Frequency Domain Wave Equation Inversion and Its Application on the Heterogeneous Reservoir Model Data

Seismic full waveform inversion seeks to make use of the full information based on full wave field modeling to extract quantitative information from seismograms. Its serious nonlinearity and high dependence on initial velocity model often results in unsatisfactory inversion results in paleo-karsts carbonate reservoir characterized by strong heterogeneity. The paper presents an improved strategy of multi-scale inversion to establish velocity field model of waveform tomography. the forward wave equation algorithm was derived in frequency domain, and then the Matrix formalism for the iterative inverse methods is derived by gradient methods to speed up calculation and to avoid convergence to local minimum value. After massive amount of frequencies tests, the appropriate bandwidth are extracted, and the velocity field calculated at low frequency is used as the input of the high frequency. After the iteration, the accurate velocity field is inverted. Finally, frequency domain wave equation full waveform inversion in mathematical and physical models is conducted in order to verify the inverse program. The method of selecting the inverse frequencies is proved to be effective.


Introduction
Seismic waves bring to the surface information gathered on the physical properties of the earth.Seismic full waveform inversion seeks to make use of the full information based on full wave-field modeling to extract quantitative information from seismograms (Dessa et al. 2007;Operto et al. 2009).Lailly (1983) and Tarantola (1984) recast the migration imaging principle of Claerbout (1976) as a local optimization problem in Born approximation, the aim of which is least squares minimization of the misfit between recorded and modeled data.They show that the gradient of the misfit function along which the perturbation model is searched can be built by cross-correlating the incident wave-field emitted from the source and the back propagated residual wave-fields (Thierry et al. 1999;Brenders et al. 2007).The perturbation model obtained after the first iteration of the local optimization looks like a migrated image obtained by reverse-time migration (Sirgue et al. 2009).One difference is that the seismic wave-field recorded at the receiver is back propagated in reverse time-migration, whereas the data misfit is back propagated in the waveform inversion (Robertson et al. 2007;Vigh et al. 2008).When added to the initial velocity, the velocity perturbations lead to an updated velocity model, which is used as a starting model for the next iteration of minimizing the misfit function.After the iteration, the accurate velocity field is inverted.The key factors that influence Wave equation inversion results are effective and efficient forward modeling (Sirgue et al. 2008), the gradient (Sheng et al. 2006) and Hessian matrix (BenHadjAli et al. 2008) calculation algorithm.
Carbonate reservoir is widely developed in China Tarim basin, where a large number of oil fields are discovered in the paleo-karsts Ordovician limestone reservoir (Peng et al. 2008;Sun et al. 2011).The storage spaces for the carbonate reservoir in this area are mostly secondary dissolution caves and characterized by strong heterogeneity (Zhang et al. 2008;Zeng et al. 2011;Yang et al. 2012).How to accurately image these dissolved caves plays a key role in exploiting the reservoir and reserve estimation (Zhang et al. 2011;Tang et al. 2012).Due to low signal to noise ratio, the accuracy of velocity model used in pre-stack migration is very important.Considering the question mentioned above, seismic full waveform inversion is introduced seeks to make use of the information based on full wave field modeling to extract quantitative information from seismograms.The paper presents an improved strategy of multi-scale inversion to establish accuracy depth migration velocity field as an initial input model of waveform tomography, so that decrease the serious nonlinearity.The velocity field calculated at low frequency is used as the input of the high frequency, the accurate velocity field is inverted after the iteration.In the application of the frequency domain waveform inversion approach, we use seismic data from mathematic model and a caved physical model which is supplied by CNPC (China National Petroleum Corporation) key laboratory.several critical processes that contribute to the success of the method were tested here like, the matching of amplitudes between real and synthetic data, the selection of sequence of frequencies in the inversion, and the relationship between inversion velocity model and wave number reconstruction.

Waveform inversion method
The correction of full waveform inversion relies on the accuracy of its forward modeling wave equation.It can get good result only when forward modeling is approximate with actual process of wave propagate.The forward wave equation is derived in frequency domain.The pressure is computed using a staggered-grid, explicit finite-difference method.The Matrix formalism for the iterative inverse methods is derived which include gradient and Gauss Newton methods to speed up calculation and to avoid convergence to local minimum value.After massive amount of frequencies tests, the appropriate bandwidth are extracted, and the velocity field calculated at low frequency is used as the input of the high frequency.After the iteration, the accurate velocity field is inverted.Here we deal with a 2D acoustic wave equation written in the frequency domain as, where ρ is the density, K the complex bulk modulus, ω the frequency, p the pressure field and g is the source.
In the frequency domain, the wave equation can be compactly written as where B is the so called impedance matrix.Solving equation ( 2) can be performed through LU factorization of B (Virieux et al. 2009).
We define the misfit vector We use the least square norm which is easier to manipulate from a mathematical point of view is given by (3) where † denotes the complex conjugate, m the model parameters.
The gradient of the misfit function in equation ( 3) with respect to slowness perturbation is computed by the zero-lag correlation between the forward-propagated wavefields and the back-projected wavefield residuals, The velocity model is iteratively updated along the conjugate directions defined by where iterations max 1, 2,..., α is the step length, which is computed by a line search that ensures sufficient decrease of k f , and k H is an approximation of the inverse of the Hessian (Brossier et al. 2009).At each iteration, one forward propagation and one back projection are needed for computing the gradient direction.
esr.ccsenet.C key lab thod is similar ure 6.The dis length, 3000m t materials, nam and mixed ma 1/10000.del is Hz, a nd the en in oling ident gy of multi-scale inversion by selecting the inverse frequencies is proven to be effective for the strong heterogeneity reservoir, suggested by the application of mathematic and physical model data.

Applica
vector ( ) C m referred to as the misfit function.
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