Efficient inversion of 2.5D electrical resistivity data using the discrete adjoint method

Geophysics ◽  
2021 ◽  
pp. 1-54
Author(s):  
Diego Domenzain ◽  
John Bradford ◽  
Jodi Mead
Keyword(s):  
Electrical Resistivity ◽  
Gradient Descent ◽  
Adjoint Method ◽  
Sparse Matrices ◽  
Joint Inversion ◽  
Geophysical Data ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Model Parameters ◽  
Inversion Algorithm

We present a memory and operation-count efficient 2.5D inversion algorithm of electrical resistivity (ER) data that can handle fine discretization domains imposed by other geophysical (e.g, ground penetrating radar or seismic) data. Due to numerical stability criteria and available computational memory, joint inversion of different types of geophysical data can impose different grid discretization constraints on the model parameters. Our algorithm enables the ER data sensitivities to be directly joined with other geophysical data without the need of interpolating or coarsening the discretization. We employ the adjoint method directly in the discretized Maxwell's steady state equation in order to compute the data sensitivity to the conductivity. In doing so, we make no finite difference approximation on the Jacobian of the data and avoid the need to store large and dense matrices. Rather, we exploit matrix-vector multiplication of sparse matrices and find successful convergence using gradient descent for our inversion routine without having to resort to the Hessian of the objective function. By assuming a 2.5D subsurface, we are able to linearly reduce memory requirements when compared to a 3D gradient descent inversion, and by a power of two when compared to storing a 2D Hessian. Moreover, our method linearly outperforms operation counts when compared to 3D Gauss-Newton conjugate-gradient schemes, which scales cubically in our favor with respect to the thickness of the 3D domain. We physically appraise the domain of the recovered conductivity using a cut-off of the electric current density present in our survey. We present two case studies in order to assess the validity of our algorithm. First, on a 2.5D synthetic example, and then on field data acquired in a controlled alluvial aquifer, where we were able match the recovered conductivity to borehole observations.

scholarly journals A Numerical Model of Temperate Glacier Flow Applied to the Quiescent Phase of a Surge-Type Glacier

1982 ◽  
Vol 28 (99) ◽  
pp. 239-265 ◽  
Author(s):  
Robert Bindschadler
Keyword(s):  
Shear Stress ◽  
Numerical Model ◽  
Stress Gradient ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Model Parameters ◽  
Small Contribution ◽  
Base Shear ◽  
Quiescent Phase ◽  
Glacier Flow

AbstractA time-dependent numerical model of temperate glacier flow without sliding is developed and applied to the quiescent phase of surge-type Variegated Glacier, Alaska. The model is based on a one-dimensional continuity equation but the transverse channel shape is explicitly included allowing the complex geometries of real glaciers to be modelled. Velocities and volume fluxes are calculated from the glacier geometry. Transverse stress is taken into account by shape factors which are fitted to measurements of geometry and velocity and are chosen to be insensitive to changes in geometry. Longitudinal stress gradients are taken into account by use of a large-scale surface slope. A Crank-Nicholson finite-difference approximation is used and it is unconditionally stable when a small contribution from the local slope is added to the average slope.Model parameters are fitted to extensive data collected on Variegated Glacier in 1973 and 1974. Predictions of the model over a four year interval agree well with field measurements. Predictions of the current quiescent phase (1965–84) indicate depth increases in the upper glacier of more than 75 m with a twenty-fold increase in the volume flux. During this interval the base shear stress increases 40% in the upper glacier and decreases 20% in the lower glacier. During the mid to late quiescent phase, ice motion becomes more important than mass balance in the redistribution of mass over the central region of the glacier. If normal flow were to persist, the predicted steady-state profile would be an average of 100 m deeper and 41% more voluminous than in 1973.The predicted base shear-stress gradient is never negative enough to satisfy Robin and Weertman’s (1973) condition for blockage of subglacial water flow. The annual rate of water production by dissipation of mechanical straining at the bed remains two orders of magnitude below that produced by summer surface melt. The predicted fractional increase in base stress during the quiescent phase is a maximum in the region believed to be the trigger zone of the surges.

Global optimization to retrieve borehole-derived petrophysical properties of carbonates

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. D75-D82
Author(s):  
Alireza Shahin ◽  
Mike Myers ◽  
Lori Hathon
Keyword(s):  
Dielectric Constant ◽  
Electrical Resistivity ◽  
Global Optimization ◽  
Joint Inversion ◽  
Joint Modeling ◽  
Carbonate Reservoir ◽  
Dual Porosity ◽  
Model Parameters ◽  
Well Log ◽  
Petrophysical Properties

Joint modeling and inversion of frequency-dependent dielectric constant and electrical resistivity well-log measurements has been addressed in literature in recent years. However, this problem is not studied for dual-porosity carbonate formations. Besides, the salinity and matrix dielectric constant are presumed to be known in previous studies. We have combined a model for brine dielectric constant and two laboratory-supported models for the electrical resistivity and dielectric constant of dual-porosity carbonates. Using this methodology, we replicate electrical resistivity and dielectric well-log measurements. Using a stochastic global optimization algorithm, we formulate a joint inversion workflow to estimate petrophysical properties of interest. For a constructed dual-porosity carbonate reservoir, we determine that the inversion workflow matches the forward-modeled data for the oil column, water column, and transition zone. We also found that our inversion workflow is capable to retrieve local model parameters (water-filled intergranular porosity and water-filled vuggy porosity) and global model parameters (matrix dielectric constant, lithology exponents for intergranular and vuggy pores, and salinity) with reasonable accuracy.

Properties of signature partner superdeformed bands in mercury nuclei

2015 ◽  
Vol 11 (1) ◽  
pp. 2918-2926 ◽  
Author(s):  
Mahmoud Abokilla ◽  
Hayam Yassin ◽  
Eman R. Abo Elyazeed
Keyword(s):  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Model Parameters ◽  
Transition Energies ◽  
Fitting Procedure ◽  
Higher Order Terms ◽  
Fourth Derivative ◽  
The Difference ◽  
Experimental Transition ◽  
Search Program

The properties of superdeformed (SD) bands of five pairs signature partners in mercury nuclei have been systematically analyzed in framework of four parameters formula including higher order terms of Bohr-Mottelson collective rotational energies. The level spins and the model parameters are determined by fitting procedure using a computer simulated search program in order to obtain minimum root mean square deviations between the calculated and the experimental transition energies.The best fitted parameters have been used to calculate the transition energies Eγ, the rotational frequencies , the kinematic J(1) and dynamic J(2) moments of inertia. The calculated results agree excellently with the experimental data. J(2) is significantly larger than J(1) for all values of  . Also J(2) show a smooth increase with increasing . The appearance of ΔI = 1 and ΔI = 2 staggering in γ-ray transition energies have been examined by using the five-points formula representing the finite difference approximation to the fourth derivative of the γ-ray transition energies at a given spin. The signature partners in Hg nuclei show large amplitude staggering. Also to appear the ΔI = 1 staggering, the transition energies relative to a rigid rotor with a moment of inertia J = 128.219 are plotted against spins for each signature partner pairs. The difference in transition energies between transitions in the two SD bands 191Hg(SD3)and 193Hg(SD3) are small, therefore, these two bands have been considered as identical bands.

scholarly journals Testing alternative uses of electromagnetic data to reduce the prediction error of groundwater models

2016 ◽  
Vol 20 (5) ◽  
pp. 1925-1946 ◽  
Author(s):  
Nikolaj Kruse Christensen ◽  
Steen Christensen ◽  
Ty Paul A. Ferre
Keyword(s):  
Electrical Resistivity ◽  
Hydraulic Conductivity ◽  
Prediction Error ◽  
Model Calibration ◽  
Hydrologic Model ◽  
Geophysical Data ◽  
Model Parameters ◽  
Calibration Data ◽  
Groundwater Model ◽  
And Inversion

Abstract. In spite of geophysics being used increasingly, it is often unclear how and when the integration of geophysical data and models can best improve the construction and predictive capability of groundwater models. This paper uses a newly developed HYdrogeophysical TEst-Bench (HYTEB) that is a collection of geological, groundwater and geophysical modeling and inversion software to demonstrate alternative uses of electromagnetic (EM) data for groundwater modeling in a hydrogeological environment consisting of various types of glacial deposits with typical hydraulic conductivities and electrical resistivities covering impermeable bedrock with low resistivity (clay). The synthetic 3-D reference system is designed so that there is a perfect relationship between hydraulic conductivity and electrical resistivity. For this system it is investigated to what extent groundwater model calibration and, often more importantly, model predictions can be improved by including in the calibration process electrical resistivity estimates obtained from TEM data. In all calibration cases, the hydraulic conductivity field is highly parameterized and the estimation is stabilized by (in most cases) geophysics-based regularization. For the studied system and inversion approaches it is found that resistivities estimated by sequential hydrogeophysical inversion (SHI) or joint hydrogeophysical inversion (JHI) should be used with caution as estimators of hydraulic conductivity or as regularization means for subsequent hydrological inversion. The limited groundwater model improvement obtained by using the geophysical data probably mainly arises from the way these data are used here: the alternative inversion approaches propagate geophysical estimation errors into the hydrologic model parameters. It was expected that JHI would compensate for this, but the hydrologic data were apparently insufficient to secure such compensation. With respect to reducing model prediction error, it depends on the type of prediction whether it has value to include geophysics in a joint or sequential hydrogeophysical model calibration. It is found that all calibrated models are good predictors of hydraulic head. When the stress situation is changed from that of the hydrologic calibration data, then all models make biased predictions of head change. All calibrated models turn out to be very poor predictors of the pumping well's recharge area and groundwater age. The reason for this is that distributed recharge is parameterized as depending on estimated hydraulic conductivity of the upper model layer, which tends to be underestimated. Another important insight from our analysis is thus that either recharge should be parameterized and estimated in a different way, or other types of data should be added to better constrain the recharge estimates.

Structural joint inversion on irregular meshes

2019 ◽  
Vol 220 (3) ◽  
pp. 1995-2008 ◽  
Author(s):  
C Jordi ◽  
J Doetsch ◽  
T Günther ◽  
C Schmelzbach ◽  
H Maurer ◽  
...  
Keyword(s):  
Electrical Resistivity ◽  
Joint Inversion ◽  
A Priori ◽  
Structural Similarity ◽  
P Wave ◽  
Parameter Distribution ◽  
Data Sets ◽  
Inversion Algorithm ◽  
Structural Joint ◽  
Irregular Meshes

SUMMARY Structural joint inversion of several data sets on an irregular mesh requires appropriate coupling operators. To date, joint inversion algorithms are primarily designed for the use on regular rectilinear grids and impose structural similarity in the direct neighbourhood of a cell only. We introduce a novel scheme for calculating cross-gradient operators based on a correlation model that allows to define the operator size by imposing physical length scales. We demonstrate that the proposed cross-gradient operators are largely decoupled from the discretization of the modelling domain, which is particularly important for irregular meshes where cell sizes vary. Our structural joint inversion algorithm is applied to a synthetic electrical resistivity tomography and ground penetrating radar 3-D cross-well experiment aiming at imaging two anomalous bodies and extracting the parameter distribution of the geostatistical background models. For both tasks, joint inversion produced superior results compared with individual inversions of the two data sets. Finally, we applied structural joint inversion to two field data sets recorded over a karstified limestone area. By including geological a priori information via the correlation-based operators into the joint inversion, we find P-wave velocity and electrical resistivity tomograms that are in accordance with the expected subsurface geology.

Joint MT and CSEM data inversion using a multiplicative cost function approach

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. F203-F214 ◽  
Author(s):  
A. Abubakar ◽  
M. Li ◽  
G. Pan ◽  
J. Liu ◽  
T. M. Habashy
Keyword(s):  
Cost Function ◽  
Large Scale ◽  
Joint Inversion ◽  
A Priori ◽  
Upper And Lower Bounds ◽  
Model Parameters ◽  
Data Sets ◽  
Inversion Algorithm ◽  
Earth Resistivity ◽  
The Cost

We have developed an inversion algorithm for jointly inverting controlled-source electromagnetic (CSEM) data and magnetotelluric (MT) data. It is well known that CSEM and MT data provide complementary information about the subsurface resistivity distribution; hence, it is useful to derive earth resistivity models that simultaneously and consistently fit both data sets. Because we are dealing with a large-scale computational problem, one usually uses an iterative technique in which a predefined cost function is optimized. One of the issues of this simultaneous joint inversion approach is how to assign the relative weights on the CSEM and MT data in constructing the cost function. We propose a multiplicative cost function instead of the traditional additive one. This function does not require an a priori choice of the relative weights between these two data sets. It will adaptively put CSEM and MT data on equal footing in the inversion process. The inversion is accomplished with a regularized Gauss-Newton minimization scheme where the model parameters are forced to lie within their upper and lower bounds by a nonlinear transformation procedure. We use a line search scheme to enforce a reduction of the cost function at each iteration. We tested our joint inversion approach on synthetic and field data.

Joint inversion of multiple geophysical data using guided fuzzy c-means clustering

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. ID37-ID57 ◽  
Author(s):  
Jiajia Sun ◽  
Yaoguo Li
Keyword(s):  
Joint Inversion ◽  
A Priori ◽  
Physical Property ◽  
Geophysical Data ◽  
Inversion Algorithm ◽  
Unified Framework ◽  
Parameter Domain ◽  
Fuzzy C Means ◽  
Fcm Clustering ◽  
Petrophysical Data

Joint inversion of multiple geophysical data has become an active area of research due to its potential to greatly enhance the fidelity of inverted models. Many open questions and challenges still remain. One of them is how to effectively incorporate into joint inversion multimodal petrophysical information that describes the statistical behavior of physical property values in the parameter domain (i.e., in a crossplot). We have regarded the multimodal petrophysical data as different clusters in the parameter domain and developed an approach that handles multimodal petrophysical information through guided fuzzy c-means (FCM) clustering in the parameter domain. We inverted the petrophysical data in the parameter domain in a similar manner to and simultaneously with the geophysical data in the spatial domain through minimizing one common objective function. Numerical examples have determined that the resulting models from this multidomain joint-inversion strategy are able to reproduce both the geophysical and the petrophysical data. In addition to incorporating a priori multimodal petrophysical information into inversion, guided FCM clustering also allows us to integrate geology differentiation and geophysical inversion into one unified framework that makes these two components positively affect each other. Geology differentiation results were obtained as a direct output from joint inversion. We have also developed a strategy for imposing different clustering constraints in different model regions, allowing region-specific a priori petrophysical information to be incorporated into inversion. We have applied our joint-inversion algorithm to the SEG/EAGE salt model in four different scenarios, and we found that the proposed algorithm produced much better geophysical models and geology differentiation results than separate inversions.

scholarly journals Pareto Joint Inversion of 2D magnetometric and gravity data- synthetic study

2019 ◽  
Vol 133 ◽  
pp. 01009
Author(s):  
Tomasz Danek ◽  
Andrzej Leśniak ◽  
Katarzyna Miernik ◽  
Elżbieta Śledź
Keyword(s):  
Pareto Front ◽  
Joint Inversion ◽  
Gravity Data ◽  
Group Analysis ◽  
Geophysical Methods ◽  
Model Parameters ◽  
Data Sets ◽  
Inversion Algorithm ◽  
Promising Tool ◽  
Real Model

Pareto joint inversion for two or more data sets is an attractive and promising tool which eliminates target functions weighing and scaling, providing a set of acceptable solutions composing a Pareto front. In former author’s study MARIA (Modular Approach Robust Inversion Algorithm) was created as a flexible software based on global optimization engine (PSO) to obtain model parameters in process of Pareto joint inversion of two geophysical data sets. 2D magnetotelluric and gravity data were used for preliminary tests, but the software is ready to handle data from more than two geophysical methods. In this contribution, the authors’ magnetometric forward solver was implemented and integrated with MARIA. The gravity and magnetometry forward solver was verified on synthetic models. The tests were performed for different models of a dyke and showed, that even when the starting model is a homogeneous area without anomaly, it is possible to recover the shape of a small detail of the real model. Results showed that the group analysis of models on the Pareto front gives more information than the single best model. The final stage of interpretation is the raster map of Pareto front solutions analysis.

scholarly journals Bed Topography and Mass-Balance Distribution of Columbia Glacier, Alaska, U.S.A., Determined from Sequential Aerial Photography

1988 ◽  
Vol 34 (117) ◽  
pp. 208-216 ◽  
Author(s):  
L. A. Rasmussen
Keyword(s):  
Mass Balance ◽  
Aerial Photography ◽  
Continuity Equation ◽  
The Other ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Model Parameters ◽  
Bed Topography ◽  
Displacement Vectors ◽  
Grid Nodes

AbstractAn internally consistent data set of geometry and flow variables for the lower part of Columbia Glacier, south-central Alaska, is derived entirely from vertical aerial photography. The principle of mass conservation is imposed on the data in the form of a centered finite-difference approximation of the continuity equation. It is applied on a 120-node section of a square grid covering the 15 km long, high-velocity stretch ending at the grounded, heavily calving terminus of this large glacier.Photography was obtained 22 times between June 1977 and September 1981. Surface altitudes on the dates of the flights and the displacement vectors between pairs of flights were determined photogrammetrically. Natural features on the glacier surface were sufficiently prominent and enduring to be followed from the date of one flight to the next.Because both the altitude points and displacement vectors were irregularly positioned spatially, interpolation was necessary to get values on the grid nodes. The points had already been subjected to the method of optimum interpolation to get surface altitudes on the grid nodes. The displacement vectors are subjected here to a constrained–interpolation method to get velocity vectors at the grid nodes that are consistent, through the continuity equation, with the other variables.The other variables needed to achieve closure of the variable set are bed topography and mass-balance distribution. The latter was taken to be a separate linear function of altitude for each time interval. Values for bed altitudes at 120 nodes and two coefficients of each 21 balance functions were inferred as the 162 model parameters in a non-linear minimization problem having 4305 observed velocity components as its data.

scholarly journals A Numerical Model of Temperate Glacier Flow Applied to the Quiescent Phase of a Surge-Type Glacier

1982 ◽  
Vol 28 (99) ◽  
pp. 239-265 ◽  
Author(s):  
Robert Bindschadler
Keyword(s):  
Shear Stress ◽  
Numerical Model ◽  
Stress Gradient ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Model Parameters ◽  
Small Contribution ◽  
Base Shear ◽  
Quiescent Phase ◽  
Glacier Flow

AbstractA time-dependent numerical model of temperate glacier flow without sliding is developed and applied to the quiescent phase of surge-type Variegated Glacier, Alaska. The model is based on a one-dimensional continuity equation but the transverse channel shape is explicitly included allowing the complex geometries of real glaciers to be modelled. Velocities and volume fluxes are calculated from the glacier geometry. Transverse stress is taken into account by shape factors which are fitted to measurements of geometry and velocity and are chosen to be insensitive to changes in geometry. Longitudinal stress gradients are taken into account by use of a large-scale surface slope. A Crank-Nicholson finite-difference approximation is used and it is unconditionally stable when a small contribution from the local slope is added to the average slope.Model parameters are fitted to extensive data collected on Variegated Glacier in 1973 and 1974. Predictions of the model over a four year interval agree well with field measurements. Predictions of the current quiescent phase (1965–84) indicate depth increases in the upper glacier of more than 75 m with a twenty-fold increase in the volume flux. During this interval the base shear stress increases 40% in the upper glacier and decreases 20% in the lower glacier. During the mid to late quiescent phase, ice motion becomes more important than mass balance in the redistribution of mass over the central region of the glacier. If normal flow were to persist, the predicted steady-state profile would be an average of 100 m deeper and 41% more voluminous than in 1973.The predicted base shear-stress gradient is never negative enough to satisfy Robin and Weertman’s (1973) condition for blockage of subglacial water flow. The annual rate of water production by dissipation of mechanical straining at the bed remains two orders of magnitude below that produced by summer surface melt. The predicted fractional increase in base stress during the quiescent phase is a maximum in the region believed to be the trigger zone of the surges.

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