Towards seismic moment tensor inversion for source mechanism

Transcription

1 Towards seismic moment tensor inversion for source mechanism Faranak Mahmoudian and Kristopher A. Innanen CREWE, University of Calary ummary The objective of this work is to obtain seismic moment tensor, M pq, from amplitude inversion of multicomponent microseismic data. M pq describes source mechanism and can be decomposed into doublecouple, isotropic and compensated-linear-vector-dipole (related to tensile fracturin) components - the tensile components miht show a correlation to hydrocarbon production rate. The retrieved M pq are sensitive to the accuracy of the source location, the accuracy of the velocity model, and the receiver array eometry. Excludin the first two factors, the determination of the proper observation eometry for which the full moment tensor is resolvable was souht. Havin two vertical or surface receiver arrays, or a combination of them, M pq may be fully determined usin the inversion of - and -wave first arrival amplitudes. This confiuration, which is sufficient for this purpose, includes the receiver arrays located in the vertical and horizontal part of a sinle deviated well. To avoid the cumbersome task of pickin firstarrival amplitudes, we adopted a waveform inversion scheme based on the method proposed by Vavrycük and Kühn (01). This method combines inversions in both the time and frequency domains. The first requirement was the inversion for the source-time function in the frequency domain. econd, a time domain inversion for M pq usin the source-time function calculated in the first step was initiated. For this waveform inversion, we have not yet achieved an accurate retrieval of M pq, but we have been able to obtain an accurate source-time function estimate. Introduction Determination of seismic moment tensor is a routine procedure in earthquake seismoloy (e.., Dziewonski et al., 1981; Jost and Hermann, 1996; Šílený, 1998; hearer, 1999). The seismic moment tensor provides a eneral representation of the seismic source and can be determined from inversion of seismic amplitudes detected on surface/downhole receiver arrays. M pq is commonly decomposed into a double-couple tensor, produced by shear faultin, and a non-double couple tensor, produced by the openin or closin of faults (tensile fracturin) (Vavryčuk, 001). Hence M pq provides information related to the physical processes at the source (e.., Ross et al., 1999; Maxwell and Urbancic, 001; Fouler et al., 004; Vavryčuk, 007). A point source with a eneral radiation pattern can be represented by the seismic moment tensor. In a Cartesian coordinate system, the seismic moment tensor is symmetric matrix of six independent force-couples M pq (M 11, M, M, M 1 ). A force-couple M pq is defined as a pair of opposin forces pointin in the p direction, separated in the q direction. The -components of the displacement wavefield observed at a eophone located at due to a source located at (which described by its moment tensor), are expressed as (Aki and Richards, 00, equation.) where is convolution symbol, n =1,,, summation convention is applied, and of source-receiver offset. m pq (t) is the time-dependent seismic moment tensor. GeoConvention (1) is the vector is the spatial derivative of the Green s function which describes the properties of the medium. For a point source, the time-dependent moment tensor can be factorized as m pq (t) = M pq (t), where (t) is the source-time function and M pq is the seismic moment tensor which are tryin to invert for.

2 Determination of the M pq is a demandin procedure influenced by several factors. The most important of these are: 1) Observations from many stations (ood station coverae of the focal zone), which here is termed receiver eometry effect. ) Derivin the representative source-time function. ) An accurate knowlede of the velocity model and focal location, in order to calculate the Green s functions. 4) The availability of quality data, to enable event pickin in the source location step. We report the effect of receiver eometry on retrieval of the seismic moment tensor, for vertical or surface arrays while inorin the other three factors listed above. As was established by others (e.., Vavryčuk, 007; Rodriuez et al., 011), we found that the full moment tensor may be determined usin - and -wave first-arrival amplitudes from two receiver arrays (vertical or surface) observin the source from two distinctive azimuths. Usin data from a sinle vertical or surface array, five terms may be resolved. Havin receivers in both vertical and horizontal part of a deviated well, we found, reardless of the azimuth of the well, the full moment tensor is resolvable when usin - and -wave amplitude data. Another key factor in recoverin seismic moment tensor is the determination of the time variation of the source, (t). The source-time function is often assumed to have a simple form, a sinle pulse appearin in all elements of the moment tensor. While this is an oversimplified assumption for earthquake sources, it is a reasonable assumption for a point source directly applicable to microseismic events. With this assumption, the determination of M pq and the source-time function is still a non-linear problem (from equations ) and requires non-linear inversions of the seismic waveform (Šílený, 1998). Avoidin a nonlinear inversion, we follow Vavryčuk and Kühn (01) and apply a two-step linear waveform inversion first to estimate the source-time function in frequency domain, and then resolvin the M pq in time domain. Moment tensor inversion of - and - wave amplitudes to investiate receiver eometry effect The -components of the total far-field displacement wavefield (equation ) can be written as the sum of the far-field - and -wave displacements, which for a homoeneous medium are expressed as (Aki and Richards, 00) () () r, a, and b are the density, -velocity, and -velocity respectively. distance. Comparin equations - to equation 1, the Green s function can be defined as is the source-receiver. (4) Equation () can be written as, which for simplicity we took A = 1 4pra assumed that the source-time function is delta function with unit amplitude. This has the form of matrix multiplication; hence equation () can be written as u u = A M 11 M 1 M 1 M 1 M M M M Doin this matrix multiplication, the -wave displacement can be written as a linear function of the six independent M pq as. and (5) GeoConvention 017

3 u u 1 = A 1 1 imilarly, from equation, the -wave displacement can be written as u u = B M 11 M M M M 1. (6) which for simplicity B is defined similar to A. Usin equations 6 and 7, - and -wave first arrival amplitudes from multiple receivers can be used to set up a linear system of equations D = GM, where D is data vector (- and -wave first arrival amplitudes), G is the eometry matrix, and M the model parameters (M 11, M, M, M 1 ); we solved it usin damped least-squares method (the dampin factor is decided based on the ratio of the max/min sinular values of the matrix G. We tested the moment tensor inversion on C synthetics seismorams for a homoeneous isotropic medium with α = 000 m/s, β = 000 m/s, and ρ = 000 K/m. We enerated the C synthetics data by convolvin a source-time function with the Green s functions. The synthetics data are recorded at multiple borehole/surface arrays and used in the moment tensor inversion described above. For all of our tests, the moment tensor source of M = (M 11, M, M, M 1 ) = (1,-, 4,-1,0.5,6) has been used. The resolvability of the full moment tensor, under sinle and multiple vertical/surface receiver array eometries, is examined. Employin - and -wave amplitudes simultaneously in the moment tensor inversion, we found that 1) Havin a sinle vertical or surface array, all elements but the M will be resolved and there exists only one null sinular values. ) Havin two vertical or surface array, the full moment tensor will be resolved. ) The resolvability of the model parameters is insensitive to the distance between the array of receivers and the source. As most of the microseismic monitorin are based on a sinle well confiuration, the use of receiver arrays in a deviated well which receivers are laid out in both vertical and nearly horizontal part of the well is suested. Hence a simultaneous amplitude inversion, usin data from a deviated well should satisfy the resolvability of full moment tensor. This deviated well confiuration was suested by Rodriuez et al. (011). Fiure 6 shows a deviated well, with receivers in both vertical and horizontal part of the well, that full moment tensor is retrievable. M 11 M M M M 1, (7) Fiure 1: Moment tensor inversion for a deviated well eometry, displayin the resolution matrix and sinular values GeoConvention 017

4 resented above, the amplitude data were picked from synthetics noise-free seismorams; we inored all the complexity in pickin the - and -wave first arrivals. For real data, first arrival amplitude pickin is a cumbersome task and normally performed on data after bein rotated into radial and transverse components. Additionally, knowin the source-time function that was used in eneratin the numerical modeled data, we applied the accurate scalar to remove the effect of source-time function. However, knowlede of the source-time function is never achieved in real data prior to the moment tensor inversion. Next followin Vavrycuk and Kuhn (01), we apply a waveform inversion to first estimate the source-time function and second estimate the full moment tensor in a time-frequency approach. No amplitude pickin is required for the followin waveform inversion. Moment tensor inversion of waveforms tep 1: The pair of indices in the Green s function (equation 4) can be reduced to define the G kj (k =1:, j =1: 6) functions as: G k1 = k1,1, G k = k,, G k = k,, G k4 = k, + k,, G k5 = k1, + k1,, and G k6 = k1, + k,1. Then in frequency domain equation 1 will transform For each frequency, ω, equation 8 could be used to construct a linear system of equations "d = Gm" to invert for the m(w) = (m 11 (w),m (w), m (w),m (w),m 1 (w),m 1 (w)), which the data vector is the Fourier transform of the recordins at each receive. We solved this linear system usin a least-squares method. Then takin Fourier transform of each column of this m(ω) matrix will result in the matrix of the six timedependent moment tensor vectors, m(t) = (m 11 (t),m (t),m (t),m (t),m 1 (t),m 1 (t)). We took the sinular value decomposition of this m(t) matrix, the eienvector associated with the larest sinular value is the source-time function, (t) (Vasco,1989). tep : Now that the source-time function is estimated, it will be taken out of the displacement wavefield. In this reard, the elementary seismorams and their spatial derivatives are defined as ubstitutin these elementary functions into equation 1, the displacement components become: With the same fashion as practiced in definin the G kj functions, these elementary functions can be treated to obtain E kj functions. By constructin the E kj functions, a linear system of equation can be set up to solve for M pq. The data vector is the recorded trace at each receiver,. Aain, this timedomain inversion is solved usin a damped least-squares method. We applied this time-domain waveform inversion on the synthetics microseismic data (enerated usin TIGER software from INTEF etroleum Research) invertin for the source moment tensor, and obtained promisin results but not the exact moment tensor that we used in the modelin. However, we obtained the source-time function that we used in the modelin (Fiure ). Retrievin the exact M pq is expected for such noise free data with accurate source location and velocity model, which requires more research at this point. (8) (9) (10) Fiure : Estimated source-time function from frequency domain inversion in step 1. GeoConvention 017 4

5 Acknowledments: We thank NERC (Natural cience and Enineerin Research Council of Canada) and the sponsors of CREWE for their financial support. We thank INTEFF for providin access to TIGER modelin software. We appreciate Dr. Gary Marrave for several discussions and uidance. Drs..F. Daley and Joe Won are reatly appreciated for their uidance. References: Aki, K., and Richards,. G., 00, Quantitative eismoloy: W. H. Freeman and Company, an Francisco. Dziewonski, A., Chou, T., and Woodhouse, J., 1981, Determination of earthquake source parameters from waveform data for studies of lobal and reional seismicity: J. Geophys. Res., 86, No. B4, Fouler, G. R., Julian, B. R., Hill, D.., itt, A. M., and Malin,. E., 004, Non-double-couple micro earthquakes at lon valley caldera, California, provide evidence for hydraulic fracturin: Journal of Volcanoloy and Geothermal Research, 1, Jost, M. L., and Hermann, R. B., 1996, A student s uide to and review of moment tensors: eismoloical Research Letters, 60, No., Maxwell,. C., and Urbancic, T. I., 001, The role of passive microseismic monitorin in the instrumented oil field: The Leadin Ede, 0, Rodriuez, I. V., Gu, Y. J., and acchi, M. D., 011, Resolution of seismic moment tensor inversions from a sinle array of receivers: Bulletin of the eismoloical ociety of America, 101, No. 6. Ross, A., Fouler, G. R., and Julian, B. R., 1999, ource processes of industrially-induced earthquakes at the eysers eothermal area, California: Geophysics, 64, hearer,. M., 1999, Introduction to eismoloy: Cambride U ress. Šílený, J., 1998, Earthquake source parameters and their confidence reions by a enetic alorithm with a memory : Geophysical Journal International, 14, 8 4. Vasco, D. W., 1989, Derivin source-time functions usin principal component analysis: Bull. eism. oc. Am., 79, No., Vavryčuk, V., 001, Inversion for parameters of tensile earthquakes: Journal of Geophysical Research, 106, No. B8, 16,9 16,55. Vavryčuk, V., 007, On the retrieval of moment tenors from borehole data: Geophysical rospectin, 5, Vavryčuk, V., and Ku hn, D., 01, Moment tensor inversion of waveforms: a two-step time-frequency approach: Geophysical Journal International, 190, GeoConvention 017 5