3-D full waveform inversion of seismic data; Part I. Theory - eScholarship (2003)

3-D full waveform inversion of seismic data; Part I. Theory Ki Ha Lee*, Ernest Orlando Lawrence Berkeley National Laboratory, USA, KHLee@lbl.gov Summary Full waveform inversion of seismic data is a challenging subject partly because of the lack of precise knowledge of the source. Since currently available approaches involve some form of approximations to the source, inversion results are subject to the quality and the choice of the source information used. A new full waveform inversion scheme has been introduced (Lee and Kim, 2003) using normalized wavefield for simple two-dimensional (2-D) scalar problems. The method does not require source information, so potential inversion errors due to source estimation may be eliminated. A gather of seismic traces is first Fourier-transformed into the frequency domain and a normalized wavefield is obtained for each trace in the frequency domain. Normalization is done with respect to the frequency response of a reference trace selected from the gather, so the complex-valued normalized wavefield is source-independent and dimensionless. The inversion algorithm minimizes misfits between measured normalized wavefield and numerically computed normalized wavefield. In this paper the full waveform inversion is extended to three-dimensional (3-D) problems. Introduction It is a common practice in seismic industry to estimate subsurface velocity structure by analyzing the traveltimes of the seismic signals. In crosshole and surface-to-borehole applications, typical approaches involve ray tomography (e.g., Peterson et al., 1985; Nolet, 1985, Humphreys and Clayton, 1988; Scales et al., 1988; Vasco, 1991) and more recently Fresnel volume tomography (e.g., Cerveny and Soares, 1992; Vasco et al., 1995). Traveltime tomographies using ray tracing require high-frequency approximation, with maximum resolution on the order of a wavelength (Sheng and Schuster, 2000), or a fraction (5 %) of the well separation in some practical cases. Due to lack of resolution, however, usefulness of ray tomography may be limited if the objective is to better understand the petrophysical and hydrological properties of soils and rocks. Such understanding is important in characterizing petroleum and geothermal reservoirs and in environmental applications of various scales. An alternative to traveltime tomography is full waveform inversion. Recent studies (e.g., Sen and Stoffa, 1991; Kormendi and Dietrich, 1991; Minkoff and Symes, 1997; Zhou et al., 1997; Plessix and Bork, 1998; Pratt, 1999a, 1999b) suggest that full waveform inversion can provide improved resolution of the velocity and density structures. Amplitudes and phases of waveforms are sensitive to the petrophysical property of the materials through which the wave propagates. Therefore, full waveform analyses may be used as tools in investigating hydrological and petrophysical properties of the medium. There is, however, one major difficulty to overcome in full waveform inversion. In all field applications, the effective source waveform, the coupling between the source and the medium, and the coupling between the receivers and the medium, are not very well understood. The problem can be alleviated to some extent with a good velocity approximation (Pratt, 1999a), but the measured signals cannot be properly calibrated in general, rendering full waveform inversion technically difficult to apply. To overcome the above difficulty a simple inversion scheme has been proposed and tested using the normalized wavefield (Lee and Kim, 2003). The approach first transforms seismic data into the frequency domain and a set of normalized wavefield is constructed. The normalized wavefield is independent of the spectrum of the source, so the method allows full waveform inversion without requiring the knowledge of the source signature. Normalized wavefield Let us assume a field survey involving NS source positions and NG receiver positions. For general 3-D problems let us consider three component measurements at each receiver position and three source functions at each source position. It is assumed that measurements are in x, y, and z, and that the vector composition of each of three source functions is unknown. The measurements may be in the form of pressure, displacement, velocity or acceleration, and are described in general as d D d ji ( t ) = R j ( t ) ⊗ P ji ( t ) ⊗ S i ( t ) , j = 1, 2,…, NG, i = 1, 2,…, NS, where the superscript d indicates data and ⊗ signifies convolution in time. Here D dji ( t ) is the trace recorded at the j-th receiver position due to the i-th source. The source function S i (t) includes source system function and the radiation pattern caused by source-medium coupling, and d therefore is an effective source. P ji ( t ) is the impulse response of the medium at the j-th receiver position due to the i-th source. The receiver function R j (t) includes the effects of the receiver system function and the medium- receiver coupling. In the following analysis R j (t) will be ignored by assuming that receiver (geophone) calibration is