dipdependent wavelet effects. In areas where the wavelet is Limitations will come from the spatial and depth sampling of the PSF locations. This sampling should match the wavelength of the 3D wavelet variations embedded in the data. In presence of strong boundaries, such as the top salt in the presented case study, the linear interpolation assumption is not valid anymore and the results will not be as reliable around such a strong contrast. This effect can be mitigated by increasing the density of PSF and by implementing more adequate non-linear interpolation schemes. Conclusions The depth domain inversion workflow illustrated in this paper uses point spread functions to capture the dip dependent effects due to acquisition geometry and complex geology. The amplitude inversion is performed in the depth domain. Comparisons of the acoustic impedance results with conventional time domain inversion results highlight a more balanced amplitude behavior in areas originally affected by strong dip dependent wavelet effects. The comparison with conventional time domain inversion leads to a cost versus expected benefits balance and the answer depends on the target area and faced challenges. Depth domain inversion provides a mechanism to produce a higher fidelity acoustic impedance and reflectivity image in presence of strong spatial and/or dip dependent wavelet effects due to complex geology or acquisition geometry. Acknowledgements The authors would like to thank Petrobras and Schlumberger for the permission to publish this work and for the permission to use the data.

4 Figure 4: Comparison of time domain inversion acoustic impedance results (stretched back to depth) and depth domain inversion acoustic impedance at well location. In the highlighted area and above, the amplitude variations are more balanced in the depth domain inversion results, with a better well match in the highlighted area. Figure 5: Spectral analysis comparing the original RTM image, the time domain reflectivity output (reflectivity TDI) and the depth domain reflectivity output (reflectivity DDI) showing the increased bandwidth in the reflectivity images. All volumes are stretched to time for the analysis and all spectra are normalized to the maximum amplitude.

5 EDITED REFERENCES Note: This reference list is a copyedited version of the reference list submitted by the author. Reference lists for the 2015 SEG Technical Program Expanded Abstracts have been copyedited 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 Fletcher, R. P., S. H. Archer, D. Nichols, and W. Mao, 2012, Inversion after depth imaging: Presented at the 82nd Annual International Meeting, SEG. Letki, L., J. Tang, and X. Du, 2015, Depth domain inversion case study in complex subsalt area: 77th Annual International Conference and Exhibition, EAGE, Extended Abstracts, SEG SEG New Orleans Annual Meeting