Joint stochastic inversion of multi‐component pre‐stack seismic data

Joint inversion is an important quantitative interpretation tool for integrated geophysics. The present study situation and progress of joint inversion using gravity and seismic data are reviewed and discussed in the paper first. Then, the improved very fast simulated annealing algorithm (VFSA) with its obvious advantage of finding overall optimal solution in inverse problems is applied. So the constrained synchronous joint inversion of gravity and seismic data using the improved VFSA is put forward. The model with uncommon interface of density and velocity is designed. Through the model test and field observed data processing and interpretation, this method is proved effective and practicable, especially the application in Xuwen area shows that the density and velocity structures of the underground complicated interfaces could be exactly determined and revealed. With the constrained known information, this joint inversion method is clearly superior to the single inversion of gravity data.

Summary Estimation of reservoir parameters is im portant for prediction and optimization of oil production. Time-lapse seismic data, commonly acquired in mature oil fields to identify profitable bypassed zones, can be used to estimate reservoir parameters by posing the history matching process as a joint inversion problem. Due to the size of the problem and existence of multiple minima (of the error function), only global optimization techniques, such as Very Fast Simulated Annealing (VFSA) have an opportunity to find the global minimum. However, even for the VFSA technique, the computational cost of the forward modeling (particularly, the reservoir simulation) can make the inversion process impractical in real applications if we consider that many inversions are required to estimate the solution uncertainty. To improve the convergence characteristics of the VFSA technique, we have developed a modification of this technique where a regulation of the local temperature associated with the model variables is driven by the local changes of the error function. We call this modification Local Thermal Regulation (LTR) and show that the time (or number of iterations) spent for convergence can be reduced by 25 to 50%, while maintaining the same degree of ergodicity of the conventional VFSA technique.

ABSTRACTWe present results from the resolution and sensitivity analysis of 1D DC resistivity and IP sounding data using a non‐linear inversion. The inversion scheme uses a theoretically correct Metropolis–Gibbs' sampling technique and an approximate method using numerous models sampled by a global optimization algorithm called very fast simulated annealing (VFSA). VFSA has recently been found to be computationally efficient in several geophysical parameter estimation problems. Unlike conventional simulated annealing (SA), in VFSA the perturbations are generated from the model parameters according to a Cauchy‐like distribution whose shape changes with each iteration. This results in an algorithm that converges much faster than a standard SA. In the course of finding the optimal solution, VFSA samples several models from the search space. All these models can be used to obtain estimates of uncertainty in the derived solution. This method makes no assumptions about the shape of an a posteriori probability density function in the model space. Here, we carry out a VFSA‐based sensitivity analysis with several synthetic and field sounding data sets for resistivity and IP. The resolution capability of the VFSA algorithm as seen from the sensitivity analysis is satisfactory. The interpretation of VES and IP sounding data by VFSA, incorporating resolution, sensitivity and uncertainty of layer parameters, would generally be more useful than the conventional best‐fit techniques.

Proppant mapping is critical for optimizing fracture treatment design and improving wells’ productivity. An electrode-based resistivity tool concept was developed earlier for proppant mapping in cased-hole wells. An array of insulating gaps is installed and cemented in place as a permanent part of the casing string. The electrical measurements are performed by imposing a voltage across each insulating gap, one at a time, before and after hydraulic fracture operations. The voltages across other insulating gaps near the transmitter gap are recorded. The method relies on direct excitation of the casing, which is expected to overcome the severe limitations of induction tools in cased-hole wells. A forward model based on a finite volume method has been developed to simulate the tool’s response to one or multiple fractures. To enable the implementation of such a practical system in multistage fractured horizontal wells, a fast and robust inversion approach is required. To that end, we have developed a divide-and-conquer approach based on a global optimization algorithm very fast simulated annealing (VFSA). Specifically, the original inverse problem is divided into subproblems and each subproblem can be solved separately using VFSA. The results indicate that our approach can invert the data and output widths and radii of multiple fractures without requiring a large number of forward simulations. The robustness of the inverse solver is also tested by adding Gaussian noise to the synthetic data. We tested example cases that demonstrate that when up to 5% noise is introduced, VFSA still provides very accurate inversion results with moderate uncertainties. Inversion results with some more realistic conditions, e.g., tilted fractures, complex fractures, and so on, are also presented.

In large loop transient electromagnetic method (TEM), the late time apparent resistivity formula cannot truly reflect the geoelectric model, thus it needs to define the all-time apparent resistivity with the position information of measuring point. Utilizing very fast simulated annealing (VFSA) to fit the theoretical electromagnetic force (EMF) and measured EMF could obtain the all-time apparent resistivity of the measuring points in rectangular transmitting loop. The selective cope of initial model of VFSA could be confirmed by taking the late time apparent resistivity of transient electromagnetic method as the prior information. For verifying the correctness, the all-time apparent resistivities of the geoelectric models were calculated by VFSA and dichotomy, respectively. The results indicate that the relative differences of apparent resistivities calculated by these two methods are within 3%. The change of measuring point position has little influence on the tracing pattern of all-time apparent resistivity. The first branch of the curve of all-time apparent resistivity is close to the resistivity of the first layer medium and the last branch is close to the resistivity of the last layer medium, which proves the correctness of the arithmetics proposed.

We describe a stochastic method of wavelet estimation combining a global optimization technique based on a Very Fast Simulated Annealing (VFSA), and spline parameterization. Our method inverts for the amplitude of the wavelet at several nodes and uses a spline interpolation scheme to estimate its amplitude in between those nodes. Our method retains all the distinct advantages of VFSA in that it searches a wider model space, chooses a random starting model and thereby has minimal dependence on the initial model, and does not require assumptions for the phase. Work is ongoing to incorporate statistical methods of computing uncertainties associated with the estimated wavelets which will provide an additional advantage to our method. We illustrate our method in this paper with example estimations of zero and hybrid phase wavelets. We also compute the mean of estimated zero phase wavelets resulting from several VFSA runs. Our results in their current form demonstrate that the method is stable in appropriately constructing the wavelet.

[1] We present a new tomography method based on the local beam semblance and the very fast simulated annealing (VFSA) global optimization method. The data space is the local beam semblance calculated using local slant stacks for overlapping offset windows, i.e. beam windows, of the original common-shot or common-receiver gathers. On each beam semblance panel, the first coherency peak can be identified with a particular ray parameter, first-arrival traveltime and beam center position. The forward problem can be solved with any ray tracer to find arrivals matching the identified peaks. Our inversion scheme uses VFSA to find the maximum-a-posteriori (MAP) solution and estimates the uncertainty by applying Bayesian analysis of all the sampled models for a specified model parameterization. This integration of automatic local semblance evaluation instead of first-arrival picking and a fast forward modeling method combined with VFSA to determine the optimal model makes our method robust, efficient and accurate.

Inverse modeling and uncertainty assessment of magnetic data from 2D thick dipping dyke and application for mineral exploration

A nonlinear stochastic inversion scheme, called very fast simulated annealing (VFSA), was applied to the time-domain electromagnetic data generated from a horizontal electric dipole. The forward formulation of the vertical magnetic field was expressed in the Laplace domain by applying the Hankel integral transform. Time-domain transformation was performed by applying the inverse Laplace transform using the Gaver–Stehfest algorithm. In this study, for noise-free synthetic data, the VFSA scheme yielded the smallest misfit and an inverted resistivity model that resembled the test model. The addition of 5% random noise to the synthetic data produced the same level of misfit and a model that still mimicked the test model. However, the addition of 10% noise to the synthetic data resulted in a misfit value that was three times that of the first two values and a resistivity model with a large discrepancy with the test model, particularly at large depths. These results indicate the efficacy of the VFSA inversion scheme for inferring the subsurface resistivity structure from the electromagnetic data. This inversion scheme was applied to field data measured in a volcanic environment. The general pattern of the resistivity structure inferred by the VFSA inversion is consistent with the structure obtained previously by using a deterministic inversion scheme.

Historically, velocity model building for PS imaging has been done sequentially, starting with estimation of P-velocity and anisotropic parameters using PP data alone, followed by updates of S-velocity to match PS and PP depth images. A better option is to include both PP and PS data in model building from the start and find the model that satisfies both data types, producing flat events on gathers and PP and PS images with consistent depths. We implemented a new method for joint PP/PS tomography in which moveout picks on PP and PS depth gathers are combined with floating event constraints between PP and PS images to simultaneously solve for Vp, Vs, and anisotropic parameters (Mathewson et al, 2013). Our method is an improvement on previous approaches that require separate treatment of PP and PS data in the early stages of model building, allowing the use of combined PP and PS data throughout the model-building process. Benefits of the joint PP/PS tomography include improved efficiency, reduced uncertainty, and better resolution of anisotropy.

Multicomponent seismic exploration provides more valuable information for the prediction of underground structures, lithology, fluids, and fractures. Most studies on lithology estimation and fluid description using PP- and PS-wave seismic data mainly focus on relatively shallow targets with high porosity. Focusing on deep carbonate (dolomite) reservoirs with low porosity (generally less than 5%), we first analyze the characteristics of the seismic response difference between PP and PS waves, and then we combine logging data and seismic forward modeling to implement the identification of fluids using the amplitude difference between PP and PS waves. Data sets are acquired over grain beach dolomite reservoirs in the Cambrian Longwangmiao Formation, the central Sichuan Basin, Western China. The reservoirs are filled with different types of fluids (e.g., gas, water, bitumen, etc.). Based on comparisons between seismic responses of PP and PS waves, we observe that, at the location of the gas-bearing layer, PP-wave amplitude exhibits strong peaks and PS-wave amplitude finds relatively weak peaks, which is different from that obtained for the water-saturated layer. However, at the location of bitumen-filled reservoirs, PP- and PS-wave amplitudes exhibit strong peaks, and the reflection amplitude of PP waves is stronger than that of PS waves. Therefore, the amplitude difference of PP and PS waves provides valuable information and feasibility for fluid identification. We verify that the maximum peak amplitude attribute of PS waves may better characterize the distribution of porous dolomite reservoirs than that of PP waves, and using the attribute of maximum peak amplitude difference between PP and PS waves, we may distinguish gas-bearing and water-saturated layers. Comparing with actual drilling results, we conclude that our PP/PS interpretation technique is feasible for fluid identification in deep carbonate reservoirs.

A good starting model is imperative in full-waveform inversion (FWI) because it solves a least-squares inversion problem using a local gradient-based optimization method. A suboptimal starting model can result in cycle skipping leading to poor convergence and incorrect estimation of subsurface properties. This problem is especially crucial for salt models because the strong velocity contrasts create substantial time shifts in the modeled seismogram. Incorrect estimation of salt bodies leads to velocity inaccuracies in the sediments because the least-squares gradient aims to reduce traveltime differences without considering the sharp velocity jump between sediments and salt. We have developed a technique to estimate velocity models containing salt bodies using a combination of global and local optimization techniques. To stabilize the global optimization algorithm and keep it computationally tractable, we reduce the number of model parameters by using sparse parameterization formulations. The sparse formulation represents sediments using a set of interfaces and velocities across them, whereas a set of ellipses represents the salt body. We use very fast simulated annealing (VFSA) to minimize the misfit between the observed and synthetic data and estimate an optimal model in the sparsely parameterized space. The VFSA inverted model is then used as a starting model in FWI in which the sediments and salt body are updated in the least-squares sense. We partition model updates into sediment and salt updates in which the sediments are updated like conventional FWI, whereas the shape of the salt is updated by taking the zero crossing of an evolving level set surface. Our algorithm is tested on two 2D synthetic salt models, namely, the Sigsbee 2A model and a modified SEG Advanced Modeling Program (SEAM) Phase I model while fixing the top of the salt. We determine the efficiency of the VFSA inversion and imaging improvements from the level set FWI approach and evaluate a few sources of uncertainty in the estimation of salt shapes.

Summary The Simulated Annealing (SA) algorithm has been widely used in geophysical data processing. However, the seri al SA algorithm is limited by low computing efficiency. In this paper, the annealing schedule has been improved in the SA algorithm, and a constrained inversion of magnetotelluric (MT) data using the parallel SA algorithm has been applied. The test of synthetic data shows that the improved Very Fast Simulated Annealing (VFSA) method is significantly better than the traditional method. Additionally, the computing efficiency has been improved using the parallel SA inversion algorithm. This method has also been applied in MT data processing for geothermal exploration and good geological results have been achieved.

The present study evaluates the benefits of using PS seismic data to improve the seismic reservoir characterization, as well as, proposes a 3D-3C seismic survey design, which guarantees to record PP and PS wave appropriately. As a result, it was obtained for the target reservoirs that PS seismic data present a shorter wavelength than PP data; therefore, a better vertical seismic resolution can be obtained using PS seismic data. Additionally, it was determined in the study area that some geological interfaces produce stronger PS than PP reflection coefficients, so it could be advantageous to use PS post-stack seismic data to map these interfaces. Furthermore, it was obtained that a key attribute to make an effective seismic lithology discrimination is density. Therefore, due to density can be better estimated using simultaneously PP and PS data, PS seismic represent an important data to improve the seismic lithology discrimination. Finally, it was proposed a 3D-3C seismic survey design that takes in consideration a maximum offset to allow a good PS amplitude response and a suitable bin size for PP and PS image requirements, as well as, smooth fold of coverage for PP data and not zero fold striping PS fold distribution.

Joint inversion of gravity and vertical gradient data based on modified structural similarity index for the structural and petrophysical consistency constraint