On the generalization of seismic tomography algorithms. The case where t 0 and s 1 is called the standard cauchy distribution. Elastic impedance inversion for effective porefluid bulk. A study of multiscale seismic data joint inversion method. So here is the inverse transform method for generating a rv x having. 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. Energies free fulltext acoustic impedance inversion.
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. Create pd by fitting a probability distribution to sample data from the. One interesting fact about a cauchy random variable is that none of its moments are finite. The standard cauchy distribution has pdf and cdf fx 1. 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 following is the plot of the cauchy cumulative distribution function. Therefore a 2d cauchy probability density function pdf in the killing angles domain is defined 3,4. 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. The standard cauchy distribution has pdf and cdf f x 1. Random variate generation raj jain washington university in saint louis saint louis, mo 63. 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. Compared with commonly used gauss distribution and cauchy distribution, modified cauchy prior constraint fits the real data well, and can effectively protect weak reflectivity. Inverse survival function, the formula for the cauchy inverse survival function is.
Cauchy prior distributionbased avo elastic parameter. The inverse cdf method for simulating from a distribution. 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. For large n the individual generation of the xis is unfeasible, and the inversionofabetavariate is potentially inaccurate. Proof let the random variable x have the standard cauchy distribution. Figures 5a, 5b, and 5c are results of the inversion using trivariate cauchy as prior distribution. 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. 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.
Simulation lecture 2 inversion and transformation methods. The cauchy distribution, named after augustin cauchy, is a continuous probability distribution. Theorem theinverseofastandardcauchyrandomvariable x. The gap theorem in this section we shall obtain gap theorems for the cauchy and cauchy stieltjes integral. Example exponential distribution for an exponential distribution with rate, we have. The rst general method that we present is called the inverse transform method. The preliminary results show the potential of the method. Note that it is easy to sample from the cauchy distribution.
Nonlinear multiparameter inversion using a hybrid global. The standard cauchy distribution has probability density function fx 1. 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. Whe cauchy distribution has heavier tails than the gaussian distribution. A computational quasireversibility method for cauchy. When x has a continuous distribution then it has a probability density function fx.
This method works when you have distributions such as exponential, cauchy, geometric, pareto, logistic, extreme value weibull, etc. We tested the method on synthetic data and analyzed the feasibility. Theorem theinverseofastandardcauchyrandomvariablex isalsostandardcauchy. Cauchy inver sion is also frequently used in the geophysical inversion as a robust optimization method 3. We demonstate our inversion method based on the cauchytype integral for several synthetic models. Chapter 2 inversion method, transformation methods and rejection sampling. Inverse problems in science and engineering 8, 119. This method can be used for any distribution in theory.
But it is particularly useful for random variates that their inverse function can be easily solved. 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. The method is implemented in the framework of the 1dvar retrieval with simulated atovs data. Using the transformation technique, y gx 1x is a 11 transformation from. There is also a variant of boxmuller that does not require the use of expensive trigonometric. The inversion for the fluid indicator is formulated in bayesian framework with the cauchy distribution as a prior constraint. For example, to use the normal distribution, include coder. Rejection sampling 2 inversion method the inversion method requires that we nd the inverse of the cumulative distribution function cdf of f. Also, assume f is continuous and strictly increasing as a function. Introduction to simulation using r free textbook course. Another important result was developing a novel method of inversion of gravity data to recover the depth to basement, based on the 3d cauchytypeintegralrepresentation. Summary the reliable estimation of properties of deep reservoirs from seismic data is a difficult and longstanding problem. Non linear multiparameter inversion 993 among local sampling of the model space.
Numerical laplace transform inversion methods with. Modified prp conjugate gradient method is a stable method, and it has faster convergent rate than conventional method. Elastic impedance variation with angle inversion for. Numerical inversion methods timeline the development of accurate numerical inversion laplace transform methods is a long standing problem. Thus, if u has a cauchy distribution, eu, and all higher order moments, do not exist.
The cauchy distribution is used to model random variables with very heavy tails. The proof of this is a straightforward calculation. Normal random numbers can also be generated using the general inverse transform method e. The results show that the new method is capable of providing highresolution depth estimation for. We will look at general principles like inversion and acceptancerejection sampling. To sample the bernoulli distribution by inversion, take x 1 1 u6p. Suppose g has a geometric distribution, so the mass function is. Distribution inverse transformation method at least in principle, there is a way to convert a uniform distribution to any other distribution. Avaf inversion method for detecting hydrocarbons journal. Using inversion, we can sample the cauchy dis tribution by. Finding a fast and effective method for achieving the temperature distribution measurement is of great importance.
Cauchy distribution is able to obtain strong reflections with large amplitudes and has superior noise handling performance. For these theorems we need an assumption on the curve r. Inverse cumulative distribution function matlab icdf. Here we will use capital f to denote the cdf and lowercase f to denote the probability density function pdf. On one side, the method is of variational due to the use of the adjoint technique. At the same time, the relations among the model parameters are assumed. You could try other methods such as rejection method. This method is sometimes called the inversion method. The input argument name must be a compiletime constant. 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. We whall call a simple, directed curve t, smooth 2, pp.
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. The probability density function pdf technique, univariate. 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. The assumptions above can be weakened to some extent. Gap and inversion formulas for the cauchy integral. Paper open access bayesian acoustic impedance inversion. The cauchy distribution has the probability density function pdf. Theorem the inverse of a standard cauchy random variable x is. Understand the theoretical foundations and convergence properties of these methods. Alemie and sacchi obtained sparse highresolution inversion results based on the prior trivariate cauchy probability distribution. Although powerful, this inverse cdf method can be computationally expensive unless you have a formula for the inverse cdf. However, the sampling function of the traditional mh sampling is a fixed parameter distribution. For example you can not find a closed form for a normal distribution.
Proof lettherandomvariablex havethestandardcauchydistribution. In this case, the cauchy distribution of data noise is assumed. Modified cauchyconstrained seismic blind deconvolution. 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. Chapter 4 generating nonuniform random variables arizona math. Pdf elastic impedance inversion for effective porefluid. Ournumericalstudiesdetermined that the new method is much faster than.
With this equation as forward solver, a prestack seismic bayesian inversion method is proposed to estimate the fluid factor of heterogeneous media. Stat 451 lecture notes 0512 simulating random variables. In this paper, we discuss and compare the corrected inversion method, the lognntail method and the. Suppose g has a geometric distribution, so the mass function is pg g 1. We compare the proposed tomography reconstruction method to. Seismic wave scattering inversion for fluid factor of. Generating random numbers from any distribution using transformations cdf inverse, boxmuller method. Prestack ava inversion of exact zoeppritz equations based. 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. Inversion of gravity and gravity gradiometry data for.