Journal of Applied Geophysics

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1 Journal of Applied Geophysics 123 (2015) Contents lists available at ScienceDirect Journal of Applied Geophysics journal homepage: Improved estimation of P-wave velocity, S-wave velocity, and attenuation factor by iterative structural joint inversion of crosswell seismic data Tieyuan Zhu a,, Jerry M. Harris b a The University of Texas at Austin, Jackson School of Geosciences, Austin, TX 78712, USA b Stanford University, Department of Geophysics, Stanford, CA, 94305, USA article info abstract Article history: Received 14 October 2014 Received in revised form 20 August 2015 Accepted 4 September 2015 Available online xxxx Keywords: Joint inversion Cross well seismic Reservoir characterization We present an iterative joint inversion approach for improving the consistence of estimated P-wave velocity, S- wave velocity and attenuation factor models. This type of inversion scheme links two or more independent inversions using a joint constraint, which is constructed by the cross-gradient function in this paper. The primary advantages of this joint inversion strategy are: avoiding weighting for different datasets in conventional simultaneous joint inversion, flexible for incorporating prior information, and relatively easy to code. We demonstrate the algorithm with two synthetic examples and two field datasets. The inversions for P- and S-wave velocity are based on ray traveltime tomography. The results of the first synthetic example show that the iterative joint inversion take advantages of both P- and S-wave sensitivity to resolve their ambiguities as well as improve structural similarity between P- and S-wave velocity models. In the second synthetic and field examples, joint inversion of P- and S-wave traveltimes results in an improved Vs velocity model that shows better structural correlation with the Vp model. More importantly, the resultant V P /V S ratio map has fewer artifacts and is better correlated for use in geological interpretation than the independent inversions. The second field example illustrates that the flexible joint inversion algorithm using frequency-shift data gives a structurally improved attenuation factor map constrained by a prior V P tomogram Elsevier B.V. All rights reserved. 1. Introduction Due to data deficiency and complex characteristic of geological system, e.g. multiple fluids in the reservoir, single geophysical model might be difficult to characterize the geological target fully. For example, compressional waves are very sensitive to gas-saturated rocks while shear waves are not. Attenuation is potentially more sensitive than velocity to the amount of gas in a rock (Winkler and Murphy, 1995). The ratio of Vp/Vs is more sensitive to changes of fluid type than Vp or Vs separately (Dvorkin et al., 1999; Hamada, 2004). It is also believed that attenuation factor is closely related to permeability (Pride et al., 2003). We can see that different geophysical models tend to reflect complementary characters of reservoir. It is natural to combine several types of geophysical data collected over the same reservoir region to reduce ambiguity in inversion results, leading to more reliable models for reservoir characterization. A type of joint inversion refers to combining several different types of geophysical datasets in a single inversion algorithm and then simultaneously or iteratively solving a least-squares problem (Vozoff and Corresponding author. addresses: tzhu@jsg.utexas.edu (T. Zhu), jerry.harris@stanford.edu (J.M. Harris). Jupp, 1975; Haber and Oldenburg, 1997; Julia et al., 2000; Gallardo and Meju, 2003). Simultaneous joint inversion approaches have been successfully applied for different geophysical data to provide improved geophysical models (e.g., Gallardo and Meju, 2004; De Stefano, 2007; Linde et al., 2008; Doetsch et al., 2010; De Stefano et al., 2011; Gao et al., 2012; Lelievre et al., 2012). However, coupling two or more datasets in a single inversion still face some difficulties, especially large-scale problem: first, the huge coupled Jacobian and/or Hessian matrices for the different data inversions have to be computed and/or stored for simultaneous use (Hu et al., 2009); second, the determination of suitable relative weighting between different objective functions can be challenging (Gallardo and Meju, 2007; Moorkamp et al., 2011). In this paper, we discuss an alternative approach to simultaneous joint inversion for the tomography problem that is quite similar to the ones of Hu et al. (2009) and Heincke et al. (2010). The iterative joint inversion couples independent inversions through iterations with a crossconstraint term. At every iteration, we still run an independent inversion by minimizing an objective function with the additional cross-constraint term. The presented approach overcomes the memory issue and the determination of relative weighting of different data sets. The cross constraint could be a direct parameter relation or a structural link. A direct parameter relation for different models based on the empirical or rock-physics relations (e.g., Carcione et al., 2007) may be limited in / 2015 Elsevier B.V. All rights reserved.

2 72 T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) some specific places. Instead, we implement a structural link the crossgradient function which measures the structural similarity between the different models instead of the direct parameter relation (Gallardo and Meju, 2003; Zhu and Harris, 2011; Um et al., 2014). Another advantage of the iterative joint inversion algorithm with the cross-gradient structural constraint is its flexibility to incorporate prior models into the independent algorithms. For example, a prior lithologic map (e.g., from reflection-migrated images) could be applied to constrain other parameters in the cross-gradient function. This is easy to implement as an additional regularization term in an independent inversion within a single unknown, but it is difficult to use for multiple unknowns, whose unknowns may not be on the same order of magnitude (e.g., velocity and resistivity). We begin by presenting a flexible iterative joint inversion framework that allows us to test different geophysical datasets. We then give an overview of the different parts of the joint inversion framework: the objective function definition, the cross-gradient function, and the determination of regularization weights. We test the algorithm with two synthetic examples for jointly inverting V p and V s models. Finally, we apply the approach to two seismic cross-well field datasets acquired at the west Texas for reservoir characterization. 2. Methodology The inverse problem is formulated as an optimization that minimizes an objective function Φ, which combines a measure of data misfit, Φ d, a regularization measure Φ m : minφðmþ ¼ Φ d ðmþþλφ m ðmþ; ð1þ where the model vector m is a spatial function m(x,y,z), and λ is a regularization parameter, which is used to adjust contributions for data misfit from the model regularization term and the constraint function. The objective functions of the iterative joint inversion of two datasets with a cross constraint ψ cc (m 1,m 2 )are defined as Φ 1 ¼ Φ d ðm 1 Þþλ 1 Φ m ðm 1 Þþβ 1 ψ cc ðm 1 ; m 2 Þ; Φ 2 ¼ Φ d ðm 2 Þþλ 2 Φ m ðm 2 Þþβ 2 ψ cc ðm 1 ; m 2 Þ; where m 1 and m 2 denote two models for two corresponding datasets. In such an iterative joint inversion, we still run two inversions separately. In each independent inversion, the cross constraint is functional as a new regularization and includes complementary information from a joint model during iterations. The coefficient β controls the influence from other models on the solution through the cross constraint. Through the constraint ψ cc (m 1,m 2 ), two independent inversions in Eq. (2) exchange information (e.g., geologic structure) during iterations. The data misfit Φ d and the regularization terms Φ m are written as Φ d ðm i Þ ¼ W d Gm ð i Þ d obs ; ð3þ L2 i ð2þ where a x,a y and a z are relatively weights applied to x,y,andz spatial components of the discrete gradient (G x,g y,g z )(Pidlisecky et al., 2007), L is the discretized Laplacian matrix (Aster et al., 2005), and a lap is the weighting value. For our problem, the cross-gradient function is chosen as the constraint functional ψ cc =ψ cg (m 1,m 2 ). The constraint functional is defined as ψ cg (m 1,m 2 )= t 2 2, where the cross-gradient function t is defined in Gallardo and Meju (2003): tx; ð y; z Þ ¼ m 1 ðx; y; zþ m 2 ðx; y; zþ; ð6þ where is the gradient in the x,y and z directions. The structural similarity requires t = 0, which means that any spatial changes occurring in both m 1 and m 2 must point in the same or opposite direction, or no spatial changes in one of m 1 and m 2 (Gallardo and Meju, 2004). The derivatives of the cross-gradient term with respect to the model parameters are given in 2D (Gallardo and Meju, 2004) and 3D (Tryggvason and Linde, 2006). The Jacobian matrix J xg is then obtained. Each row of Jacobian matrix has six nonzero elements of 2N m (N m is the model size) (cf. Gallardo and Meju, 2004, Eq. (9)). In our synthetic and field examples, we carefully choose λ through several tests to balance model misfit and data misfit in the independent inversion. When λ is obtained, we use this value for the iterative joint inversion. We determine the β value by the experienced rule given by Hu et al. (2009) h i β ¼ 10 L jφ m j 2 = N jψ cc j 2 þ δ 2 ; ð7þ where N=N x N y N z and δ is a small value. L usually ranges 0bLb5and depends on which model is superior, i.e., the superior model has relatively small weights. N x, N y and N z are the number of discretized grid points in the x,y and z directions. Fig. 1 shows the flowchart of our iterative procedure. The procedure begins with two input datasets and their corresponding initial models m 0 =(m 1,m 2 ), which are usually homogenous in our tomography algorithm. In the first iterative, we run two independent inversions (box A and B ) form 1 1 and m 2 1. The superscript denotes the iteration number. When we obtained updated models m 1 1 and m 2 1 from flows A and B Φ m ðmþ ¼ 1 2 kwmk2 2 ð4þ where 2 2 represents an L 2 -norm, and all quantities written in bold represent vectors. The subscript i refers to the index of multiple models (dataset), G(m) is the forward functional, d obs is the observed data vector, and m is the unknown model vector. W d is the data weighting matrix, which ensure the data by giving appropriate weights in the inversion (see Eq. (15) in Pidlisecky et al., 2007). The regularization term W is chosen as the first- and second-order spatial derivatives (Zhu and Harris, 2015). A finite-difference approximation of the W in 3D results in the sparse matrix W ¼ a x G x þ a y G y þ a z G z þ a lap L ð5þ Fig. 1. Flowchart of iterative joint inversion scheme. The boxes A and B represent the independent inversions. The box J represents the joint constraint term between A and B.

3 T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) respectively, the cross constraint box J is updated from updated models m 1 1 and m 2 1 for the next iteration. Then we have the new objective functions with new constraint functions for each independent inversion in A and B. The procedure is repeated iteratively until the predefined iteration number or the data error tolerances for both independent inversions are satisfied. The output inversion results are the optimal solutions m optimal =(m 1,m 2 ). All following examples come from crosswell seismic tomography. The forward modeling of refraction traveltimes is used to solve the eikonal equation by the finite-difference method (Vidale, 1990; Zelt and Barton, 1998). Times are calculated away from a source on the sides of an expanding cube, one side being completed before the next is considered. We use a Gauss Newton strategy to solve the independent inverse problem. Computation details of gradient and Hessian Fig. 2. a) True P-wave velocity model, b) S-wave velocity model, c) Vp/Vs ratio model, and d) cross-gradient map between two models. e) - h): Corresponding inverted models by independent inversions. i) l): Corresponding inverted models by iterative joint inversion.

4 74 T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) of the regularization term can be found in Zhu and Harris (2015). The linearized Hessian of the cross-gradient term is approximated by J T xg J xg. 3. Synthetic examples We first test the algorithm on a reservoir model in the crosswell geometry. Fig. 2a and b show P- and S-wave velocity models. A gas water saturated reservoir is embedded in the second layer. Note that the P- wave velocity contrast between gas and water saturated zone is high at 24%. The relatively small differences in S-wave velocities of the gassaturated sand and the water-saturated sand is only 6%, which makes it hard to identify the gas-water contact in the reconstructed S-wave velocity model. The joint inversion of P and S data is expected to resolve this ambiguity. In the first synthetic example, the model dimension is 500 m by 1210 m. We set up the source-receiver geometry with 29 shots in the right well with a spacing of 40 m and 116 receivers in the left well with a spacing of 10 m. The well distance is about 450 m. We calculated P- and S-wave traveltime data from the true models in Fig. 2a andbby solving the eikonal equation by the finite-difference method (Vidale, 1990). No noise is added. The starting models for P- and S-wave inversions are homogeneous, with mean values of the true models. The regularization parameters λ are and for the P- and S- wave model inversion algorithms, respectively. We ran ten iterations for both inversions, which is sufficient for convergence of the Gauss- Newton method. Fig. 2e h show inverted P- and S-wave velocity models by independent inversions, Vp/Vs ratio model, and its cross-gradient values. The inverted P-wave model is quite good, especially at the gas-water contact because of high velocity contrast. But the lateral geometry of the objects is not well recovered. The reason probably is limited ray aperture in the crosswell geometry. Fig. 2f shows the inverted S-wave velocity model. The geometry of the gas-water reservoir zone is better defined because rays fairly pass through this zone. However, the gas-water contact is not delineated. We can see this model is difficult for either the P-wave or S- wave method alone to resolve. The right panel (Fig. 2h) shows the structural similarity (cross-gradient) of the inverted P-wave velocity (e) and S-wave velocity models (f). Next, we ran the iterative joint inversion algorithm for the P- and S- wave models. The same regularization λ are used as in the independent inversions. Fig. 2i l show the joint inversion results. Overall, we can see that the joint inversion results tend to remove artifacts seen in independent inversion results. Notably, the gas-water contact in the inverted S- wave velocity model (Fig. 2j) is better resolved and the edge of the reservoir in the P-wave velocity model (Fig. 2i) is better delineated. Below 50 m, there are improvements in S-wave velocity structure but P-wave velocity along the dipping channel become slightly smoother. Fig. 2k displays the Vp/Vs ratio model. The cross-gradient values from joint inversion (Fig. 2l) are closer to zeros, as designed. The root mean square sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (RMS) value (defined as NxNz i¼1 t 2 i =N xn z 1e6, where t(x,y,z)isdefined at Eq. (6)), at the final step decreases to ~0.09 from ~0.25 for independent inversion results. This implies that the joint inversion with the cross-gradient constraint produces structurally similar P- and S-wave models. Fig. 3 presents the cross-plot of the P- and S-wave velocities. The cross-plot values from true P- and S-wave velocity are shown in seven red crosses that represent seven different blocks. The blue circles are independent inversion values, while the yellow triangles indicate joint inversion results. The P and S velocity values of the joint inversion models are somewhat less dispersed than those obtained from the independent inversion results. All model and data misfits are listed in Table 1. The data misfit between the observed data d obs and the calculated data d cal is defined as d cal d obs 2 / d obs 2. The normalized RMS model misfit is m est m true 2 / m true 2. Both model and data misfits are decreasing to Fig. 3. Crossplot of V P and V S obtained from independent inversion, joint inversion, and true model. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) slightly smaller values by joint inversion than by independent inversions. In the second example, we test the algorithm in a more realistic model taken from west Texas well logs. The V P log and its corresponding velocity model are shown in Fig. 4a. The S-wave and density logs can be found in van Schaack et al. (1995). We set up the source-receiver geometry of a west Texas field dataset (Harris et al., 1995) with 201 shots in the right well with a spacing of 2.5 m and 201 receivers in the left well with a spacing of 2.5 m. The source function is a Ricker wavelet with a central frequency of 800 Hz. We use an elastic finite-difference solver (Wu et al., 2005) to generate the synthetic data set. Fig. 4bshowsacommon shot gather at depth m. The P- and S-wave picks are easily identified in Fig. 4b. We manually picked P- and S-wave traveltimes. Note that S-wave picks are not pickable in the near offset, due to the source radiation pattern (Harris et al., 1995). The regularization parameter λ = is chosen for independent P- and S-wave inversions. Fig. 5a d show the synthetic P- and S-wave velocity models, V P /V S ratio model, and their cross-gradient values that are zeros. The corresponding models obtained from independent inversions of these new data are shown in Fig. 5e h. The V P model recovers many of the structural features of the test model but the bottom edge is not well resolved (Fig. 5e). However, the V S model is smooth, without much detail (Fig. 5f). The S-wave model shows a curved high-velocity layer between depths about 100 m to 200 m, possibly because of incomplete small-offset data. This curved interface may come from the ray bending in this part of the model and the overlying low-velocity zone above depth 100 m becomes poorly resolved. The normalized RMS model misfits are 5.5% for inverted Vp model (Fig. 5e) and 5.8% for inverted Vs model (Fig. 5f). Fig. 5g shows the resultant V P /V S ratio map with strong artifacts. The independently inverted P- and S-wave velocity models show non-similar structures, with an RMS value of (Fig. 5f). Then we ran the joint inversion of the P- and S-wave traveltime data with cross-gradient constraints. We used the same starting Vp and Vs models as well as the same regularization λ as the independent inversions. Fig. 5i shows Vp model inverted by joint inversion that visually similar to Fig. 5e. Note that the bottom layer is resulted correctly in Fig. 5i. But it somehow appears curved layers, which might be caused by curved Vs layers. It reminds us that less confidence Vs model may degrade the Vp model through joint constraints. The normalized RMS model misfit is 4.8% for Vp model (see Table 2). Through joint inversion, the V S model (Fig. 5j) shows large improvements in comparison with the independent inversion. For example, the S-wave low-velocity Table 1 Final model misfit, data misfit, and RMS of cross-gradient functions for synthetic dataset I. Inversions Normalized RMS of model misfit Normalized RMS of data misfit P-wave S-wave P-wave S-wave Independent Joint RMS of cross-gradient value

5 T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) Fig. 4. (a) P-wave velocity model interpreted from the sonic log (black line). Black triangles denote receivers while red stars represent sources. (b) Shot gather at m depth. P-waves are easily picked and constitute a complete dataset. S-wave picks are relatively easy to make at the far offset but are more difficult in the near offset, so the S-wave traveltime pick dataset is incomplete. layer above depth 100 m is better resolved and the high-velocity layer is flat. The jointly inverted models give the vertical stratification better than do the separately inverted models. The normalized RMS model misfit is 4.5%for inverted Vs model(fig. 5j). Fig. 5k shows the resultant V P /V S ratio map with fewer artifacts and appears flatter for use in geological interpretation than the map from the independent inversions. The RMS of the cross-gradient values is The cross-gradient maps (Fig. 5h and l) imply that Vp and Vs models from the joint inversion are more structurally similar than they do from independent inversions. The V P V S cross-plots obtained from the independent and joint inverted models are shown in Fig. 6. Similar to the previous example, the estimated values are less scattered from the jointly inverted models than the independently inverted models. 4. Field examples 4.1. Joint inversion of V P and V S in the McElroy field We now use the joint inversion algorithm to characterize Vp and Vs models using seismic P- and S-wave traveltime data collected between wells in west Texas. The baseline data recorded in 1993 before CO 2 injection is chosen for this study, since it has relatively high-quality S- wave picks. The source-receiver system in McElroy field (Harris et al., 1995) consists 241 shots in the right well with a spacing of 2.5 ft. and 241 receivers in the left well with a spacing of 2.5 ft. The real sourcereceiver geometry is in 3D with the maximum deviation of 20 ft. in the Y direction (perpendicular to X-Z plane). Compared to the size of X (horizontal) and Z (depth) directions, we simplify the tomography problem in an approximate 2D geometry (see Fig. 7a). The piezoelectric source is used. The profile contains approximately 58,081 traces. Preprocessing of the crosswell data included manual traveltime picking and correction of the source-receiver positions in the deviated boreholes. Seismic P-wave traveltimes range between 8.9 and 15.4 ms, and S-wave traveltimes range between 14.7 and 34.5 ms, with estimated picking errors for both data sets of less than 1 ms. The original P- and S-wave traveltime data are shown in Fig. 6 in Harris et al. (1995). Due to radiation pattern, S-wave energy is missing in the near offset but pickable in the far offset, shown in Fig. 7b, which is same as the second synthetic example. So we limit the effective S-wave picks with incidence angles larger than 45 for inversion in this paper. The S-wave traveltime data contains 40,722 picks compared to 58,081 P-wave picks. The 2D inverse domain size is , for the total number of 21,648 unknowns. The grid spacings in the horizontal and vertical directions are 2.5 ft. From the previous studies (Harris et al., 1995; Lazaratos and Marion, 1997), we know that the lithology is quite flat, so we used 10 times more horizontal constraint than vertical (see Eq. (5)). The regularization parameter λ = for P- and S-wave models is kept constant during the independent and joint inversion procedures. The coefficient of the cross-gradient term β is computed by setting L =1 in the V P inversion and L =4intheV S inversion (see Eq. (7)), implying larger constraint in the V S inversion than the V P inversion using this cross-gradient term. Fig. 8a d show the results of independent inversions for V P,V S, V P /V S, and the resulting cross-gradient models, respectively. The P- and S-velocity models show a high-velocity layer between 2750 ft (838 m) and 2850 ft (868 m) and possibly another between 2600 (792 m) and 2700 ft (823 m). However, the S-velocity model shown less geologically shaped features. The V P /V S ratios (Fig. 8c) are difficult to use for geological prediction, as they have lots of variables, some of which are artifacts. Fig. 8e h show the results of the joint inversion, for which the V P and V S models correlate better. The high-velocity layer between depths ft is more pronounced in the S-velocity model (Fig. 8f). Interestingly, the resultant V P /V S ratio map has fewer artifacts and appears flatter for use in geological interpretation than the map from the independent inversions. Fig. 8h shows the improvement in the structural similarity between V P and V S images as judged by the computed RMS value of the cross-gradient function (RMS = 89.3), which is smaller than that of the separately inverted models (RMS = 309.3). The data misfits are shown in Table 3. Joint inversion leads to the anomalously increased data misfit of P-wave traveltime data than that of independent inversion. There are possible reasons: using a 2D geometry in our inversion could increase nonlinearity of this tomography with realistic 3D geometry. This is true for both independent and joint inversion. But the weighting parameter on the joint structural constraint is likely not to effectively reduce data misfit butmakes inversion focus on enforcing

6 76 T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) Fig. 5. The first row: synthetic V P model (a), V S model (b), V P /V S ratio (c), and cross-gradient values between V P and V S (d). The second row: inverted V P model (e), V S model (f), V P /V S ratio (g), and cross-gradient values between V P and V S (h) reconstructed by independent inversions of P- and S-wave traveltime data. The third row: inverted V P model (i), V S model (j), V P /V S ratio (k), and cross-gradient values between V P and V S (l) reconstructed by iterative joint inversion of P- and S-wave traveltime data. structural similarity. Or, this similar-structure assumption is not truly satisfied in the studying area. Fig. 9 shows the scatter plot of V P V S values from the inverted models and the sonic logs. The V P -V S trend from the joint inversion is less scattered and shows better agreement with that from sonic logs (blue). Similar observations using simultaneous joint inversion of Vp and Vs against independent inversions are also observed using local earthquake data by Tryggvason and Linde (2006). Table 2 Final model misfit, data misfit, and RMS of cross-gradient functions for synthetic dataset II. Inversions Normalized RMS of model misfit Normalized RMS of data misfit P-wave S-wave P-wave S-wave Independent Joint RMS of cross-gradient value 4.2. Constrained inversion of attenuation factor in the King Mountain site In the second field example, we setup an inversion of the attenuation factor constrained by using the V P cross-gradient constraint. The difference from the first example is that we fixthev P model during iterative joint inversion. The prior V P map is well obtained from the previous studies (Langan et al., 1997; Zhu and Harris, 2015). The crosswell data were collected for 201 sources spaced at approximately 1.5 m (5 ft) depth intervals and 203 receivers, also at 1.5 m spacing. Thus, we have approximately 40,000 traces of raw data. The 2D inverse domain size is , for the total number of 4326 unknowns. The grid spacings in the horizontal and vertical directions are 9.5 and 1.5 m (32 and 5 ft), respectively. We estimate the frequency-independent attenuation factor from the first arrivals in the crosswell field data collected in the King Mountain site in west Texas. The attenuation factor estimation is based on the centroid frequency-shift method (Quan and Harris, 1997). The centroid

7 T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) Fig. 6. Crossplot of V P and V S obtained from independent inversion, joint inversion, and true model. frequency-shift method is attractive for crosswell data because it is less sensitive to wave behaviors that affect amplitudes and has a broad frequency band. The idea is simple. The centroid frequency-shift of the wavelet between the source position and the receiver position is proportional to the integral of the attenuation factor over the path. Therefore, the input data for inversion is the centroid frequency shift between the reference wavelet and each trace. In practice, we can measure the receiver centroid frequency from recorded seismograms, but may not directly measure the source centroid frequency and its variance. For simplicity (to obtain relative attenuation), the source centroid frequency is chosen as the maximum value of the receiver centroid frequency at the receivers. The source variance is chosen as the average of the receiver variance at the receivers (Quan and Harris, 1997). For the ray-based approach, the ray path matrix is determined from velocity inversion. We took the velocity traveltime tomography code (see Zhu and Harris, 2015) using centroid frequency shift data rather than traveltime for estimating the attenuation factor. We performed two inversions: independent and joint. Two inversions stop at the tenth iteration. The starting models for both attenuation inversions are homogeneous. Fig. 10a shows the independent V P model taken from Fig. 5 of Zhu and Harris (2015). Fig. 10bshowstheattenuation factor model estimated by independent inversion. The attenuation factor tomogram shows lateral heterogeneities in the reservoir depth from 8700 to 9000 ft (~2651 m to 2712 m). However, the reservoir zone may be hard to delineate from Fig. 10b. Moreover, the independent V P and attenuation factor tomograms give inconsistent geological information about this area. The high-velocity layer between depths 8400 ft (2560 m) and 8500 ft (2590 m) is almost horizontal, while the corresponding low attenuation layer is slightly dipping. In the implementation of the constrained joint inversion, since the velocity model is fairly good, we put a large weight on the cross-gradient term in the objective function of attenuation factor inversion. Fig. 10c shows the resulting jointly estimated attenuation factor tomogram, which gives a consistent (structurally similar) image of the velocity model, especially the carbonate reservoir body. The final data misfit and cross-gradient values are shown in Table 4. Integrating velocity and attenuation factor results, we see that the reservoir zone exhibits the strongest attenuation, and corresponds to the low velocity, around 16,000 ft./s (4876 m/s). The shale layers (between depths 8500 ft (2590 m) and 8700 ft (2651 m) and between depths 9100 ft (2773 m) and 9200 ft (2804 m)) exhibit the proportional relation between velocity and attenuation, which means that regions with low/ medium velocity correspond to medium/low attenuation. We remark the fact that the present iterative joint inversion algorithm with a flexible constraint would make attenuation factor model to be structurally similar to V P model, which might be more geologically interpretable. 5. Discussion Fig. 7. Source-receiver crosswell geometry (a) and a common shot gather data with the shot location at 2840 ft. (b). We can see that P-wave traveltimes are easily picked while S-wave traveltimes are not complete due to source radiation pattern, missing in the near offset. The first synthetic example shows that the presented joint inversion of P- and S-wave traveltime data apparently improves the P-wave and S-wave velocity models (See Fig. 1). In the case, the ambiguity in reservoir geometry from P-wave velocity inversion is caused by limited ray path through reservoir. The reservoir geometry is well solved in the S- wave velocity inversion. On the other hand, the ambiguity of S-wave velocity inversion is caused by relative weak contrast between the gas-saturated sand and the water-saturated sand while P-wave velocity between two sands exhibits high contrast. Therefore, taking advantages of two models in the iterative joint inversion of P- and S-wave traveltimes data can give improved P- and S-wave velocity models by mitigating their ambiguities. However, this is not the case in the rest of examples. Either S-wave traveltimes in the specific crosswell geometry are incomplete or frequency-shift attenuation data is with uncertainty when using the first arrivals' waveform that is always interfered by later arrivals. Two models inverted from such an imperfect dataset are believed to have larger uncertainties than P-wave model inverted from complete and robust P-wave first arrival traveltimes. In this situation, the iterative joint inversion takes efforts to improve the low-confident S-wave velocity and attenuation factor models by incorporating P-wave velocity model but not much improvement in the high-confident P-wave velocity model.

8 78 T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) Fig. 8. Inverted models by independent (a d) and joint inversions (e h). a) d): Inverted P-wave, S-wave velocity model, V P /V S and cross-gradient values between two models by independent inversion. e) h): Corresponding results by joint inversion. Although iterative joint inversion scheme avoid weighting between multiple datasets in single objective function, it still requires the weighting between the data, the regularization, and the cross-gradient functional (i.e., λ,β), which adjust the tradeoff among structural similarity, model misfit, and data misfit. In this study, we specify the same weights for the cross-gradient approach as independent inversions for comparisons, which may impact the joint inversion results. For example, the joint inversion results show improved models with similarstructure, but P-wave model seems not to be improved consistently (e.g., see the dipping channel in Fig. 2i in the first synthetic example and P-wave model in Fig. 5i in the second synthetic example). This might be due to the un-optimal weights for the regularization term and cross-gradient constrained term, which reminds us to investigate the weightings for the joint inversion approach in the future. It is worthy to note that structural constraint using the cross-gradient function is effective in our synthetic examples, provided that Table 3 Final data misfit and RMS of cross-gradient functions for field dataset I. Inversions Normalized RMS of P-wave traveltime data Normalized RMS of S-wave traveltime data RMS of cross-gradient value Independent Joint

9 T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) Table 4 Final data misfit and RMS of cross-gradient functions for field dataset II. Inversions Normalized RMS of frequency-shift data Independent Joint RMS of cross gradient value show our iterative joint inversion algorithm with the cross-gradient structural constraint, other constraints can also be incorporated into this iterative joint inversion framework. 6. Conclusions Fig. 9. Crossplot of V P and V S obtained from independent inversion (red), joint inversion (yellow), and the sonic log (blue). The trend indicated by yellow dots is closer to the sonic log. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) structural similarity exists between physical parameters. When it violates, the convergence of joint inversion with structural constraint may be slow or even fail. If this is the case, other constraints, e.g., petrophysical and empirical relation, may be the choice. Although we We have reported an iterative joint inversion approach for inverting P-wave velocity, S-wave velocity and attenuation factor models using a structural constraint. It is different from simultaneous joint inversion. Simultaneous joint inversion couples independent inversions in a single iterative domain with a joint cross constraint. Careful attention must be taken for the regularization coefficients and relative weights between multiple models for convergence. The iterative joint inversion framework avoids this selection of relative weights for different datasets and flexibly allows us to change the model parameterization and number of objective functions in order to investigate different coupling approaches. It is also flexible to incorporate a prior model as a structural constraint in our iterative joint inversion. In addition, the iterative Fig. 10. Inverted attenuation factor models by independent (b) and joint inversions (c). Inverted P-velocity model (a) is used as a cross-gradient constraint in the iterative joint inversion.

10 80 T. Zhu, J.M. Harris / Journal of Applied Geophysics 123 (2015) joint inversion is very easy to code without many modifications in the original independent inversion. This inversion framework is straightforwardly extended for inverting multiple datasets. The joint constraint can also use other constraints, e.g., petrophysical and empirical relations. We demonstrate the algorithm's feasibility using two synthetic examples and two different field datasets from west Texas. Our results demonstrate the benefits of using iterative joint inversion are: (1) better similarity in the geologic structural features between Vp and Vs models and between Vp and attenuation factor models; (2) moderate improvement in estimated values, e.g., improved V P -Vs relationships compared with those determined by well logs; (3) flexibility for constraining a lower-confidence model with the use of a higher-confidence model. 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