Theorem theinverseofastandardcauchyrandomvariable x. The cauchy distribution, named after augustin cauchy, is a continuous probability distribution. Inversion of gravity and gravity gradiometry data for. Paper open access bayesian acoustic impedance inversion. In industrial fields, temperature distribution information is regarded as an important parameter for monitoring different industrial processes, improvement of the system efficiency, reduction of pollutant emission, etc. The standard cauchy distribution has probability density function fx 1.
Here we will use capital f to denote the cdf and lowercase f to denote the probability density function pdf. Generating random numbers from any distribution using transformations cdf inverse, boxmuller method. The standard cauchy distribution has pdf and cdf fx 1. Compared with commonly used gauss distribution and cauchy distribution, modified cauchy prior constraint fits the real data well, and can effectively protect weak reflectivity. Chapter 4 generating nonuniform random variables arizona math. So here is the inverse transform method for generating a rv x having. We compare the proposed tomography reconstruction method to. Seismic wave scattering inversion for fluid factor of. The proof of this is a straightforward calculation. Normal random numbers can also be generated using the general inverse transform method e. Suppose g has a geometric distribution, so the mass function is pg g 1. Although powerful, this inverse cdf method can be computationally expensive unless you have a formula for the inverse cdf. Request pdf modified cauchyconstrained seismic blind deconvolution starting from bayes sparse inversion theory, the paper adopted modified cauchy criterion to build up reflectivity sparse. In this method, cauchy distribution is utilized to the ratios of fluid factors, shear moduli and densities of perturbation media and background homogeneous media, respectively.
The results show that the new method is capable of providing highresolution depth estimation for. Summary the reliable estimation of properties of deep reservoirs from seismic data is a difficult and longstanding problem. A study of multiscale seismic data joint inversion method. Posts formula 1930 based on asymptotic expansion laplaces method of the forward integral post 1930, gaver 1966, valkoabate 2004 weeks method 1966 laguerre polynomial expansion method.
Avaf inversion method for detecting hydrocarbons journal. Create pd by fitting a probability distribution to sample data from the. Gap and inversion formulas for the cauchy integral. Introduction to simulation using r free textbook course. Note that it is easy to sample from the cauchy distribution. With this equation as forward solver, a prestack seismic bayesian inversion method is proposed to estimate the fluid factor of heterogeneous media. Distribution inverse transformation method at least in principle, there is a way to convert a uniform distribution to any other distribution. The inverse cdf method for simulating from a distribution. We demonstate our inversion method based on the cauchytype integral for several synthetic models. Cauchy priori distributionbased bayesian avo reflectivity inversion may lead to sparse estimates that are sensitive to large reflectivities. On the other hand, the method is robust by taking into account the nongaussian feature of data. Prestack ava inversion of exact zoeppritz equations based. A computational quasireversibility method for cauchy.
Constantnormal in the args value of codegen the input argument pd can be a fitted probability distribution object for beta, exponential, extreme value, lognormal, normal, and weibull distributions. Ournumericalstudiesdetermined that the new method is much faster than. On one side, the method is of variational due to the use of the adjoint technique. The inversion for the fluid indicator is formulated in bayesian framework with the cauchy distribution as a prior constraint. Cauchy inver sion is also frequently used in the geophysical inversion as a robust optimization method 3. Pdf elastic impedance inversion for effective porefluid. Theorem theinverseofastandardcauchyrandomvariablex isalsostandardcauchy. Alemie and sacchi obtained sparse highresolution inversion results based on the prior trivariate cauchy probability distribution. Modified prp conjugate gradient method is a stable method, and it has faster convergent rate than conventional method. Theorem the inverse of a standard cauchy random variable x is. The same trend is observed using both the gaussian and trivariate cauchy priors that reflects they do a similar job in incorporating the welllog information thereby stabilizing the inversion. In this paper, we discuss and compare the corrected inversion method, the lognntail method and the. Inverse survival function, the formula for the cauchy inverse survival function is.
Rejection sampling 2 inversion method the inversion method requires that we nd the inverse of the cumulative distribution function cdf of f. At the same time, the relations among the model parameters are assumed. The probability density function pdf technique, univariate. The cauchy distribution is used to model random variables with very heavy tails. Whe cauchy distribution has heavier tails than the gaussian distribution. Proof lettherandomvariablex havethestandardcauchydistribution. The input argument name must be a compiletime constant. Using inversion, we can sample the cauchy dis tribution by. However, the sampling function of the traditional mh sampling is a fixed parameter distribution. Understand the theoretical foundations and convergence properties of these methods. Finding a fast and effective method for achieving the temperature distribution measurement is of great importance. The equation for the standard cauchy distribution reduces to. Example exponential distribution for an exponential distribution with rate, we have.
Also, assume f is continuous and strictly increasing as a function. Cauchy distribution is able to obtain strong reflections with large amplitudes and has superior noise handling performance. Non linear multiparameter inversion 993 among local sampling of the model space. Random variate generation raj jain washington university in saint louis saint louis, mo 63. In this case the usual inverse function need not be defined.
Numerical inversion methods timeline the development of accurate numerical inversion laplace transform methods is a long standing problem. Suppose g has a geometric distribution, so the mass function is. We will look at general principles like inversion and acceptancerejection sampling. But it is particularly useful for random variates that their inverse function can be easily solved. For example you can not find a closed form for a normal distribution. The cauchy distribution has the probability density function pdf. The method is implemented in the framework of the 1dvar retrieval with simulated atovs data. To sample the bernoulli distribution by inversion, take x 1 1 u6p. For this method, aprioriinformation that total variation tv data of seismic data satisfy the laplace distribution, a form of nongaussian distribution, is adopted to control the noise reduction. Inverse cumulative distribution function matlab icdf. Modified cauchyconstrained seismic blind deconvolution. Reconstruction method for inversion problems in an. Proof let the random variable x have the standard cauchy distribution.
One interesting fact about a cauchy random variable is that none of its moments are finite. The rst general method that we present is called the inverse transform method. The standard cauchy distribution has pdf and cdf f x 1. The following is the plot of the cauchy cumulative distribution function. The gap theorem in this section we shall obtain gap theorems for the cauchy and cauchy stieltjes integral. Numerical laplace transform inversion methods with. The assumptions above can be weakened to some extent. When x has a continuous distribution then it has a probability density function fx.
Through the test from the field and model data, this method is feasible and this model based on gamma distribution has a result with tall resolution. For example, to use the normal distribution, include coder. Thus, if u has a cauchy distribution, eu, and all higher order moments, do not exist. For the inversion, the computation of the covariance matrix and regularized terms requires prior estimation of model parameters, which makes the iterative inversion weakly nonlinear. Using the transformation technique, y gx 1x is a 11 transformation from.
Mallick et al 2000 show that when there is 2% random noise in the observed data, the linear fit method proposed by cambois 2000 is shown to be dramatically unstable. Figures 5a, 5b, and 5c are results of the inversion using trivariate cauchy as prior distribution. For large n the individual generation of the xis is unfeasible, and the inversionofabetavariate is potentially inaccurate. Inverse problems in science and engineering 8, 119. Stat 451 lecture notes 0512 simulating random variables. Chapter 2 inversion method, transformation methods and rejection sampling. Elastic impedance variation with angle inversion for. This method works when you have distributions such as exponential, cauchy, geometric, pareto, logistic, extreme value weibull, etc.
With the increasing burial depth of the reservoir, the seismic data acquisition is difficult to obtain data in large incident angle with high sn ratio, the result of large angle data inversion turns out to be in poor quality. Nonlinear multiparameter inversion using a hybrid global. For these theorems we need an assumption on the curve r. In sas the quantile function implements the inverse cdf function, but for many distributions it has to numerically solve for the root of the equation fx u the inverse cdf technique is particularly useful when you want to generate data from a truncated distribution. Elastic impedance inversion for effective porefluid bulk. Alemie and sacchi 2011 proposed a highresolution threeterm avo inversion method based on the trivariate cauchy probability distribution by comparing it with the multivariate gaussian distribution. Another important result was developing a novel method of inversion of gravity data to recover the depth to basement, based on the 3d cauchytypeintegralrepresentation. Simulation lecture 2 inversion and transformation methods. There is also a variant of boxmuller that does not require the use of expensive trigonometric. We whall call a simple, directed curve t, smooth 2, pp. The inversion method can also be applied to discrete random variables. A bayesian impedance inversion method is proposed, under the assumption that the prior distribution is a gamma distribution. The case where t 0 and s 1 is called the standard cauchy distribution.
This method can be used for any distribution in theory. Therefore a 2d cauchy probability density function pdf in the killing angles domain is defined 3,4. We tested the method on synthetic data and analyzed the feasibility. On the generalization of seismic tomography algorithms. The markov chain monte carlo mcmc method based on metropolishastings mh sampling is a popular approach in solving seismic acoustic impedance ai inversion problem, as it can improve the inversion resolution by statistical prior information.
Due to the positivity constraints 3 the killing angles lie in the domain. Compared with commonly used gauss distribution and cauchy distribution, modified cauchy prior constraint fits the real data well, and can effectively protect weak, a study of multiscale seismic data joint inversion method. In this case, the cauchy distribution of data noise is assumed. The preliminary results show the potential of the method. This method is sometimes called the inversion method. Energies free fulltext acoustic impedance inversion. Combining equations and, it is easy to obtain the absolute ei for each angle range, which means that the wavelet influence in the seismic data has been stripped. Cauchy prior distributionbased avo elastic parameter. You could try other methods such as rejection method.
676 458 320 552 1531 295 1441 1394 1410 1229 578 1464 995 359 366 871 1332 284 1270 684 821 434 204 919 478 114 169 302 133 1023 1185 138 220 337