Temporal windowing and inverse transform of the wavefield in the Laplace-Fourier domain

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1 GEOPHYSICS, VOL. 7, NO. 5 (SEPTEMBER-OCTOBER ); P. R7 R, FIGS..9/GEO-9. Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at Temporal windowing and inverse transform of the wavefield in the Laplace-Fourier domain Sangmin Kwak, Hyunggu Jun, Wansoo Ha, and Changsoo Shin ABSTRACT Temporal windowing is a valuable process, which can help us to focus on a specific event in a seismogram. However, applying the time window is difficult outside the time domain. We suggest a windowing method which is applicable in the Laplace-Fourier domain. The window function we adopt is defined as a product of a gain function and an exponential damping function. The Fourier transform of a seismogram windowed by this function is equivalent to the partial derivative of the Laplace-Fourier domain wavefield with respect to the complex damping constant. Therefore, we can obtain a windowed seismogram using the partial derivatives of the Laplace-Fourier domain wavefield. We exploit the time-windowed wavefield, which is modeled directly in the Laplace-Fourier domain, to reconstruct subsurface velocity model by waveform inversion in the Laplace-Fourier domain. We present the windowed seismograms by introducing an inverse Laplace-Fourier transform technique and demonstrate the effect of temporal windowing in a synthetic Laplace-Fourier domain waveform inversion example. INTRODUCTION Full-waveform inversion is a method to recover subsurface parameters by minimizing the difference between the observed and modeled wavefields (Tarantola, 9). The algorithm can be constructed in the time domain (Gauthier et al., 9; Kolb et al., 9; Mora, 97) or frequency domain (Pratt et al., 99; Pratt, 999). Shin and Cha (, 9) introduced the algorithm to the Laplace and Laplace-Fourier domains. Although the method exploits the full waveforms, several preprocessing steps that modify the observed signal can be applied for various purposes. A muting or a band-pass filtering process can improve the signal-to-noise ratio (Operto et al., ). An amplitude scaling can be used to match the amplitude of the observed and modeled data (Brenders and Pratt, 7). The offset and time windowing can eliminate ground rolls (Operto et al., ). The time windowing can be more versatile depending on the inversion domain. It can remove the elastic waves such as shear wave and mode-converted waves before the inversion, when we use the acoustic wave equation in the forward modeling step (Brenders and Pratt, 7). Moreover, a selective time windowing can be a part of an inversion algorithm in the time domain. Shipp and Singh () applied the process to focus on different events in their sequential inversion stages. The selective windowing process enables weak but important events overshadowed by strong signals to make significant contribution to inversion result (Shipp and Singh, ). Sheng et al. () used early arrivals only in their waveform tomography method. On the other hand, it is not possible to rigorously apply the time windowing to the modeled monochromatic wavefields in the frequency domain (Ravaut et al., ). The time-function gain is not a preferable process in the fullwaveform inversion because it changes the waveforms significantly. However, if we can model a gained wavefield directly, it can be used in the inversion as Ha et al. () showed in the Laplace domain. Applying a gain function expressed as a power of time is equivalent to taking the partial derivative of the Laplace-domain wavefield with respect to the damping constant (Kreyszig, ). Therefore, we can use the gain process to amplify the late arrival signals corresponding with deep reflections in the Laplace-Fourier domain. In this study, we apply the gain function in the Laplace-Fourier domain modeling. We show that the application of the gain function in the Laplace domain is equivalent to applying a time windowing in the frequency domain. This windowing is a weighted windowing and is different from the conventional windowing using step Manuscript received by the Editor July ; revised manuscript received May ; published online September. Seoul National University, Department of Energy Systems Engineering, Seoul, Republic of Korea. kwak.sangmin@gmail.com; h.jun@ gmail.com; css@model.snu.ac.kr. Pukyong National University, Department of Energy Resources Engineering, Busan, Republic of Korea. wansooha@gmail.com. Society of Exploration Geophysicists. All rights reserved. R7

2 R Kwak et al. Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at functions. We suggest an inverse Laplace-Fourier transform technique that uses the partial derivatives of the Laplace-Fourier domain wavefields with respect to the damping constant. The time window we propose cannot change with varying offsets, therefore, this temporal windowing is helpful only in some specific situations. However, this windowing method can be useful when the target reflection signals are in a certain time window. We use the technique to demonstrate the effect of the window function on the wavefield. Then, we apply the windowing method to the Laplace-Fourier domain waveform inversion. LAPLACE-FOURIER DOMAIN WAVEFIELD AND TEMPORAL WINDOWS We show that the Laplace-Fourier transform of a gained signal is equivalent to the Fourier transform of the windowed signal when the gain function is gðtþ ¼t n ; () where n is a positive integer and the window function is wðtþ ¼t n e σt ; () where σ is a positive real damping constant. The Laplace-Fourier domain wavefield can be thought of as a Fourier transform of a damped time-domain wavefield. It can be expressed as ~uðsþ ¼ Z uðtþe st dt ¼ Z uðtþe σt e iωt dt; () where uðtþ is the time-domain seismogram, s is a complex damping constant defined as s ¼ σ iω; () σ is a positive damping constant, and ω is an angular frequency (Shin and Cha, 9). We replaced the lower limit of the integral from minus infinity to zero because the time-domain wavefield is a causal function. If we think of the damping function e σt as a window function, equation is equivalent to a Fourier transform of a time-windowed seismogram. The damping function shown in Figure is very limited as a window function because it always damps out the late arrivals. We can control the size and range of the window by multiplying the damping function with a gain function t n as ~uðs; nþ ¼ Z uðtþt n e st dt ¼ Z uðtþt n e σt e iωt dt: (5) In the equation above, t n e σt behaves as a window function. The function can have various sizes and shapes according to the combination of σ and n as shown in Figure. Increasing σ moves the window to the early time whereas increasing n moves the window to the opposite direction. The window functions have the maximum amplitudes at the time of ðn σþ. Note that this window function applies a weighted window to the signal. Figure shows an example seismogram and its windowed seismograms for different σ and n. Accordingly, the Laplace-Fourier transform of a gained signal is equivalent to the Fourier transform of the windowed signal when the gain and window functions are given as equations and. Because introducing the gain function changes the Laplace-Fourier domain wavefield, an appropriate modification is required to the wave equation in the Laplace-Fourier domain. MODELING IN THE LAPLACE-FOURIER DOMAIN The acoustic wave equation in the Laplace-Fourier domain is expressed using a matrix notation as S ~uðsþ ¼ ~ fðsþ; () Figure. Exponentially decaying window functions. Figure. Window functions given in equation : ( t e t, ( t 5 e 5t, and ( t e t.

3 Time windowing in the Laplace-Fourier domain R9 Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at where S is the complex impedance matrix defined as S ¼ s M þ K, c c is the P-wave velocity in the medium, M is the mass matrix, K is the stiffness matrix, Z ~uðsþ ¼ uðtþe st dt; (7) ~fðsþ ¼ Z fðtþe st dt; () and fðtþ is a time source function (Shin and Cha, 9). Because a Laplace-transform of a gained signal (equation 5) can be expressed as a partial derivative of the Laplace-transformed signal with respect to the damping constant (Kreyszig, ; Ha et al., ), we can rewrite equation 5 as ~uðs; nþ ¼ Z Z uðtþt n e st dt ¼ð Þ n n s n uðtþe st dt ¼ð Þ n n ~uðsþ s n : (9) This implies that we can obtain the time-windowed wavefield in the Laplace-Fourier domain by taking the partial derivative of the wave equation recursively as s c M þ K M þ K c s s c M þ K n ~u s n ~u s ~u s ¼ ~ f s s M ~u; c ¼ f ~ s M ~u s c.. s c M ~u; ¼ n ~ f s n ns c M n ~u s n nðn Þ c M n ~u s n ; n. () Note that the order of the derivative is equal to the power in the gain function (equation ). To obtain the nth order partial derivative wavefield, we have to calculate the derivative wavefields from th to (n )th order in advance. This calculation requires n times of the wave propagation when the maximum order of derivative wavefield we want to obtain is n. We can transform this Laplace-Fourier domain partial derivative wavefields to the time domain by the inverse Laplace-Fourier transform. Figure. ( A shot gather and windowed shot gathers when the window functions are ( t e t, ( t 5 e 5t, and ( t e t.

4 R Kwak et al. Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at INVERSE LAPLACE-FOURIER TRANSFORM We derive a method to invert a time seismogram using the Laplace-Fourier domain partial-derivative wavefields (equation ). A discrete time signal can be expressed using the Dirac s delta functions as follows uðtþ ¼a δðt t Þþa δðt t Þþ þa nt δðt t nt Þ; () where a i ði ¼ ; ;ntþ is the amplitude of the signal at ith time sample and nt is the number of time samples. The Laplace-Fourier transform (equation ) of the discrete signal yields ~uðsþ ¼a e st þ a e st þ þant e st nt : () The partial-derivative wavefield (equation 9) can be expressed as Figure. A velocity profile extracted from the Marmousi velocity model. n ~uðsþ s n ¼ a ð t Þ n e st þ a ð t Þ n e st þ þ a nt ð t nt Þ n e st nt ; n ¼ ; ; np; () where np is the maximum order of the derivative. We can construct a matrix equation using this relationship as Re½TŠ Im½TŠ a ¼ Re½dŠ Im½dŠ ; () where Re½ Šand Im½ Šdenotes the real and imaginary parts of the elements of the matrix or vector. Because the matrix T is independent to the wavefield, equation can be solved as a system of Figure 5. The wavefield obtained by frequency domain modeling. Figure. ( The inverse-transformed wavefield obtained using an impulse source and ( the wavefield convolved with a first derivative Gaussian source wavelet.

5 Time windowing in the Laplace-Fourier domain R Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at equations with multiple right-hand-sides including the whole wavefield. The matrix and vectors are defined as T d T d a.. a T ¼ T ; d ¼ n d ; a ¼ n. ; (5) a 5 nt T np d np where ð t Þ n expð s t Þ ð t Þ n expð s t Þ ::: ð t nt Þ n expð s t nt Þ ð t Þ n expð s t Þ ð t Þ n expð s t Þ ð t nt Þ n expð s t nt Þ T n ¼ ; 7 5 ð t Þ n expð s ns t Þ ð t Þ n expð s ns t Þ ::: ð t nt Þ n expð s ns t nt Þ d n ¼ n ~uðs Þ s n n ~uðs Þ s n n ~uðs ns Þ T; n¼ ; ;np; () s n and ns is the number of complex damping constants. Note that ns ¼ nσ nf, where nσ is the number of positive real damping constants and nf is the number of temporal angular frequencies (equation ). The real and imaginary parts of an element of the matrix corresponding to ith time sample, jth damping constant, kth angular frequency, and nth order derivative are Figure 7. Traces extracted from Figures 5 (Fourier) and b (Laplace-Fourier) at the depth of (. and (. km. ð t i Þ n expð σ j t i Þcos ω k t i ; ð t i Þ n expð σ j t i Þsin ω k t i ; i ¼ ;; ;nt; j ¼ ;; ;nσ; k¼ ;. ;nf; and n ¼ ;;; ;np: ð7þ Therefore, each row of the matrix is the window function (equation ) multiplied by the cosine or sine function. The number of rows of the matrix (equation ) is ns ðnp þ Þ ¼ nσ nf ðnp þ Þ and the number of columns is nt. Because the matrix is not square, we solve it using the singular value decomposition which is a powerful set of techniques for dealing with sets of equations that are either singular or numerically very close to singular (Press et al., 7). If the numbers of the rows and columns satisfy the following condition, the inverse problem is an overdetermined problem. nσ nf ðnp þ Þ nt: () The problem can be solved to obtain reasonable solution even if it is an underdetermined problem. However, the problem with larger Row 9 Column Figure. ( The inverse-transformed wavefield obtained using only one window function (t 5 e 5t ). The shape of window function is attached on the right. ( The corresponding inverse transform matrix (equation ).

6 Kwak et al. R Figure 9. The inverse Laplace-Fourier transformed wavefields and the corresponding window functions ( t e t, ( t e 5t, ( t e t, and ( t e 7t. Figure. The inverse Laplace-Fourier transformed wavefields and the corresponding window functions ( t e t, ( t e t, ( t e 5t, and ( t e t. Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at

7 Time windowing in the Laplace-Fourier domain R Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at number of rows will give better solution, which has more accurate amplitudes when compared to the corresponding time domain seismogram. The number of rows of the matrix T in equation is nf to generate the time-windowed seismogram using a window function with a single damping constant and a single order of the gain function. Before we solve the matrix equation, we normalize each row in equation by the maximum value of the elements to compensate for the different weighting in the window function (Figure ). We can construct the matrix equation without the partial derivatives as " # " # Re½T Š Re½d Š a ¼ : (9) Im½T Š Im½d Š However, this equation cannot be solved properly because the exponential damping function without gains decreases the amplitudes of the signals in the late arrivals. We demonstrate the inverse Laplace-Fourier transform using a D example. NUMERICAL EXAMPLES OF INVERSE LAPLACE-FOURIER TRANSFORM Figure shows a velocity profile extracted from the Marmousi velocity model (Versteeg, 99). We placed one source at the surface and 75 receivers along the profile with the interval of m. We applied the free surface boundary condition at the surface and an absorbing boundary condition (Reynolds, 97) at the bottom. Figure 5 shows a reference seismogram modeled in the frequency domain with the time sample interval ms. We used frequencies with the interval of.5 Hz. The maximum frequency was 5 Hz. We modeled the wave propagation in the Laplace-Fourier domain using 5 positive damping constants ranging from to and angular frequencies ranging from.5 to 5 Hz. The maximum order of the partial derivative with respect to s was. We inverse transformed the modeled wavefield to the time domain using equation. Figure shows the inverse-transformed wavefield. We used a delta function as the source wavelet in the Laplace-Fourier domain and convolved the inverse-transformed wavefield (Figure with the source wavelet used in the frequency domain modeling (Figure. In the inverse Laplace-Fourier transform procedure, we used the singular value decomposition to control the singularity of the matrix T in equation. We ignored singular values smaller than a threshold defined as % of the maximum singular value because the matrix has singular values close to zero. This threshold was obtained for general usage by various numerical experiments. The solution of the singular value decomposition is not an exact solution (Press et al., 7). Therefore, the inverse-transformed wavefield has errors when compared with the reference seismogram (Figure 7). We can control the time range recovered by the inverse Laplace- Fourier transform by limiting the number of damping constants and the order of the partial derivative. Recall that each row of the inverse transform matrix corresponds to the weighted time window multiplied by the cosine or sine function (equation 7). Therefore, we can make elements associated with the early or late time arrivals significant or negligible using appropriate window functions. Figure a shows the windowed inverse transform results for the window function shown in Figure b. We used only one damping constant (σ ¼ 5) and one order of the partial derivative (n ¼ 5) to inverse transform this seismogram. As shown in Figure b, the matrix Figure. The inverse Laplace-Fourier transformed wavefields and the corresponding window function t 9 e t. ( and ( angular frequencies are used in the inverse Laplace-Fourier transform.. Figure. The reference velocity model for waveform inversion test

8 R Kwak et al. Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at Figure. Seismograms of common shot gather obtained by wave propagation modeling by shots at distance of (.75, ( 7.5, and (.5 km. corresponding to the window function has very small value at early time and relatively large value at late time in column. Accordingly, we can diminish the wavefield at early time. Figure 9 shows several window functions and the inversetransformed wavefields. The time windows of the wavefields have small time delay when compared with their window functions due to the time delay in the first-derivative Gaussian source wavelet we used. To design a window to focus on a specific time, we can set the time which of the window function has maximum amplitude and the width of window. A window function has maximum amplitude at t ¼ n σ, and the larger values of the gain function s order n and positive damping constant σ form windows of narrower width. Figure shows the examples of the window functions which have the maximum amplitude at. s, and the widths of windows are different because of the gain function s order and the damping constant value. When we inverse-transform the Laplace-Fourier domain wavefield, the accuracy of the inverse-transformed wavefield depends on the number of rows in the matrix T. A larger number of rows give a better windowed result because of the large number of frequency used in the inverse transform, as Figure shows. LAPLACE FOURIER DOMAIN WAVEFORM INVERSION USING TEMPORAL WINDOWS As we observed in the previous discussions, we can implement the temporal windowing by applying a partial derivative to the Laplace-Fourier domain wavefield with respect to the damping constant. The partial derivative wavefield is equivalent to the Laplace-Fourier transformed wavefield of the windowed seismogram. Therefore, the Laplace-Fourier domain waveform inversion Figure. A time trace at the receiver located at the distance of (.75, ( 7.5, and (.5 km in the seismogram of Figure a.

9 Time windowing in the Laplace-Fourier domain R5 Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at using temporal windows can be performed by exploiting the partial derivative wavefield. We can directly model the partial derivative wavefield, which is equivalent to the time-windowed wavefield in the Laplace-Fourier domain (equation ). The steepest descent direction to update the velocity model is p lþ k ¼ p l k αh p k E; () where p k is kth model parameter, l is the iteration number, α is the step length, H p is the pseudo-hessian matrix (Ha et al., ), and k E is the gradient direction e) f) The logarithmic objective function of the inversion using nth order partial derivative wavefields for a single complex frequency can be expressed as E ¼ Xnshot i¼ X nrcv j¼ δr ij δr ij ; () where δr ij ¼ lnð n ~u ij s n n ~ d ij s n Þ is the logarithmic residual at the jth receiver by the ith shot (Shin and Min, ), nshot and nrcv are the number of shots and receivers for the waveform inversion, respectively. The superscript * denotes the complex conjugate, ~u ij Figure 5. Real value of the shot-receiver gathers of the ( original wavefield, ( first, ( second, ( third, (e) fourth, and (f) sixth partial derivative wavefield with respect to the damping constant. Temporal frequency is. Hz and damping constant is.

10 R Kwak et al. Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at and ~d ij are modeled and observed wavefields in the Laplace-Fourier domain, respectively. The gradient direction at the kth model parameter can be expressed as k E ¼ E nshot X ¼ Re ½ðv p ik Þ T S r i Š ; () k i¼ where v ik ¼ S p k ð n ~u i sn Þ is the virtual source vector for the kth model parameter and r i is the residual vector for ith shot (Pratt et al., 99). The virtual source v ik is the product of the partial derivative of the impedance matrix with respect to the model parameter and the partial derivative wavefield with respect to the damping constant, v ik ¼ ð s M c p k Þð n ~u i sn Þ. The procedure to obtain the virtual Figure. Imaginary value of the shot-receiver gathers of the ( original wavefield, ( first, ( second, ( third, (e) fourth, and (f) sixth partial derivative wavefield with respect to the damping constant. Temporal frequency is. Hz and damping constant is. source is computationally intensive because it requires the partial derivative wavefield with respect to the damping constant and the model parameter. The recursive equations to compute the partial derivative wavefield are s c M þ K s M þ K nþ ~u c p k s ¼ n ns c M s M þ K c ~u p k s ¼ s s M c p k n ~u p k s n e) f) s p k ¼ M c p k ~u s s M c p k ~u.. n ~u s ns M n ~u n c p k s n nðn Þ M c ~u M ~u c p k s c M ~u p k nðn Þ M c n ~u p k s n ;n : n ~u p k s n ()

11 Time windowing in the Laplace-Fourier domain R7 Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at It requires n times of the wave propagation to obtain the virtual source of nth-order partial derivative wavefield for each parameter. Therefore, the process to compute the virtual source for all parameters in the model becomes much heavier as the order of partial derivative with respect to the damping constant increases. The windowing process is much more flexible and easier to implement in the time domain than that in the Laplace-Fourier domain. The window functions in the Laplace-Fourier domain are restrictive and controlled by the damping constant and the order of the derivative, i.e., power of the gain function. Moreover, the windowing process in the Laplace-Fourier domain requires heavy computational costs. However, the time windowing is not available in the Laplace-Fourier domain before we introduce the gain function to the Laplace-Fourier transform. The suggested algorithm enables temporal windowing in the Laplace-Fourier domain waveform inversion to enhance the inversion result, as shown below. NUMERICAL EXAMPLES OF WAVEFORM INVERSION USING TEMPORAL WINDOWS We performed waveform inversion tests using temporal windows in the Laplace-Fourier domain for a D reference velocity model. The reference model has. km s homogeneous background and three circular structures with the velocity of. km s (Figure ). We synthesized observed seismograms by using the finite difference method in the time domain for s with time sample interval ms. The peak frequency of the source wavelet to generate the synthetic data is 5 Hz, the number of shots is 5 and the spatial interval of Figure 7. Time windowed seismograms of the common shot gather (Figure when the temporal window is ( e t, ( te t, ( t e t, ( t e t, and (e) t e t.

12 R Kwak et al. Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at Figure. The first gradient direction of waveform inversion in the Laplace-Fourier domain using the ( original wavefield, ( first, ( second, ( third, and (e) fourth order partial derivative wavefield with respect to the damping constant. Three temporal frequencies ranging from. to. Hz and a damping constant are used e) e) Figure 9. The tenth updated velocity model of waveform inversion in the Laplace-Fourier domain using the ( original wavefield, ( first, ( second, ( third, and (e) fourth order partial derivative wavefield with respect to the damping constant. Three temporal frequencies ranging from. to. Hz and a damping constant are used.

13 Time windowing in the Laplace-Fourier domain R9.. Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at e) Figure. The th updated velocity model of waveform inversion in the Laplace-Fourier domain using the ( original wavefield, ( first, ( second, ( third, and (e) fourth order partial derivative wavefield with respect to the damping constant. Three temporal frequencies ranging from. to. Hz and a damping constant are used. e) Figure. The first gradient direction of waveform inversion in the Laplace-Fourier domain using the ( original wavefield, ( first, ( second, ( third, and (e) fourth order partial derivative wavefield with respect to the damping constant. Five temporal frequencies ranging from. to. Hz and two damping constants and are used.

14 R Kwak et al. Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at each shot is m. The seismograms of common shot gathers at distance of.75, 7.5, and.5 km show the reflection signals around. 7. s (Figure ). The amplitude of the signal reflected by the deepest circle at the far-offset receiver (Figure is relatively small compared to that reflected by the shallowest circle at the near-offset receiver (Figure. We present shot-receiver gathers of original and partial derivative wavefields in the Laplace-Fourier domain (Figures 5 and ). The temporal angular frequency is. Hz and the damping constant is.. The shot-receiver gather of the original wavefield has the maximum amplitude in the diagonal line, at the shot positions. The shotreceiver gathers of partial derivative wavefields show fluctuations at the locations of the high-velocity circles. Higher order of the partial derivative with respect to the damping constant leads to larger amplitude of the wavefield for the high-velocity circle in the deep part because the maximum weighting of the window function moves to the late arrivals. This fact implies that the waveform inversion using partial derivative wavefield has the potential to reconstruct the deeper part of the velocity model compared to that using original wavefield only. We demonstrate the effect of the temporal windowing by performing waveform inversion of seismic data containing weak events caused by a circle in the deep part of the model. The starting velocity model is a. km s homogeneous velocity model with grid spacing of 5 m. We used 5 shots for the inversion and the space interval of each shot is m. The step length is. km s, and the constraints for velocity model update are minimum. km s and maximum. km s. We used three frequencies ranging from. to. Hz and one damping constant. for this inversion test. The windowed seismograms of Figure b are described in Figure 7. Laplace or Laplace-Fourier domain waveform inversion exploits the damped seismogram by the exponential damping function e σt. Figure 7a shows the seismogram and the damping function applied in the conventional waveform inversion in the Laplace-Fourier domain. It damps out most of the late arrival signals in the time domain. In this study, we take advantage of the Laplace-Fourier domain wavefield, which is time-windowed by a window function of t n e σt. Figure 7b 7f show the seismograms and the window functions applied in the Laplace-Fourier domain waveform inversion using partial derivative wavefield with respect to the damping constant. They focus on the reflection signals as the time window moves toward late arrivals. We compared conventional waveform inversion using original wavefield and the time-windowed waveform inversion using partial derivative wavefield with respect to the damping constant. Because the reflection signals are around. 7. s for all shot gathers, it is e) Figure. The tenth updated velocity model of waveform inversion in the Laplace-Fourier domain using the ( original wavefield, ( first, ( second, ( third, and (e) fourth order partial derivative wavefield with respect to the damping constant. Five temporal frequencies ranging from. to. Hz and two damping constants and are used.

15 Time windowing in the Laplace-Fourier domain R Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at ideal to apply the temporal window that has the maximum amplitude around these reflection signals. We applied the temporal window that has the maximum amplitude around.. s by controlling the partial derivative order as. Because the high derivative order shifts the temporal window to the late arrival for the same damping constant, the waveform inversion with temporal windows can focus on the reflection signals. We can confirm that the first gradient direction shows the possibility of time-windowed waveform inversion to reconstruct the deeper part of the model (Figure ). The first gradient direction of the conventional waveform inversion shows that it can detect the left and center high-velocity circles. However, the first gradient direction of the time-windowed waveform inversion shows that it detects not only the left and center high-velocity circles but also the high-velocity circle on the right. The tendency to detect circles in the deep part becomes stronger as the partial derivative order increases because the temporal window moves toward late arrivals. The tenth updated velocity models of the waveform using original and partial derivative wavefield are compared (Figure 9). The updated velocity model using partial derivative wavefield for the temporal windowing shows that the target update area moves downward as the derivative order increases. The updated velocity model using original wavefield shows high velocity on the left and center e) The inverted velocity model using partial derivative wavefield also shows high velocity on the left and center, however, the inversion results using higher-order partial derivative wavefield can detect the circle on the right better when compared to that using original wavefield. Also, the results using partial derivative wavefields converge to the true velocity model rapidly in comparison with that using original wavefield. Figure shows th updated velocity models of the waveform inversion. In particular, waveform inversion result using fourth partial derivative wavefield focuses on the center circle compared to other inversion results. We also tested time-windowed waveform inversion using five temporal frequencies ranging from. to. Hz and two damping constants and (Figures,, and ). We used shots for the inversion and the space interval of each shot is 5 m. By introducing two damping constants, we exploited two window functions for each inversion test. The tenth updated velocity model using partial derivative wavefield shows better convergence to the true model in comparison with that using original wavefield (Figure ). The inverted velocity model using fourth-order partial derivative wavefield focuses on the left, center, and right circles simultaneously because of the two time windows (Figure e). The inversion result shows better resolution at center and right circles compared to other inversion results Figure. The th updated velocity model of waveform inversion in the Laplace-Fourier domain using the ( original wavefield, ( first, ( second, ( third, and (e) fourth order partial derivative wavefield with respect to the damping constant. Five temporal frequencies ranging from. to. Hz and two damping constants and are used.

16 R Kwak et al. Downloaded /9/ to Redistribution subject to SEG license or copyright; see Terms of Use at CONCLUSIONS We showed that we can simulate a time-windowed wavefield in the Laplace-Fourier domain. The window function is a product of a gain function and an exponential damping function. The window is a weighted window and we can control the window by adjusting the power of the gain function and the exponent of the damping function. We also presented an inverse Laplace-Fourier transform method using the window function and the partial derivative wavefields with respect to the damping constant. By changing the window function, we can modify the time range recovered by the inverse Laplace-Fourier transform using the singular value decomposition. In the numerical example, we applied temporal windows to invert a synthetic data set in the Laplace-Fourier domain. One obvious limitation of this windowing method is that we cannot change the window with varying offset. Moreover, precise control of the window is limited by the power of the gain function and the damping constant. Nevertheless, this technique can be applied to a full-waveform inversion in the Laplace-Fourier domain to amplify the weak events of late-arrival signals corresponding with deep reflections. We reconstructed velocity models from the synthetic data which has relatively small amplitude of signals caused by deep reflectors. Spatial resolution in depth is better in the inverted velocity models using appropriate time windows by controlling the damping constants and the order of derivatives. Following studies need to examine the effect of the various windowing on the full-waveform inversion in the Laplace-Fourier domain. ACKNOWLEDGMENTS This work was supported by the Energy Efficiency & Resources of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korean government s Ministry of Knowledge Economy (No. 9B). REFERENCES Brenders, A., and R. 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