In many probabilistic analysis problems, the homogeneous/nonhomogeneous nonGaussian field is represented as a mapped Gaussian field based on the Nataf translation system. We propose a new samplebased iterative procedure to estimate the underlying Gaussian correlation for homogeneous/nonhomogeneous nonGaussian vector or field. The numerical procedure takes advantage that the range of feasible correlation coefficients for nonGaussian random variables is bounded if the translation system is adopted. The estimated underlying Gaussian correlation is then employed for unconditional as well as conditional simulation of the nonGaussian vector or field according to the theory of the translation process. We then present the steps for augmenting the simulated nonGaussian field through the KarhunenLoeve expansion for a refined discretized grid of the field. In addition, the steps to extend the procedure described in the previous section to the multidimensional field are highlighted. The application of the proposed algorithms is presented through numerical examples.
Probabilistic analysis and reliability estimation often require the unconditional and conditional simulation of the correlated nonGaussian vector of random variables with the prescribed marginal probability distribution and correlation coefficients. The use of the simulated samples for uncertain propagation or reliability analysis for civil engineering problems is well illustrated in several textbooks^{[13]}. The simulation could be carried out by modeling the random variables using the Nataf translation system (or theory of translation process or normal to anything)^{[36]}, which represents the joint probability distribution of the vector of random variables by the Gaussian copula^{[7]} and probability transformation. The simulation can also be based on normal polynomials^{[811]}. A key component of the modeling is to evaluate underlying Gaussian correlation coefficients based on the prescribed correlation coefficients for the nonGaussian random variables^{[1215]}. This underlying Gaussian correlation is governed by a double integral equation. Since the analytical solution for the underlying Gaussian correlation is only available under very special circumstances, the underlying characteristics are frequently evaluated using an iterative procedure and numerical integration methods. Moreover, given a correlation coefficient for the nonGaussian random variables, there is no guarantee that one can find the underlying Gaussian correlation based on the Nataf translation system^{[12, 13, 16]}. An additional problem for the modeling that needs to be dealt with is that the nearest correlation matrix^{[17, 18]} may need to be employed since there is no guarantee that the derived underlying Gaussian correlation matrix from the translation process is positive semidefinite.
Besides simulating the nonGaussian vector of random variables, the simulation of nonstationary/nonhomogeneous and nonGaussian random fields is also of importance for structural reliability estimation^{[19]}. Two popular methods to simulate the random field are the spectral representation method (SRM) and the method based on the KarhunenLoeve (KL) expansion. SRM^{[2022]} is based on the spectral characteristics of the random field in the time domain, space domain, or both. The spectral characteristics of the field are defined by the power spectral density (PSD) function obtained by using Fourier transform. SRM was extended to simulate nonstationary/nonhomogeneous and nonGaussian fields^{[5, 2326]}. These studies use iterative procedures to find the underlying Gaussian PSD function; the PSD and autocorrelation (AC) function forms a Fourier pair (i.e., Wiener  Khintchine theorem). The sampled Gaussian field using the underlying Gaussian PSD function can then be mapped to the nonGaussian domain based on the theory of the translation process. SRM was extended for conditional simulation in several studies^{[2730]} for stationary/nonstationary processes and fields based on the conditional joined Gaussian distribution function.
The KL expansion is a special case of the orthogonal series expansion^{[31]}. The orthogonal functions in KL expansion are obtained from the eigenfunctions of a Fredholm integral equation of the second kind with the autocovariance function as the kernel^{[19]}. A random field is represented by the KL expansion with random coefficients. There are many studies focused on simulating the random field based on the KL expansion with engineering applications^{[19, 3236]}. Since the sum of the KL expansion with random coefficients leads to the Gaussian process, an iterative procedure was proposed^{[32, 33]} to simulate the nonGaussian process. It includes sampling many sets of random fields, and updating the simulated fields by modifying and shuffling random expansion coefficients based on ranking in each iteration. The adequacy of the sampled field is judged based on the closeness between the marginal distribution of the sampled field and its prescribed distribution. The use of numerical integration to evaluate the underlying Gaussian correlation function for a nonGaussian field was considered in several studies^{[6, 35]}. They sampled the Gaussian field based on the underlying Gaussian correlation function and mapped the sampled field to the nonGaussian field. The disadvantages of using different methods, including the numerical integration method, are discussed in the literature^{[6]}.
In the present study, we propose a new samplebased iterative procedure to estimate the underlying Gaussian correlation or its nearest correlation matrix for a nonGaussian vector or process. We use the estimated underlying Gaussian correlation to simulate the nonGaussian vector or field according to the theory of the translation process. We then proposed the steps for augmenting the simulated nonGaussian field through the KL expansion for a refined discretized grid of the field. We also outline the necessary steps for the conditional simulation of the nonGaussian field. In the following, we first describe the steps used to evaluate the underlying Gaussian correlation for a nonGaussian process and simulate the nonGaussian process. We then describe the steps to augment the simulated field using the KL expansion and the formulations for the conditional simulation. The proposed simulationbased iterative procedure to evaluate the underlying Gaussian correlation and its use to carry out unconditional and conditional simulation is illustrated through several numerical examples.
Consider a onedimensional random field
Since, in many practical applications, the data collection and data analysis for a random field is carried out at discrete points with a given sampling interval. Let
If the marginal CDF is Gaussian and
where
However, if the prescribed marginal distribution is not Gaussian,
where
a)
b)
c) The maximum and minimum values of feasible correlation coefficients
Once
For those elements in
We note that
A verification that the underlying Gaussian correlation matrix
Based on the above description, the proposed simulation procedure in the present study for the nonGaussian vector or field is summarized as follows:
1. Sample the
2. Evaluate
3. Calculate the underlying Gaussian correlation coefficients
4. Form the underlying Gaussian correlation matrix
5. Apply Cholesky decomposition^{[40]} to
A flowchart showing the above steps are presented in
Flowchart of the proposed simulation procedure: the left panel represents the unconditional simulation, and the right panel shows the steps for the conditional simulation of the field.
In the previous section, it is considered that the marginal CDF and the correlation function are prescribed for the nonhomogeneous nonGaussian field. Moreover,
The use of
For completeness and in order to provide the background for the conditional simulation based on the KL expansion, the use of the KL expansion is summarized below. In general, the random field (Gaussian or nonGaussian)
where
By considering that the series shown in Equation (7) is truncated to the first few terms, an approximation to
By taking advantage of the availability of
where
By applying
Consider that the field to be simulated
Let
where
This matrix can be constructed by rearranging
By generating
We also note that a method^{[34]} was developed to carry out the conditional simulation of the Gaussian field based on the KL expansion. It can be shown that their method for the underlying Gaussian random field at
where the eigenvalue
and
where
Once
In this section, we highlight steps to extend the procedure described in the previous section to the
Since the extension to the multidimensional field is straightforward, no further consideration for the
Several numerical examples are presented by considering the homogeneous/nonhomogeneous processes with weakly and strongly nonGaussian processes to illustrate the applicability of the proposed procedures depicted in
The examples in this subsection were considered in the literature^{[32, 35]}. The assumed marginal distributions and the AC function for the homogeneous nonGaussian cases (i.e., Cases 1 to 8) are listed in
Definition of the cases (see ^{[35]})



1  


2  
3  
4  
5  
6  
7  
8 
First, for each of the considered homogeneous cases, the application of the proposed procedure shown in
Comparison of the prescribed AC function, estimated AC function based on samples, and underlying Gaussian AC function for Cases 1 to 8 by considering that the marginal distribution is the beta distribution shown in
Comparison of the prescribed AC function, estimated AC function based on samples, and their differences for Cases 5 to 8. The considered marginal distribution is the beta distribution shown in
The results presented in
The plots for Cases 5 to 8 shown in
By repeating the analysis that is carried out for the results presented in
Comparison of the prescribed AC function, estimated AC function based on samples, and underlying Gaussian AC function for Cases 1 to 8 by considering that the marginal distribution is the lognormal distribution shown in
Comparison of the prescribed AC function, estimated AC function based on samples, and their differences for Cases 5 to 8. The considered marginal distribution is the lognormal distribution shown in
The plots presented in
The plots presented in
In this subsection, we consider again all cases shown in
First, the eigendecomposition is applied to
Value of
We use the obtained
Typical samples obtained by applying the KL expansions and probability transformation mapping for each of the cases listed in
Comparison of the calculated correlation coefficient from the sampled random field to
For illustrating the conditional simulation, we consider Cases 1 and 5. By assuming that the marginal distribution is the beta or lognormal distribution shown in
Five conditionally sampled fields for Cases 1 and 5. The large dots represent the conditioning points and the lines represent the sampled fields.
A samplebased numerical procedure to estimate the underlying Gaussian correlation for a homogeneous/nonhomogeneous nonGaussian field is proposed in the present study. The estimation is based on the Nataf translation framework and takes advantage that the range of feasible correlation coefficients for nonGaussian random variables by using the translation is bounded. Numerical examples show that the proposed estimation procedure is effective and can be used directly to identify whether the underlying Gaussian correlation that leads to exactly matching the prescribed correlation can be found. It is shown that by taking into account the bounds, the underlying Gaussian correlation can be easily found by using the simple bisection method or optimization algorithm. From the numerical example, it was noted that
We also present the steps to augment the simulated nonGaussian field for a refined discretized grid of the random field by applying the KL expansion and using samples obtained for a much more coarse grid system. The procedure is illustrated through numerical examples.
It was shown that the conditional simulation could easily be carried out within the established simulation framework, and the extension of the procedure to the multidimensional random field case is straightforward.
We gratefully acknowledge the financial support received from the Natural Sciences and Engineering Research Council of Canada (RGPIN201604814, for HPH), the China Scholarship Council (No. 201807980004 for MYX), and the University of Western Ontario.
Investigation, Methodology, Formal analysis, Writing  original draft, Writing  review & editing: Xiao MY
Conceptualization, Funding acquisition, Investigation, Methodology, Project administration, Resources, Supervision, Writing  original draft, Writing  review & editing: Hong H
Some or all data, models, or code generated or used during the study are available from the author by request.
The authors acknowledge the financial support received from the Natural Sciences and Engineering Research Council of Canada (RGPIN201604814, for HPH) and the China Scholarship Council (No.201807980004, for MYX).
All authors declared that there are no conflicts of interest.
Not applicable.
Not applicable.
© The Author(s) 2022.
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