Joint seismic and petrophysical nonlinear inversion with Gaussian mixture-based adaptive regularization

Geophysics ◽  
2021 ◽  
pp. 1-57
Author(s):  
Qiang Guo ◽  
Jing Ba ◽  
Li-Yun Fu ◽  
Cong Luo
Keyword(s):  
Inverse Problem ◽  
Seismic Data ◽  
Simulated Annealing Algorithm ◽  
Gaussian Mixture ◽  
Forward Model ◽  
Inversion Method ◽  
Model Parameters ◽  
Adaptive Regularization ◽  
Petrophysical Parameters ◽  
Highly Nonlinear

The estimation of reservoir parameters from seismic observations is one of the main objectives in reservoir characterization. However, the forward model relating petrophysical properties of rocks to observed seismic data is highly nonlinear, and solving the relevant inverse problem is a challenging task. We present a novel inversion method for jointly estimating elastic and petrophysical parameters of rocks from prestack seismic data. We combine a full rock-physics model and the exact Zoeppritz equation as the forward model. To overcome the ill-conditioning of the inverse problem and address the complex prior distribution of model parameters given lithofacies variations, we introduce a regularization term based on the prior Gaussian mixture model under Bayesian framework. The objective function is optimized by the fast simulated annealing algorithm, during which the Gaussian mixture-based regularization terms are adaptively and iteratively adjusted by the maximum likelihood estimator, allowing the posterior distribution to be more consistent with the observed seismic data. The adaptive regularization method improves the accuracy of petrophysical parameters compared to the sequential inversion and non-adaptive regularization methods, and the inversion result can be used for indicating gas-saturated areas when applied to field data.

scholarly journals Uncertainty reduction of interval inversion estimation results using a factor analysis approach

2020 ◽  
Vol 11 (1) ◽  
Author(s):  
Armand Abordán ◽  
Norbert Péter Szabó
Keyword(s):  
Inverse Problem ◽  
Factor Analysis ◽  
Water Saturation ◽  
Inversion Method ◽  
Model Parameters ◽  
Depth Range ◽  
Petrophysical Parameters ◽  
Shale Volume ◽  
Interval Inversion ◽  
The Impact

AbstractThis paper aims to investigate the impact of the overdetermination (data-to-unknowns) ratio on the global inversion of wireline logging data. In the course of the so-called interval inversion method, geophysical data measured in a borehole over a longer depth range is jointly inverted and the depth variation of the investigated petrophysical parameters are expanded into series using Legendre polynomials as basis functions resulting in a highly overdetermined inverse problem. A metaheuristic Particle Swarm Optimization (PSO) approach is applied as a first phase of inversion for decreasing the starting model dependence of the interval inversion procedure. In the subsequent linear inversion steps, by using the measurement error of logging tools and the covariance matrix of the estimated petrophysical parameters, we can quantify the accuracy of the model parameters. The dataset used in this study consists of nuclear, resistivity and sonic logs which are inverted to compute porosity, shale volume and water saturation along the investigated interval. For increasing the data-to-unknowns ratio of the inverse problem, shale volume is estimated separately by a PSO-based factor analysis and fixed as known parameter for the interval inversion process. Since the shale volume has been described as high degree Legendre polynomial, a significant increase of the overdetermination ratio considerably decreases the uncertainty of the remaining model parameters allowing for a more reliable calculation of hydrocarbon content.

scholarly journals Interval inversion approach for an improved interpretation of well logs

Geophysics ◽  
2016 ◽  
Vol 81 (2) ◽  
pp. D155-D167 ◽  
Author(s):  
Mihály Dobróka ◽  
Norbert Péter Szabó ◽  
József Tóth ◽  
Péter Vass
Keyword(s):  
Inverse Problem ◽  
Well Logging ◽  
Well Logs ◽  
Depth Interval ◽  
Inversion Method ◽  
Model Parameters ◽  
Data Set ◽  
Petrophysical Parameters ◽  
Inversion Procedure ◽  
Interval Inversion

The quality analysis of well-logging inversion results has always been an important part of formation evaluation. The precise calculation of hydrocarbon reserves requires the most accurate possible estimation of porosity, water saturation, and shale and rock-matrix volumes. The local inversion method conventionally used to predict the above model parameters depth by depth represents a marginally overdetermined inverse problem, which is rather sensitive to the uncertainty of observed data and limited in estimation accuracy. To reduce the harmful effect of data noise on the estimated model, we have suggested the interval inversion method, in which an increase of the overdetermination ratio allows a more accurate solution of the well-logging inverse problem. The interval inversion method inverts the data set of a longer depth interval to predict the vertical distributions of petrophysical parameters in a joint inversion procedure. In formulating the forward problem, we have extended the validity of probe response functions to a greater depth interval assuming the petrophysical parameters are depth dependent, and then we expanded the model parameters into a series using the Legendre polynomials as basis functions for modeling inhomogeneous formations. We solved the inverse problem for a much smaller number of expansion coefficients than data to derive the petrophysical parameters in a stable overdetermined inversion procedure. The added advantage of the interval inversion method is that the layer thicknesses and suitably chosen zone parameters can be estimated automatically by the inversion procedure to refine the results of inverse and forward modeling. We have defined depth-dependent model covariance and correlation matrices to compare the quality of the local and interval inversion results. A detailed study using well logs measured from a Hungarian gas-bearing unconsolidated formation revealed that the greatly overdetermined interval inversion procedure can be effectively used in reducing the estimation errors in shaly sand formations, which may refine significantly the results of reserve calculation.

A New Inversion Method Using a Modified Bat Algorithm for Analysis of Seismic Refraction Data in Dam Site Investigation

2019 ◽  
Vol 24 (2) ◽  
pp. 201-214
Author(s):  
Rashed Poormirzaee ◽  
Siamak Sarmady ◽  
Yusuf Sharghi
Keyword(s):  
Seismic Data ◽  
Seismic Refraction ◽  
Bat Algorithm ◽  
Inversion Method ◽  
Model Parameters ◽  
Inversion Algorithm ◽  
Dam Site ◽  
Seismic Refraction Data ◽  
Modified Bat Algorithm ◽  
Refraction Data

Similar to any other geophysical method, seismic refraction method faces non-uniqueness in the estimation of model parameters. Recently, different nonlinear seismic processing techniques have been introduced, particularly for seismic inversion. One of the recently developed metaheuristic algorithms is bat optimization algorithm (BA). Standard BA is usually quick at the exploitation of the solution, while its exploration ability is relatively poor. In order to improve exploration ability of BA, in the current study, a hybrid metaheuristic algorithm by inclusion a mutation operator into BA, so-called mutation based bat algorithm (MBA), is introduced to inversion of seismic refraction data. The efficiency and stability of the proposed inversion algorithm were tested on different synthetic cases. Finally, the MBA inversion algorithm was applied to a real dataset acquired from Leylanchay dam site at East-Azerbaijan province, Iran, to determine alluvium depth. Then, the performance of MBA on both synthetic and real datasets was compared with standard BA. Moreover, the dataset was further processed following a tomographic approach and the results were compared to the results of the proposed MBA inversion method. In general, the MBA inversion results were superior to standard BA inversion and results of MBA were in good agreement with available boreholes data and geological sections at the dam site. The analysis of the seismic data showed that the studied site comprises three distinct layers: a saturated alluvial, an unsaturated alluvial, and a dolomite bedrock. The measured seismic velocity across the dam site has a range of 400 to 3,500 m/s, with alluvium thickness ranging from 5 to 19 m. Findings showed that the proposed metaheuristic inversion framework is a simple, fast, and powerful tool for seismic data processing.

scholarly journals Evaluation of a Straight-Ray Forward Model for Bayesian Inversion of Crosshole Ground Penetrating Radar Data

Electronics ◽  
2019 ◽  
Vol 8 (6) ◽  
pp. 630 ◽  
Author(s):  
Hui Qin ◽  
Xiongyao Xie ◽  
Yu Tang
Keyword(s):  
Ground Penetrating Radar ◽  
Computational Cost ◽  
Measurement Data ◽  
Forward Model ◽  
Bayesian Inversion ◽  
Inversion Method ◽  
Model Parameters ◽  
Model Structure ◽  
Modeling Error ◽  
Ground Penetrating

Bayesian inversion of crosshole ground penetrating radar (GPR) data is capable of characterizing the subsurface dielectric properties and qualifying the associated uncertainties. Markov chain Monte Carlo (MCMC) simulations within the Bayesian inversion usually require thousands to millions of forward model evaluations for the parameters to hit their posterior distributions. Therefore, the CPU cost of the forward model is a key issue that influences the efficiency of the Bayesian inversion method. In this paper we implement a widely used straight-ray forward model within our Bayesian inversion framework. Based on a synthetic unit square relative permittivity model, we simulate the crosshole GPR first-arrival traveltime data using the finite-difference time-domain (FDTD) and straight-ray solver, respectively, and find that the straight-ray simulator runs 450 times faster than its FDTD counterpart, yet suffers from a modeling error that is more than 7 times larger. We also perform a series of numerical experiments to evaluate the performance of the straight-ray model within the Bayesian inversion framework. With modeling error disregarded, the inverted posterior models fit the measurement data nicely, yet converge to the wrong set of parameters at the expense of unreasonably large number of iterations. When the modeling error is accounted for, with a quarter of the computational burden, the main features of the true model can be identified from the posterior realizations although there still exist some unwanted artifacts. Finally, a smooth constraint on the model structure improves the inversion results considerably, to the extent that it enhances the inversion accuracy approximating to those of the FDTD model, and further reduces the CPU demand. Our results demonstrate that the use of the straight-ray forward model in the Bayesian inversion saves computational cost tremendously, and the modeling error correction together with the model structure constraint are the necessary amendments that ensure that the model parameters converge correctly.

Seismostratigraphic inversion: Appraisal, ambiguity, and uncertainty

Geophysics ◽  
2007 ◽  
Vol 72 (4) ◽  
pp. R51-R66 ◽  
Author(s):  
Matthias G. Imhof ◽  
Arvind K. Sharma
Keyword(s):  
Seismic Data ◽  
Individual Characteristics ◽  
Forward Model ◽  
Process Models ◽  
Model Parameters ◽  
Influx Rate ◽  
Trade Off ◽  
Geologic Time ◽  
Geological Processes ◽  
Seismic Reflectors

Geologic process models predict the geometry of geologic strata and their petrophysical properties, based on mathematical models of geological processes that affect the formation and evolution of geologic strata. Such processes include erosion, sediment transport, and deposition. The resulting forward model is typically nonlinear. Given observations and a misfit measure, one may attempt inversion of these models to estimate process parameters that yield compatible predictions. For seismostratigraphic inversion, seismic data are used as observations. We tested such an algorithm in a prograding-delta environment to examine the effect of using different seismic attributes as observations and, thus, different choices of misfit measures. The first measure,based on the degree of parallelism between seismic reflectors and modeled geologic strata, demonstrated a trade-off between geologic time and the sediment-influx rate used to parameterize the model. A second misfit measure used unwrapped seismic instantaneous phase as a crude proxy to relative geologic time, which regularized the model parameters. Then last, we combined the two measures to take advantage of their individual characteristics. For most of these inversion experiments, we obtained results that capture the geometry of the geologic strata as observed on the seismic data. With the exception of the depositional time-rate trade-off, where the same strata can be obtained in a shorter geologic interval when rates are increased, we found the inversion to be surprisingly stable, with a unique cluster of acceptable parameters, despite the nonlinearity of the geologic forward model.

Bayesian Hamiltonian Monte Carlo method for the estimation of pyrolysis parameter S1

Geophysics ◽  
2021 ◽  
Vol 86 (6) ◽  
pp. M197-M209
Author(s):  
Kun Luo ◽  
Zhaoyun Zong ◽  
Xingyao Yin ◽  
Hong Cao ◽  
Minghui Lu
Keyword(s):  
Monte Carlo ◽  
Seismic Data ◽  
Rock Physics ◽  
Gaussian Mixture ◽  
Organic Carbon Content ◽  
Shale Oil ◽  
Important Indicator ◽  
Hamiltonian Monte Carlo ◽  
Petrophysical Parameters ◽  
Sweet Spots

A Gaussian mixture Hamiltonian Monte Carlo (HMC) Bayesian method has been developed for the inversion of petrophysical parameters such as pyrolysis parameter S1, which is driven by a statistical shale rock-physics model. Pyrolysis parameter S1 can be used to indicate the content of free or adsorbed hydrocarbons in source rock, and it is an important indicator to evaluate the production of shale oil reservoirs. However, most studies on pyrolysis parameters are based on pyrolysis experiments and there is no relevant study to inverse pyrolysis parameter S1 from seismic data. In addition, compared to the total organic carbon content, pyrolysis S1 is more accurate for evaluating gas and oil in shale. In particular, high values of pyrolysis S1 can directly indicate the content of shale oil. We have developed a strategy for assessing shale oil sweet spots through estimating pyrolysis S1 and other petrophysical parameters. Based on the Gaussian mixture assumptions for the prior distribution of the model, we build a joint distribution to link the pyrolysis parameter S1 with elastic attributes, and then we derive a formulation to inverse S1 with the Bayesian model. Due to the components of the Gaussian mixture, the HMC method has been used to sample the posterior distribution. Our study finds that the HMC method for sampling can improve the efficiency and allow a more robust quantification of the uncertainty; also, application to real seismic data sets indicates that the delineation of sweet spots is more accurate combined with pyrolysis S1.

scholarly journals Seismic inversion by Newtonian machine learning

Geophysics ◽  
2020 ◽  
Vol 85 (4) ◽  
pp. WA185-WA200
Author(s):  
Yuqing Chen ◽  
Gerard T. Schuster
Keyword(s):  
Neural Network ◽  
Machine Learning ◽  
Wave Equation ◽  
Seismic Data ◽  
Velocity Model ◽  
Seismic Inversion ◽  
Low Rank ◽  
Inversion Method ◽  
Model Parameters ◽  
Latent Space

We present a wave-equation inversion method that inverts skeletonized seismic data for the subsurface velocity model. The skeletonized representation of the seismic traces consists of the low-rank latent-space variables predicted by a well-trained autoencoder neural network. The input to the autoencoder consists of seismic traces, and the implicit function theorem is used to determine the Fréchet derivative, i.e., the perturbation of the skeletonized data with respect to the velocity perturbation. The gradient is computed by migrating the shifted observed traces weighted by the skeletonized data residual, and the final velocity model is the one that best predicts the observed latent-space parameters. We denote this as inversion by Newtonian machine learning (NML) because it inverts for the model parameters by combining the forward and backward modeling of Newtonian wave propagation with the dimensional reduction capability of machine learning. Empirical results suggest that inversion by NML can sometimes mitigate the cycle-skipping problem of conventional full-waveform inversion (FWI). Numerical tests with synthetic and field data demonstrate the success of NML inversion in recovering a low-wavenumber approximation to the subsurface velocity model. The advantage of this method over other skeletonized data methods is that no manual picking of important features is required because the skeletal data are automatically selected by the autoencoder. The disadvantage is that the inverted velocity model has less resolution compared with the FWI result, but it can serve as a good initial model for FWI. Our most significant contribution is that we provide a general framework for using wave-equation inversion to invert skeletal data generated by any type of neural network. In other words, we have combined the deterministic modeling of Newtonian physics and the pattern matching capabilities of machine learning to invert seismic data by NML.

scholarly journals Fast probabilistic petrophysical mapping of reservoirs from 3D seismic data

Geophysics ◽  
2012 ◽  
Vol 77 (3) ◽  
pp. O1-O19 ◽  
Author(s):  
Mohammad S. Shahraeeni ◽  
Andrew Curtis ◽  
Gabriel Chao
Keyword(s):  
Seismic Data ◽  
Water Saturation ◽  
Random Noise ◽  
Clay Content ◽  
Inversion Method ◽  
Model Parameters ◽  
Nonlinear Inversion ◽  
Desktop Computer ◽  
Data Set ◽  
Mixture Density

A fast probabilistic inversion method for 3D petrophysical property prediction from inverted prestack seismic data has been developed and tested on a real data set. The inversion objective is to estimate the joint probability density function (PDF) of model vectors consisting of porosity, clay content, and water saturation components at each point in the reservoir, from data vectors with compressional- and shear-wave-impedance components that are obtained from the inversion of seismic data. The proposed inversion method is based on mixture density network (MDN), which is trained by a given set of training samples, and provides an estimate of the joint posterior PDF’s of the model parameters for any given data point. This method is much more time and memory efficient than conventional nonlinear inversion methods. The training data set is constructed using nonlinear petrophysical forward relations and includes different sources of uncertainty in the inverse problem such as variations in effective pressure, bulk modulus and density of hydrocarbon, and random noise in recorded data. Results showed that the standard deviations of all model parameters were reduced after inversion, which shows that the inversion process provides information about all parameters. The reduction of uncertainty in water saturation was smaller than that for porosity or clay content; nevertheless the maximum of the a posteriori (MAP) of model PDF clearly showed the boundary between brine saturated and oil saturated rocks at wellbores. The MAP estimates of different model parameters show the lateral and vertical continuity of these boundaries. Errors in the MAP estimate of different model parameters can be reduced using more accurate petrophysical forward relations. This fast, probabilistic, nonlinear inversion method can be applied to invert large seismic cubes for petrophysical parameters on a standard desktop computer.

A constrained global inversion method using an overparameterized scheme: Application to poststack seismic data

Geophysics ◽  
2001 ◽  
Vol 66 (2) ◽  
pp. 613-626 ◽  
Author(s):  
Xin‐Quan Ma
Keyword(s):  
Simulated Annealing ◽  
Seismic Data ◽  
A Priori ◽  
Constant Thickness ◽  
Inversion Method ◽  
Model Parameters ◽  
A Priori Information ◽  
Initial Model ◽  
Seismic Waveform ◽  
Priori Information

A global optimization algorithm using simulated annealing has advantages over local optimization approaches in that it can escape from being trapped in local minima and it does not require a good initial model and function derivatives to find a global minimum. It is therefore more attractive and suitable for seismic waveform inversion. I adopt an improved version of a simulated annealing algorithm to invert simultaneously for acoustic impedance and layer interfaces from poststack seismic data. The earth’s subsurface is overparameterized by a series of microlayers with constant thickness in two‐way traveltime. The algorithm is constrained using the low‐frequency impedance trend and has been made computationally more efficient using this a priori information as an initial model. A search bound of each parameter, derived directly from the a priori information, reduces the nonuniqueness problem. Application of this technique to synthetic and field data examples helps one recover the true model parameters and reveals good continuity of estimated impedance across a seismic section. This approach has the capability of revealing the high‐resolution detail needed for reservoir characterization when a reliable migrated image is available with good well ties.

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