Modeling time and spatial variability of degradation through gamma processes for structural reliability assessment

Abstract The objective of this work is to propose a spatio-temporal random field for assessing the effect of spatial variability on the degradation process in structural reliability assessment. Our model extends the classical Gamma process which is usually developed for degradation temporal variability concerns to integrate spatial variability and heterogeneity issues. A log-normal distributed spatial random scale is then introduced in the Gamma process. Mathematical models for a structure degradation in space and time and estimation procedures are developed in this paper. Approximations and simulations are given to evaluate the failure time distribution and to characterize the residual lifetime after inspection. Simulation results are performed using pseudo-random data based on Monte-Carlo simulations to fit the model and the inference of their parameters. The two proposed estimation method – Method of Moments and Pseudo Maximum Likelihood – are numerically compared according to statistical measures. The accuracy of the methods are also discussed by numerical examples given the approximations of quantities of interests.


Introduction
In recent years, the scientific material and structural engineering community pays a lot of attention to the elaboration of mathematical degradation models for integrating spatial variability and uncertainty. It is obvious that the deterioration of structures exposed to environmental conditions is spatially and 5 temporally varying. The variation along the space is caused by the inherent uncertainty through the material at several positions on the structure and the physical parameters involved in the deterioration mechanism. This material non-homogeneity problem is well known for steel and concrete structures and has been, e.g., studied in [1,2,3,4,5] through concrete diffusion property, 10 concrete cover and chloride external concentration.
A common way to deal with such uncertainties is to use a probabilistic framework by modeling the input data with random fields [2,10,19,21,34].
For instance, a randomized salty environment (de-icing of the sea natural salts) can be integrated in the transport equation to compute the concentration of the 15 chloride in concrete, in order to estimate the time until failure under uncertainties. In [3,5] the authors consider a two-dimensional Gaussian random field with a Gaussian correlation to compute the likelihood of corrosion-induced cracking in reinforcing steel bar. They provide an estimation of the time to first crack and time to limit crack widths. In [1], authors propose a probabilistic model 20 for steel corrosion in reinforced concrete structures considering crack effect on the corrosion mechanism, in which an empirical model for the crack propagation stage is developed by the standard gamma process and combines corrosion crack width with steel-bar cross sectional loss. The main disadvantage of these approaches is the problem of larger dimension so called "curse of dimensionality" 25 in which the resolution of a large number of deterministic problems is involved.
2 Meta-models are commonly used to tackle this problem of curse of dimensionality for degradation prediction in structural engineering. The time-dependent deterioration processes are modeled by a stochastic model where only the variation in time is studied. In particular a standard Gamma process is an appropri-30 ate mathematical model for predicting deterioration encountered in civil engineering [6,35]; such as corrosion and crack of reinforced concrete. The authors in [17,18] developed a probabilistic framework in which interaction between shocks and gradual process are combined in view to describe the deterioration process. They proposed a semi-analytic computation model to estimate with 35 less cost the time to failure. The work in [9] describes the degradation by a Gamma process and includes other dependent-parameters as covariate in the shape function. In [26], a Gamma process with a random scale following a gamma law is considered to model heterogeneity in the degradation data and obtained analytic results for reliability assessment. Markov chain is another 40 widely used approach to model cumulative damage. It is seen as a discrete Markov process where the deterioration is assumed to be a single step function.
The estimation of the one-step transition matrix requires a large number of transitions to estimate all its elements [16,28]. The drawback of this approach is that time variability is difficult to capture. 45 In [15], the authors propose the construction of the state-dependent degradation model based on the Gamma process, where the cracking of a submerged concrete structure subjected to corrosion is described by the proposed bi-variate model with a suitable parameters. In the same time, the authors in [23] introduce a state-dependent Gamma process for the degradation, the dependency 50 is modelled in the scale function instead of the shape function as proposed in [15]. The authors in [24] introduce a bi-variate spatio-temporal field to model the action induced by the walking of a small group of persons. Based on the spectral and coherence functions of the forces, they proposed an evaluation of vertical and transversal accelerations at nodes of a finite element. 55 All models based on temporal variability assume a uniform degradation and do not integrate the spatial variability through degradation process. Neverthe-less, recent studies have shown that this spatial correlation has an important and direct impact on the level of structural reliability estimates [32,3,4]. Therefore, incorporating these uncertainties in the degradation processes through mathe-60 matical modelling improves their prediction and versatility in term of maintenance and decision. On the other hand, to construct an accurate model of the degradation, a large amount of data using destructive or nondestructive testing is required from a large amount of structures [31,32]. Therefore, one way of obtaining accurate and reliable information is to embed the spatial variability in 65 the models. This allows to increase the relevance of the Meta-Model approach where the uncertainties are reduced, improving the accuracy of the inference and extending the use of the non-destructive testing.
Therefore, the major contribution of this paper is a new spatio-temporal random model based on Gamma process for predicting a single degradation 70 measure which takes into account both temporal and spatial variability. Under the stationary assumption, the spatial monitoring data of the structure contributes in the parameters estimate to increase the accuracy of the meta-model approach. Our model requires only few parameters and is thus very suitable for inference when only few components are inspected: that is the case for on-site 75 inspection of civil engineering structures or marine structures where the cost of inspection is high.
The spatio-temporal degradation model is assumed to be an observable process in space and time with limited observations (Non Destructive Testing, distributed sensors). However, such kind of database that considers both hazard 80 time and space of the degradation are not available in the literature. In order to validate the proposed inference framework, we construct a synthetic discrete degradation model through Monte Carlo simulations. Therefore, numerical experiments will be conducted and compared for identifying preliminary properties and advantages of our model in terms of statistical inference and computation 85 of quantities of interest for reliability and maintenance.
The article is organized as follows: Section 2 introduces the classical Gamma process for temporal variability, the Gaussian random field with its simulation 4 method and the construction of degradation model is detailed. Section 3 develops quantities of interest which are useful in the reliability analysis, namely the 90 distribution of the failure time and the distribution of the remaining lifetime of the unit. Section 4 compares approaches for identifying properties of the model in terms of statistical inference. Finally, Section 5 presents a numerical example illustrating the proposed methodology for model validation. The standard Gamma process (GP) is an appropriate mathematical model for modeling the degradation evolution in structural engineering, such as corrosion and cracks of materials, which are the common causes of structural failure.

Degradation Processes and Random field variability
The deterioration is supposed to take place gradually over time in a sequence of 100 tiny increments. Consider α(·) to be a non-decreasing, right-continuous, realvalued function for t ≥ 0 and vanishing at t = 0.

Definition 2.1.
A stochastic process (X t ) t≥0 is said to be a GP with shape function α(t) and identical scale parameter β > 0 if the process satisfies the following properties:

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• X 0 = 0 with probability one, • The increment X t+s −X t has a Gamma distribution Ga(α(t+s)−α(t), β), • X t has independent positive increments, where the Gamma distribution Ga(α, β) is defined by the density function: and Γ is the classical Gamma function. The process X t is said to be stationary if α(·) is a linear function and X t is non-stationary if α(·) is a non-linear function.
The mean and variance of X t are E[X t ] = α(t) β , var[X t ] = α(t) β 2 , respectively. The process X t satisfies the scaling property, γX t = Ga(α, β/γ), for each γ > 0, (2) and its logarithm log(X t ) has the following two first moments: where the function ψ is digamma function which is defined as the logarithmic 110 derivative of Γ, and ψ 1 is the trigamma function defined as the derivative of ψ.

Gaussian Spatial random field modeling
Spatial variability of material properties is classically modeled by a secondorder stationary random field (RF) given by a non-linear transformation T (·) of a Gaussian random field (GRF) T (Y ) [8,31]. For example, in the transport 115 equation, the diffusivity coefficient is modeled by a log-normal RF [1,2,19], where its distribution is obtained as a limit of physical positive quantities.
We consider Y (z, ω) to be a spatial GRF in the set D × Ω, where D is a set in R d with d = 1, 2, 3 and Ω is an abstract set of events. The GRF Y is assumed to be homogenous and then totally defined by its mean µ ∈ R and its covariance 120 function cov(r) [7,36]. The cov(r) function models the correlation between two spatial random variables on any two points separated by the distance r.
In order to simulate Y on {z 0 , z 1 , . . . , z N } ⊂ D a set of equidistant points, we choose the circulant embedding matrix approach [14] (also named by DSM discrete spectral method in [11]). This method is a very versatile approach for 125 generating GRF, the discretized RF has the same spatial correlation on the grid points. In [30] the authors develop a continuous spectral method to simulate Y where the spectral density is discretized on a uniform grid, then a Discrete Fourier Transform (DFT) is used to generate an approximation of Y . The proposed simulated GRF is asymptotically Gaussian where its correlation structure 130 is an approximation of the target correlation and its accuracy is strongly related to the regularity of Y (see [22]). Another widely used approach to generate Y is the Karhunen-Loève expansion. However, it gives only an approximation of Y by a truncation of an infinite series, where its accuracy depends strongly on the smoothness of Y and its correlation length, [20, 21,31].
The DSM is based on the matrix factorization approach of the convenient positive definite circulant correlation matrix R using DFT. It has a Toeplitz structure and defined by its first row r = (r 0 , r 1 , . . . , r N −1 , r N , r N −1 , . . . , r 1 ) where each r l = cov(z l ), for l = 0 . . . , N . For instance, in one-dimensional uniform grid z 0 = 0, . . . , z N , a realization of the random field Y is simulated by the following components: where λ k are positive and given by the DFT of the circulant vector r. The set are independent random variables with standard normal distribution N (0, 1). The random vector (Y N (z 0 ), . . . , Y N (z N )) given in (5) where ν and l c are non-negative real numbers, l c is the correlation length, r is the Euclidean distance between two points, K ν denotes the modified Bessel function of the second kind. The regularity parameter ν tackles different models: a large value of ν implies that Y is ([ν] − 1)-times differentiable and small value of ν implies that Y is rough. When ν = 1 2 , cov coincides with the exponential covariance, Y are only continuous. When ν −→ ∞, it tends to the gaussian model, which is an analytic function and so for the samples paths of Y . Both models are the most used in structural engineering applications [1,3,19].

Spatial random field scale for Gamma process
We look for modeling a degradation process by a convenient spatio-temporal 145 RF to consider its aleatory evolution in time and space. A separable model is one simple spatio-temporal model obtained through the tensorial product between a merely stochastic process (X t ) t≥0 and a spatial RF Z(z), where z is the spatial variable. This class of separable RF is extensively used even in situations in which they are not always physically justifiable since separability 150 gives important computational benefits.
We are interested herein to model the deterioration in structural engineering under the presence of spatial variability in the model. The evolution in time models the intrinsic aleatory while the spatial RF models the variability and uncertainty through the structure. Thus, for simplicity of the model, we assume that randomness in time and in space are independent. In the classical approach for modeling the deterioration over time, the GP is ideally suited to model gradual deterioration which monotonically accumulates over time. This process can be extended spatially to obtain a spatio-temporal random field by considering its parameters to be spatial random fields, i.e the shape function α(·) or the scale parameter β. The scaling property of the GP in (2) motivates to model the scale parameter with a spatial RF and then to obtain a separable spatio-random RF, where X t is the GP with a shape function α(t) and unit scale parameter Ga(α(t), 1). The positive spatial RF β(·) is assumed to be independent of X t .
The scaling property satisfied by X t in (2) motivates us to take the spatiorandom field G t (z) as GP with spatial random scale β(·).

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In practice, it is difficult to find a convenient positive distribution for the spatial RF. However, some several reasons (detailed above) suggest to choose a Log-normal distribution for the random scale coefficient, where Y is a GRF with zero mean, unit variance and defined by the correlation function in (6). The constant e µ is seeing as the deterministic contribution of the scale parameter β(·).
The Log-normal distribution appears naturally as a limit law of physical processes. The Central Limit Theorem applied to the product of positive inde-160 pendent random variables (number of measures greater than 30) ensures that the log-normal distribution occurs. Further, a Log-normal RF is completely defined by its covariance. The maximum likelihood method is a commonly used inference procedure for such field. Another remarkable properties is that e σY and e −σY have the same finite-dimensional law, which implies that Ga(α, e σY +µ ) 165 and Ga(α, e µ )e σY have the same finite-dimensional law. However, since in this work, we use only the method of moments to perform the inference procedure; we can consider any choice of positive RF.

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The marginal distribution noted by f t (·) of the model is approximated in order to compute some quantities of interest which are used in structural reliability and maintenance. The GRF Y (z) considered in this work for the spatial variability is homogeneous, thus f t (·) does not depend on the position z but only on the time t. Let ξ(y) be the density of the standard Gaussian random variable N (0, 1) and setting η = e µ , the marginal density of G t |y is a gamma distribution with the shape parameter α(t) and a scale parameter ηe σy for each t > 0. Then, the pdf f t is given for all v > 0 by the following form: The integral in (11) has a transcendental form, thus we use Gauss-Hermite quadrature formula to approximate this marginal density f t . We consider m roots {y j } m j=1 of the Hermite polynomial and their associated weights {w j } m j=1 .

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Thus, an approximation of f t writes: The convergence of the sequence f m t (v) for a fixed v > 0 is relatively fast. However, the norm of any m-derivative of the integrand depends on the value of the parameters α(t), σ and η. A large value of these parameters requires a large order m of the approximation in (12), in particular for large time t. Then, the order m is built by the following stop criterion, where ǫ > 0 is a convenient threshold value.

Failure Time Distribution
The failure time T F for a component is defined as the time at which the degradation path G t first crosses a critical level g F for any spatial location, In what follows, the critical level g F is assumed to be deterministic. For some simple path models, the distribution F T (t) := P (T F < t) of T F can be expressed in a closed form. However, this is not always possible and it can 175 be numerically computed with Monte Carlo simulations by generating multiple paths of G t (z).
For the spatio-temporal random model considered here, sample paths are monotonic. Thus the failure time cumulative distribution F T (T ) satisfies: where f t is the marginal probability density function of the degradation process G t given in (11). Therefore, we can approximate the distribution F T by using an approximation f m t of the density function f t with a suitable order m: The integral (16) can be computed accurately by Legendre-Gauss quadrature 180 formula. The derivative of (15) and (16) with respect to the time variable t provides the probability density function of T F and its approximation, respectively.
The approximation of f m t can be inaccurate when the time t or the variance become large. And the computation of the cdf (16) will require a huge cost. In this case, an estimation of F T is provided using Monte-Carlo (MC) simulations 185 where a sufficiently large number of sample paths of G t (z) are conducted.
Let N t be the desired times, N z the desired locations and M the realizations where I A represents the characteristic function of the set A, i.e I A (z) = 1 if z ∈ A and zero otherwise. Note that since G t (z) is homogeneous with respect to the spatial variable, the estimateF T can be computed also using realizations of G t (z p ) fixed at any position z p , However, estimation (18) requires more MC simulations of G t (z) than (17) since the spatial average contributes in the convergence ofF T to F T (ergodic property).

Remaining Lifetime after inspection 190
In reliability analysis and survival studies, residual lifetime after inspection is a key indicator. In the maintenance decision analysis, the current measured degradation is used to predict the remaining lifetime (RL) of the structure [33].
If t is the current time of inspection, the residual lifetime is defined by the random variable: where g F is the critical level and g t is the measured degradation at a given time t. We have implicitly g t < g F . The Markov property of the model ensures that, from any current state, future states can be predicted. However, if we suppose that a component has survived to a given time t and we have no information or measure about the current degradation path G t , then a conditional reliability 195 function gives an evaluation of the remaining lifetime: When the current degradation measure path of G t is available. The probability that the unit survival after time t + τ given its current state G t = g t at time t is: where δ τ G t := G t+τ − G t is the associated degradation increment. Let 200 denote f δτ Gt|Gt the conditional marginal density of the process δ τ G t given the The cdf F RL := P (RL t ≤ τ ) of the residual lifetime becomes: Increments of the model are independent given a realization of the random field e σY . The marginal density of the bivariate variable (δG t , G t ) given y denoted π y (u, v) for any u, v > 0 can thus be defined as the product of the density function of δG t given y and that of G t given y: where Ψ(u, v) := and η = e µ .
The joint probability density of the bivariate variable (δ τ G t , G t ) is, Therefore, the conditional density of δ τ G t given the event where f t is the marginal density of G t given in (11). When g t > 0 and t > 0, this conditional probability density is given by the formula: is the beta function. Given the approximations of is a good approximation of the conditional probability density in (26). It follows that F RL (τ ) is given by, Note that when t = 0 and g t = 0, equation (26) is not definite and the 205 cumulative probability function F RL is the function F T given in (15). The derivative of (22) with respect to the variable h provides the probability density function of the Residual lifetime RL.

Parameters inference of the model
The estimation parameters for spatio-time fields is not so thoroughly devel-

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Our degradation model is a separable random-field given by a tensorial product of Gamma process and a Log-normal RF. The estimation process is in two phases: the space parameters estimation for GRF is firts performed, and then the temporal parameters of GP. For shake of simplicity, α(·) is assumed here to be the power law: for some unknown a > 0 and a known power b > 0.

Method of Moments (MOM)
where S h l is the set of the points separated with distance h l and N h l its cardinal.
The variogram is invariant by translation with any random variable. Then, the empirical variogramΥ Y is provided by computing the variogram of the random field log(G t ). At any fixed time t, we get: Note that the estimation ofΥ Y (h l ) can be improved by the empirical average when M realizations of the model G t (z) are available. The RF Y is stationary.
So the theoretical variogram is given by, where the correlation function cov(h) is given by (6). Therefore, the spatial parameters are estimated by minimizing the quadratic error (Least square method) between the theoretical and the experimental variograms. Thus, σ 2 , lc and ν are deduced by the following minimization problem: The classical least square method is sometimes not efficient, in particular when the length of the space and the number of positions are not large enough, because the set {Υ Y (h l ) 2 } Nz l=1 are not independent and with various variance. The generalized least squares method should be preferred: where Υ Y (h l ) is given in (30) and the matrix R is the correlation matrix of the set {Υ Y (h l )} Nz l=1 (see [29], for more details). A simplified approach is the weighted method where R is restricted at the diagonal matrix defined by entries Moreover, the proposed variogram of such fields depends only on the distance between positions. And finally, the MOM given by the least square method in (31) can be used for some non-stationary RFs Y , such that a trend-stationary field with a linear trend or with stationary increments.

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The temporal parameters are the parameters of the process X t , i.e. the shape function α(t) defined in (28) and the deterministic contribution η = e µ of the scale RF β . Since we assume that the power b is known, a and η can be estimated. Note that η contains the contribution of the spatial mean of the random field Y . From now, {G i tj (z l )} is assumed to be a sequence of independent and 250 identically distributed (i.i.d.) simulations of the degradation model described in the previous section. Each i-th degradation process is observed at time t j among N t times and on location z l among N z locations.

Case of time-stationary model 255
Let consider the time-stationary case where α(t) = at. Let ζ i j,l := log(δG i tj (z l )) be the logarithm of the increments of G t (z). Using the scaling property, each variable writes the following equation, where Y i (z l ) is the i-th sample of Y at position z l . Since the process X t and Y are independent, the first two moments of ζ i j,l write from equation (3): where τ := (t j −t j−1 ) is constant. The increments (X i tj −X i tj−1 ) Nt j=1 are independent and identically distributed. So, from M N t realizations of these increments combined with a spatial average, m 1 and m 2 can be estimated by; Hence, an estimation of parameters a and η is given by: where f is given by f σ 2 the estimate of σ 2 given by (31).

Case of non-stationary model in time
Let consider now α(t) = at b where b = 1 is known. The non-stationarity of GP can be transformed to a stationary GP by performing a monotonic trans-265 formation of the operational time [12]. However, the transformed inspection times are not equidistant to perform estimates given in (38). Let define the transformed times by ν j = t b j − t b j−1 , and introduce the increments of ith sample of G i t (z l ) denoted D i j,l := δG i tj (z l ). These increments are conditionally gamma distributed and conditionally independent. Therefore, according to [12], the whereσ 2 is the estimate of σ 2 given by (31).

Maximum Likelihood (MLM) and Pseudo Maximum Likelihood (PML)
The classical Maximum-Likelihood Estimate for the spatio-temporal field is performed when the RF is Gaussian or log-normal distributed. The model G t (·) is given by the product of two independent process, the Gamma process and the spatial log-normal field. Because it is not obvious to compute the likeli- The probability density function of (G T (z 1 ), . . . , G T (z N )) given a realization of The likelihood ̺(v) can be accurately computed through the Gauss-Laguerre quadrature or the Gauss-Hermite quadrature after transforming the integral on R with the transformation y = log(z).
Therefore, from M copies of the vector degradation (G i T (z 1 ), . . . , G i T (z Nz )), for i = 1 . . . , M , the likelihood of these observed data is given by Another approach which can be combined with MOM for non-stationary temporal variability is the pseudo maximum-likelihood method (PML). It consists of maximizing the likelihood of the increments (δ 1 G, . . . , δ t N G) on a given fixed spatial position z. These increments (δ tj G = G tj − G tj−1 ) Nt j=1 are conditionally independent and their likelihood is given by the product of the marginal density of each increment, which is similarly computed as in (11), where δ tj α := α(t j ) − α(t j−1 ) and v is the observed vector of the increment (δ 1 G, . . . , δ t N G) on a given position z. When b is unknown, we can use this 290 latter approach of PML given by the maximum of (46) to estimate temporal parameters. The value of σ comes from problem (31). Then, by using an appropriate approximation of each integral as given in (12), PML estimators of temporal parameters are given by maximizing log(ℓ(v)).

Numerical simulations 295
The deterioration model is assumed to be an observable process in space and

Simulation of discrete degradation model
In this subsection, the steps for generating a path of the degradation model

Method of moments (MOM) Step 1
The first step of the method of moment consists in estimating spatial parameters, variance σ 2 and correlation length l c . The quality of the estimation is measured by the mean absolute error. This error is given by the average of 350 the absolute differences between the exact parameter and 10 estimated values calculated across MOM. Table 1 Table 1: Parameters estimation of one dimensional spatial variability by MOM.
From Table 1

Method of moments Step 2
Once spatial parameters are estimated, the second step of the MOM consists in estimating the temporal parameters given by (38). An estimation of σ 2 is inserted in (38) for each case of M and N z . Table 2

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Similarly to the spatial estimation, let fix N t = 30 and N z = (5 : 5 : 100). The figures shows that the accuracy on η depends on N z and strongly on N t , unlike of the parameter a which is nearly independent of the spatial positions.
Pseudo-maximum-likelihood (PML) estimates Table 3 gives estimates of a and η using PML which consists of maximizing The method of moments looks to be more attractive for inference procedure since it gives the parameter estimates on two separable stages. First, the spatial     In order to illustrate the impact of dealing with the spatial variability in  Figure 6 compares the cdf of the remaining lifetime for both models G t and X t (without spatial variability). This cdf is computed using the estimate parameters given in Table 4, critical level g max = 25 and observed degradation g ti = 6 at time t i = 2. Let remind that the cdf F RL X t (τ ) of the remaining 435 lifetime function for X t is given by the formula F RL X t (τ ) = 1 − P(X t+τ − X t < g max − x t ), where x t is the state at time t. From the result of Table 4 and Figure 6, we can conclude that considering the spatial variability yields to an accurate model and more predictions for reliability analysis.

MOM.
Step 1: estimation of spatial parameters Table 5   In two-dimensional variability, parameters estimation (Table 5) is more accurate than the one-dimensional estimation (even with poor realizations of G t ).

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This accuracy is explained by the use of large spatial locations in the inference (N z1 × N z2 ). Table 6 provides the estimations of the spatial parameters with total number positions N = 100 where N z1 = 20, N z2 = 5 and N z1 = 50, N z2 = 2.
Results show that the accuracy is closed to the one-dimensional case, in particular estimation using the discretization N z1 = 50, N z2 = 2. Therefore, when the model is isotropic, estimations using data on a single direction lead to the same accuracy as one-dimensional model.

MOM.
Step 2: estimation of temporal parameters Table 7 summarizes estimations of a and η. Results show that accuracy 465 depends more on N t than N z1 × N z2 . An acceptable accuracy is reached with few realizations M when N t and N z1 , N z2 are significantly large. Estimation of η depends strongly on the number of positions since each position contains a stochastic contribution of the RF Y given by its mean µ.
In Table 8, as in one-dimensional case, we estimate temporal parameters with 470 total number positions N = 100 where N z1 = 20 and N z2 = 5 or (N z1 = 50 and N z2 = 2. Estimation provides nearly the same accuracy as in the onedimensional case (Table 2), since it depends strongly on N t and significantly on the total number of positions. Therefore, these results combined with Table 6 suggest that when the model is isotropic, the spatio-temporal variability can be 475 appropriately modeled by one-dimensional model.     field G t (z). Figure 8  The PML method consists of maximizing (46), the marginal density of the increments of G t (z) in an arbitrary positionz. Table 9 gives estimates of a and  ( Table 5). Results of these estimates show that the PML estimators depend strongly on M independent copies of N t increments of G t (z).

Discussion
The comparison between estimates given by MOM and by PMLmethod leads to highlight the interest of MOM. The PML providesonly the temporal parame-495 ters by using the increments of the model in any fixed position, yieldingto a set of acceptable estimations. MOM makes the inference procedureon two stages.
First, it determines the spatial parameters and then the temporal ones. Under the property ofspatial ergodicity , the use by MOMof the spatial data in both stages leads to improve the inference accuracy by exploiting inspection 500 results. When the temporal variability is characterized by a non-homogeneous GP, MOM requires some additional transformations in the time step to obtain a stationary process. This transformation is not clear when the non-linear shape function is defined by unknown parameters. Thus,MOM and PML can be combined for more practical inference procedure in such case. This combi-505 nation consists in, first, estimatingthe spatial parameters with MOM, second, computing compute the temporal parameters for the non-homogenous GP with PML.

Conclusion
We have developed a spatio-temporal degradation model that tackles both 510 the inert spatial uncertainty and heterogeneity across structural component. It is based on the classical gamma process and non-negative spatial random field.
The temporal paths of the process are monotonic with conditionally independent increments, the random field scale follows a log-normal distribution. The on the state dependent Gamma process. Such spatio-temporal field is not sep-arable when the spatial aleatory is considered in the shape function, it seems computationally more difficult and is still a challenge.