Probabilistic inversion

Another alternative to standard iterative linearized localization methods is the probabilistic inversion approach of Tarantola and Valette [1982]. In recent years, software packages have become available that combine efficient, nonlinear, global search algorithms (e.g. NonLinLoc, Lomax et al. 2000). This method gained in importance for the precise localization of... 

Monte Carlo simulation can involve several methods for using a pseudo-random number generator to simulate random values from the probability distribution of each model input. The conceptually simplest method is the inverse cumulative distribution function (CDF) method, in which each pseudo-random number represents a percentile of the CDF of the model input. The corresponding numerical value of the model input, or fractile, is then sampled and entered into the model for one iteration of the model. For a given model iteration, one random number is sampled in a similar way for all probabilistic inputs to the model. For example, if there are 10 inputs with probability distributions, there will be one random sample drawn from each of the 10 and entered into the model, to produce one estimate of the model output of interest. This process is repeated perhaps hundreds or thousands of times to arrive at many estimates of the model output. These estimates are used to describe an empirical CDF of the model output. From the empirical CDF, any statistic of interest can be inferred, such as a particular fractile, the mean, the variance and so on. However, in practice, the inverse CDF method is just one of several methods used by Monte Carlo simulation software in order to generate samples from model inputs. Others include the composition and the function of random variable methods (e.g. Ang Tang, 1984). However, the details of the random number generation process are typically contained within the chosen Monte Carlo simulation software and thus are not usually chosen by the user. 

Given prior information in terms of a lower and upper bound, a prior bias, and constraints in terms of measured data, the MRE provides exact expressions for the posterior pdf and expected value of the inverse problem. The plume source is also characterized by a pdf The problem solved in their study is the same as Skaggs and Kabala s problem. For the noise-free data, MRE was able to reconstruct the plume evolution history indistinguishable from the true history. As for data with noise, the MRE method managed to recover the salient features of the source history. Another advantage using the MRE approach is that once the plume source history is reconstructed, future behavior of the plume can be easily predicted due to the probabilistic framework of MRE. Woodbury et al. [71] extended the MRE approach to reconstruct a 3D plume source within a ID constant velocity field and constant dispersivity system. 

The function g is designed for exploiting positivity and atomicity of the electron density (the same information exploited in reciprocal space by the probabilistic formulae estimating structure invariants). Therefore the inversion of p is expected to produce better phase values than those used for calculating the map p. 

If we decide to treat the estimation problem using the nonlinear model, the problem becomes more challenging. As we will see, the parameter estimation becomes a nonlinear optimization that must be solved numerically instead of a linear matrix inversion that can be solved analytically as in Equation 9.8. Moreover, the confidence intervals become more difficult to compute, and they lose their strict probabilistic interpretation as a-level confidence regions. As we will see, however, the approximate confidence intervals remain very useful in nonlinear problems. The numerical challenges for nonlinear models... 

Models can be categorized in various ways. Predictive models forecast the future behavior of a system, whereas conceptual models are used to understand relationships between system parts and processes. Deterministic models are constructed from mathematical functions that imambiguously relate cause and effect so that a particular set of input parameters produces a clearly related set of predicted results. Probabilistic models use statistical data to estimate the chance that an event or condition will occur. Forward models predict the future behavior of a system, whereas inverse (or reverse) models are used to extract fundamental data or mathematical relationships from past observations. 

Assessment procedures capable of giving an estimate of condition, probability of failure and serviceable lifetime while the plant is operating are based on calculation (Fig. 2.1). An inverse design (e.g. API RP-530) deterministic approach can be adopted, but this invariably produces conservative results. Thus, in line with the current trend for risk-based information, the more accurate approach is to deploy probabilistic techniques and a more rigorous... 

Carvalho, E., J. Cruz, P. Barahona (2013). Probabilistic constraints for nonlinear inverse problems. Constraints 7(9(3), 344-376. 

All linearized location algorithms originate from Geiger s method (1910). The differences are typically in the details, such as how the inverse of the G matrix is obtained, what kind of weighting scheme is applied, and how the formal uncertainties are calculated. Expressed in a probabilistic framework and assuming independent, normally distributed data, linearized location algorithms maximize the likelihood function... 

Carmona, R., Hernandez, J., Marcos, J., Minetto, C., Arro, R., 2010. Applicatirm of RMN using a probabilistic method into the Echoes inversion. In Petroleum Wmld, Latin American Energy, Oil Gas, Venezuela, Bolivia, Trinidad, Peru, August 29. http //www.petroleumworld.com/ sf09110101.htm. 

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