Naturally smooth inversions with a priori information from well logs

Geophysics ◽  
1991 ◽  
Vol 56 (11) ◽  
pp. 1811-1818 ◽  
Author(s):  
M. Pilkington ◽  
J. P. Todoeschuck
Keyword(s):  
Inverse Problem ◽  
A Priori ◽  
Power Spectral Analysis ◽  
Well Logs ◽  
Noise Model ◽  
Model Parameters ◽  
Scaling Exponent ◽  
A Priori Information ◽  
Power Spectral ◽  
Priori Information

Regularization is usually necessary to guarantee a solution to a given inverse problem. When constructing a model that gives an adequate fit to the data, some suitable method of regularization which provides numerical stability can be used. When investigating the resolution and variance of the computed model parameters, the character of regularization should be specified by the a priori information available. This avoids arbitrary variation of the damping to suit the interpreter. For geophysical inverse problems we determine the appropriate level of regularization (in the form of parameter covariances) from power spectral analysis of well‐log measurements. For resistivity data, well logs indicate that the spatial variation with depth can be described adequately by a scaling noise model, that is, one in which the power spectral density is proportional to some power (α) of the frequency. We show that α, the scaling exponent, controls the smoothness of the final model. For α < 0, the model becomes smoother as α becomes more negative. As a specific example, this approach is applied to the magnetotelluric inverse problem. A synthetic example illustrates the smoothing effect of α on inversion. Comparison between the scaling noise approach and a previous Backus‐Gilbert type inversion on some field data shows that using the appropriate value of α (−1.8 for this example) results in a model which is structurally simple and contains only those features well resolved by the data.

scholarly journals The Neuroelectromagnetic Inverse Problem and the Zero Dipole Localization Error

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Rolando Grave de Peralta ◽  
Olaf Hauk ◽  
Sara L. Gonzalez
Keyword(s):  
Inverse Problem ◽  
Unique Solution ◽  
A Priori ◽  
Inverse Solution ◽  
Localization Error ◽  
A Priori Information ◽  
Dipole Localization ◽  
Inverse Solutions ◽  
Priori Information ◽  
Additional Constraints

A tomography of neural sources could be constructed from EEG/MEG recordings once the neuroelectromagnetic inverse problem (NIP) is solved. Unfortunately the NIP lacks a unique solution and therefore additional constraints are needed to achieve uniqueness. Researchers are then confronted with the dilemma of choosing one solution on the basis of the advantages publicized by their authors. This study aims to help researchers to better guide their choices by clarifying what is hidden behind inverse solutions oversold by their apparently optimal properties to localize single sources. Here, we introduce an inverse solution (ANA) attaining perfect localization of single sources to illustrate how spurious sources emerge and destroy the reconstruction of simultaneously active sources. Although ANA is probably the simplest and robust alternative for data generated by a single dominant source plus noise, the main contribution of this manuscript is to show that zero localization error of single sources is a trivial and largely uninformative property unable to predict the performance of an inverse solution in presence of simultaneously active sources. We recommend as the most logical strategy for solving the NIP the incorporation of sound additional a priori information about neural generators that supplements the information contained in the data.

scholarly journals Linear inverse Gaussian theory and geostatistics

Geophysics ◽  
2006 ◽  
Vol 71 (6) ◽  
pp. R101-R111 ◽  
Author(s):  
Thomas Mejer Hansen ◽  
Andre G. Journel ◽  
Albert Tarantola ◽  
Klaus Mosegaard
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Covariance Function ◽  
A Priori ◽  
Model Space ◽  
Sequential Gaussian Simulation ◽  
A Priori Information ◽  
Sequential Approach ◽  
Linear Inverse Problem ◽  
Priori Information

Inverse problems in geophysics require the introduction of complex a priori information and are solved using computationally expensive Monte Carlo techniques (where large portions of the model space are explored). The geostatistical method allows for fast integration of complex a priori information in the form of covariance functions and training images. We combine geostatistical methods and inverse problem theory to generate realizations of the posterior probability density function of any Gaussian linear inverse problem, honoring a priori information in the form of a covariance function describing the spatial connectivity of the model space parameters. This is achieved using sequential Gaussian simulation, a well-known, noniterative geostatisticalmethod for generating samples of a Gaussian random field with a given covariance function. This work is a contribution to both linear inverse problem theory and geostatistics. Our main result is an efficient method to generate realizations, actual solutions rather than the conventional least-squares-based approach, to any Gaussian linear inverse problem using a noniterative method. The sequential approach to solving linear and weakly nonlinear problems is computationally efficient compared with traditional least-squares-based inversion. The sequential approach also allows one to solve the inverse problem in only a small part of the model space while conditioned to all available data. From a geostatistical point of view, the method can be used to condition realizations of Gaussian random fields to the possibly noisy linear average observations of the model space.

BAYESIAN METHODS IN THE EVALUATION OF SCATTERING DATA

2006 ◽  
Vol 15 (02) ◽  
pp. 354-361
Author(s):  
H. J. KRAPPE
Keyword(s):  
Elastic Scattering ◽  
Bayesian Methods ◽  
Optical Model ◽  
A Priori ◽  
Scattering Data ◽  
Model Parameters ◽  
A Priori Information ◽  
Model Independent ◽  
Priori Information

It is shown how a priori information can be introduced in an optimal way to extract optical-model parameters in a model-independent way from incomplete elastic scattering data.

scholarly journals Aircraft Turbine Engine Automatic Control Based on Adaptive Dynamic Models

2020 ◽  
Vol 2020 (4) ◽  
pp. 61-70
Author(s):  
Sergiy Yepifanov
Keyword(s):  
Automatic Control ◽  
Thermodynamic Model ◽  
Dynamic Models ◽  
A Priori ◽  
Static Model ◽  
Measuring System ◽  
Model Parameters ◽  
A Priori Information ◽  
Static Models ◽  
Priori Information

AbstractOne of the most perspective development directions of the aircraft engine is the application of adaptive digital automatic control systems (ACS). The significant element of the adaptation is the correction of mathematical models of both engine and its executive, measuring devices. These models help to solve tasks of control and are a combination of static models and dynamic models, as static models describe relations between parameters at steady-state modes, and dynamic ones characterize deviations of the parameters from static values.The work considers problems of the models’ correction using parametric identification methods. It is shown that the main problem of the precise engine simulation is the correction of the static model. A robust procedure that is based on a wide application of a priori information about performances of the engine and its measuring system is proposed for this purpose. One of many variants of this procedure provides an application of the non-linear thermodynamic model of the working process and estimation of individual corrections to the engine components’ characteristics with further substitution of the thermodynamic model by approximating on-board static model. Physically grounded estimates are obtained based on a priori information setting about the estimated parameters and engine performances, using fuzzy sets.Executive devices (actuators) and the most inertial temperature sensors require correction to their dynamic models. Researches showed, in case that the data for identification are collected during regular operation of ACS, the estimates of dynamic model parameters can be strongly correlated that reasons inadmissible errors.The reason is inside the substantial limitations on transients’ intensity that contain regular algorithms of acceleration/deceleration control. Therefore, test actions on the engine are required. Their character and minimum composition are determined using the derived relations between errors in model coefficients, measurement process, and control action parameters.

On an inverse spectral problem for one integro-differential operator of fractional order

2019 ◽  
Vol 27 (1) ◽  
pp. 17-23 ◽  
Author(s):  
Mikhail Ignatiev
Keyword(s):  
Inverse Problem ◽  
Differential Operator ◽  
Spectral Problem ◽  
Fractional Order ◽  
Inverse Spectral Problem ◽  
A Priori ◽  
A Priori Information ◽  
Inverse Spectral ◽  
Priori Information

Abstract An inverse spectral problem for some integro-differential operator of fractional order {\alpha\in(1,2)} is studied. We show that the specification of the spectrum together with a certain a priori information about the structure of the operator determines such operator uniquely. The proof is constructive and provides a procedure for solving the inverse problem.

Should blood transport kinetics be modeled in metabolic processes?

1984 ◽  
Vol 247 (5) ◽  
pp. R895-R900
Author(s):  
G. Belforte ◽  
B. Bona ◽  
G. Molino
Keyword(s):  
A Priori ◽  
Model Parameters ◽  
Physiological System ◽  
A Priori Information ◽  
Design Task ◽  
Single Compartment ◽  
Priori Information ◽  
Blood Transport ◽  
Insight Into ◽  
Drug Kinetics

We analyze the interaction between blood transport phenomena and uptake processes when drug kinetics are studied with compartmental models. Relevant advantages in the physiological interpretation of the model parameters are obtained when blood transport is explicitly included in the model. This is done by aggregating into a single compartment all the blood spaces where no exchange with extravascular spaces takes place and separating into different blood compartments those spaces where some uptake and/or return occurs. The proposed strategy extensively uses all available a priori information about the physiological system, instead of considering only the information available in the measurements. This modeling approach has three main advantages: it provides greater insight into the identified quantities; it allows the introduction of quantitative a priori information; and it facilitates the experiment design task.

Particle swarm optimization of 2D magnetotelluric data

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. E125-E141 ◽  
Author(s):  
Francesca Pace ◽  
Alessandro Santilano ◽  
Alberto Godio
Keyword(s):  
Inverse Problem ◽  
Particle Swarm Optimization ◽  
A Priori ◽  
Particle Swarm ◽  
Stability And Convergence ◽  
Real Field ◽  
A Priori Information ◽  
Data Set ◽  
Swarm Optimization ◽  
Priori Information

We implement the particle swarm optimization (PSO) algorithm for the two-dimensional (2D) magnetotelluric (MT) inverse problem. We first validate PSO on two synthetic models of different complexity and then apply it to an MT benchmark for real-field data, the COPROD2 data set (Canada). We pay particular attention to the selection of the PSO input parameters to properly address the complexity of the 2D MT inverse problem. We enhance the stability and convergence of the solution of the geophysical problem by applying the hierarchical PSO with time-varying acceleration coefficients (HPSO-TVAC). Moreover, we parallelize the code to reduce the computation time because PSO is a computationally demanding global search algorithm. The inverse problem was solved for the synthetic data both by giving a priori information at the beginning and by using a random initialization. The a priori information was given to a small number of particles as the initial position within the search space of solutions, so that the swarming behavior was only slightly influenced. We have demonstrated that there is no need for the a priori initialization to obtain robust 2D models because the results are largely comparable with the results from randomly initialized PSO. The optimization of the COPROD2 data set provides a resistivity model of the earth in line with results from previous interpretations. Our results suggest that the 2D MT inverse problem can be successfully addressed by means of computational swarm intelligence.

scholarly journals Monte Carlo full-waveform inversion of crosshole GPR data using multiple-point geostatistical a priori information

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. H19-H31 ◽  
Author(s):  
Knud Skou Cordua ◽  
Thomas Mejer Hansen ◽  
Klaus Mosegaard
Keyword(s):  
Inverse Problem ◽  
Monte Carlo ◽  
A Priori ◽  
Waveform Inversion ◽  
Multiple Point ◽  
Full Waveform Inversion ◽  
A Posteriori ◽  
A Priori Information ◽  
Full Waveform ◽  
Priori Information

We present a general Monte Carlo full-waveform inversion strategy that integrates a priori information described by geostatistical algorithms with Bayesian inverse problem theory. The extended Metropolis algorithm can be used to sample the a posteriori probability density of highly nonlinear inverse problems, such as full-waveform inversion. Sequential Gibbs sampling is a method that allows efficient sampling of a priori probability densities described by geostatistical algorithms based on either two-point (e.g., Gaussian) or multiple-point statistics. We outline the theoretical framework for a full-waveform inversion strategy that integrates the extended Metropolis algorithm with sequential Gibbs sampling such that arbitrary complex geostatistically defined a priori information can be included. At the same time we show how temporally and/or spatiallycorrelated data uncertainties can be taken into account during the inversion. The suggested inversion strategy is tested on synthetic tomographic crosshole ground-penetrating radar full-waveform data using multiple-point-based a priori information. This is, to our knowledge, the first example of obtaining a posteriori realizations of a full-waveform inverse problem. Benefits of the proposed methodology compared with deterministic inversion approaches include: (1) The a posteriori model variability reflects the states of information provided by the data uncertainties and a priori information, which provides a means of obtaining resolution analysis. (2) Based on a posteriori realizations, complicated statistical questions can be answered, such as the probability of connectivity across a layer. (3) Complex a priori information can be included through geostatistical algorithms. These benefits, however, require more computing resources than traditional methods do. Moreover, an adequate knowledge of data uncertainties and a priori information is required to obtain meaningful uncertainty estimates. The latter may be a key challenge when considering field experiments, which will not be addressed here.

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