Conditional simulation of non-Gaussian random fields

C o n d i t i o n a l s i m u l a t i o n of n o n Gaussian r a n d o m fields I. Elishakoff and Y. J. R e n Center,[or Applied Stochastics Research and Department oJ Mechanical Engineering, Florida Atlantic UniversiO', Boca Ramn, FL 33431-0991, USA

M. S h i n o z u k a Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ 08544, USA

The method of conditional simulation of homogeneous or inhomogeneous non-Gaussian random fields is developed. It represents a combination of two previously developed methods, namely of the iterative method for unconditional simulation of non-Gaussian random fields put forward by Shinozuka and Yamazaki, and of the Hoshiya method for conditional simulation of Gaussian random fields. To gain physical insight, several numerical examples are evaluated. High accuracy of the proposed method is demonstrated. Keywords: random fields, simulation, earthquake monitoring systems

The methods of numerical simulation of stationary or nonstationary random processes and random fields have been well established in the last two decades and have been widely applied to solve engineering problems which cannot be treated effectively by purely analytical tools. By the combination of simulation techniques and deterministic numerical methods, one can analyse complex engineering structures with uncertain geometrical and/or material parameters, and being subjected to deterministic and/or random external loads. Study of the conditional simulation technique of random fields stems from the important issue of earthquake monitoring engineering. Urban functions in highly developed modern cities are significantly influenced by the lifeline systems like electric power, gas, water, the communication network and others. Earthquake damage to these installations may cause significant impairmant or even loss of functionality. The restoration of damaged installations must be performed in the quickest possible manner. Appropriate channelling of the information on the earthquake damage becomes of paramount importance. Therefore, several seismometers are utilized in the region of facilities performing responsible functions, to obtain records for seismic ground motions. Conditional simulation can be a useful tool to provide important information on the ground motions in inaccessible locations so that emergency operation can be performed to control critical facilities. For the detailed description of issues associated with earthquake monitoring systems one may consult the definitive article by Kameda and MorikawaL So far, some interesting work has been done by several 0141-0296/94/070558-06 © 1994 Butterworth-Heinemann Ltd

558 Engng Struct. 1994, Volume 16, Number 7

investigators to develop the conditional simulation technique 2 6 The problem can be formulated as follows: let Z(x,t) be the ground motion during the earthquake, which is assumed to be a space-time random field, and its several realizations z(xi, t) have been recorded at some locations xi (i = 1, 2, ..., N) by using measurement instruments; one needs to simulate the random field Z(x,t) under these realization constraints Z(x~, t) = z(xi, t) in some other locations of engineering significance. In other words, by giving these realizations, the conditioned random field Z(x,tlaxi, t); i = 1, 2, ..., N) is to be estimated at some other locations of interest. Vanmarcke and Fenton 3 applied Fourier series representation of random fields and the kriging technique to simulate a local field of earthquake ground motion. Kameda and Morikawa ~'24 have established an analytical framework by expanding conditioned random processes into Fourier series and derived expressions of the joint probability density function of the Fourier coefficients. They also obtained expressions of conditional mathematical expectations, variances and first-passage probabilities of conditioned random processes. Hosihya 5 and Hoshiya and Maruyama 6 have split a conditional random field into a sum: its kriging estimate and the error. They have simulated the kriging estimate and the error separately, and then combined them to obtain samples of the conditional random field. Studies by both Kameda and Morikawa 2 and Hoshiya and Maruyama 6 arrived at the same conclusion: that conditional mathematical expectations depend both on the number of realization locations and values of the realizations, whereas the values of conditional variances only depend on the number of realization locations.

Conditional simulation of random fields: I. Elishakoff et al. The conditional simulation technique developed so far has been focused on Gaussian random fields. The purpose of this study is to present the conditional simulation technique for non-Gaussian random fields for which realizations at some locations are given. In this study, the iterative procedure proposed by Yamazaki and Shinozuka 7 to generate samples of unconditional non-Gaussian fields is adopted to construct a mapping between Gaussian fields and nonGaussian fields so that the existing conditional simulation technique for Gaussian random fields can be efficiently applied. The difference between the simulated and targeted correlation functions is utilized as a criterion for convergence of the iterative procedure. Moreover, the error between the simulated correlation function and targeted correlation function is decomposed into two parts, the simulation error and mapping error. The former is caused by simulation of the Gaussian random field, whereas the latter is caused by the mapping procedure between nonGaussian and Gaussian random fields. The simulation error can be reduced by the increasing number of samples. Mapping error can be eliminated by the iteration procedure. This study focuses on univariate time-independent nonGaussian random fields. Several examples of fields are given to illustrate the effectiveness of the proposed procedure for conditional simulation of non-Gaussian random fields.

Conditional simulation of Gaussian random field Suppose that G(x) is a univariate time-independent Gaussian random field with zero-mean value E[G(x)] = 0 and specified autocovariance function E[ G(x~ )G(x2) ] = Coo(x l, x2). Assume that a set of its realizations g(x~) at recorded locations xi (i = 1, 2, ..., N) are given. One is interested in simulating G(x) at unrecorded locations Xr (r = 1, 2, ..., M). The original random field G(x) can be homogeneous or nonhomogeneous, however, the conditioned field G(xlg(x~); i -- 1, 2, ..., N), which is obtained through the condition that realizations are recorded in some locations, is nonhomogeneous and possesses a nonzero mean. Following Hoshiya's approach -~, we represent the random field G(x) at unrecorded locations as

G(xD = G~k)(x~) + [G(xr) -- G
(l)

where G(k)(Xr) denotes the kriging estimate 8 of the random variable G ( x r ) at an unrecorded location Xr according to N random variables G(xi) (i = 1, 2, ..., N) at recorded locations, e ( X r ) : G(xr) - G ( k ) ( X r ) is the error between the actual field G(x) and its kriging estimate G~k)(x) at location x . The kriging estimate G(k)(x~) is interpolated as follows

G ~k' (Xr)

=

AT(Xr)G(x)

(2)

where A(xr) = [ / ~ l ( X r ) , •2(Xr), .--, AN(Ir)] T is the vector of kriging weights, G(x) = [G(xl), G(x2), ..., G(XN)] r is the vector of recorded random variables. The kriging weights Ar(xr) can be determined by the condition that the kriging technique gives the best linear unbiased estimator. Remembering that the random field G(x) is assumed to have a vanishing mean function, we only require that ~y2 = E{[ G(Xr) _ G~k)(x~) ]2} = m i n i m u m

(3)

Substitution of equation (2) into equation (3) yields (r 2 = Crr - 2 A r ( x r ) D ~ + A T ( x ~ ) C A ( x r )

(4)

where C 0 = E[G(x~)G(xj)] = Cc,o(x~, xj) ( i , j = 1, 2, ..., N, r), D~ = [C~r, C2, ..., Cur] 7- is the vector having as its elements covariances of G(x) between recorded locations x~ (i = 1, 2, ..., N) and the unrecorded location Xr, and C = [C 0] ( i , j = 1, 2, ..., N) is the covariance matrix of G(x) between recorded locations. Minimizing the right-hand-side of equation (5), we obtain CA(xr) = D r

(5)

The kriging weights A(x~) are evaluated from equation (5) provided the covariance matrix C is nonsingular. By the definitions of e(Xr) and equation (3) and using equation (5), it is easily proven that irrespective of A(xr) the following equalities hold

E(e(Xr) ] = 0

(6)

E(e(x~)G(x)] = 0

(7)

E(e(xr)e(x,)] = C ~ - A r ( x ~ ) D ~

(8)

If the unrecorded location x r coincides with one of the recorded locations xk, the solution of equation (5) is obtained simply by taking Ai(xD = 1 for i -- k and Ai(xD = 0 for i ~ k. Then E[e(xr) e(xi)] = 0 (i = 1, 2, ..., N) from equation (8). This implies that e(xr) and e(xi) are uncorrelated. Thus, to simulate G(xr) under the condition that G(xi) = g(xi) ( i = 1, 2, ..., N), one can first calculate the realization g~k~(x~) = AT(x~)g(x) of the kriging estimate G(k)(Xr) and simulate the error e(Xr) separately, and then form their sum. The simulation of the error S(Xr) is achieved based on the knowledge of its zero mean and the known correlation structure of equation (8). The error e(Xr) is Gaussian since a Gaussian field is considered. Generally, the right-hand side of equation (8) does not vanish. As a result, the errors at two unrecorded locations are mutually correlated. Instead of simulating the mutually correlated random field e(Xr) (r -- 1, 2, ..., M), for example, by means of the decomposition of the covariance matrix 9, one can use an alternative recursive procedure proposed by Hosiya 5, namely, e(Xr) (r = 1) is first simulated at a single unrecorded location based on the knowledge of E[e(x~)] = 0 and cre[e(Xr)] = C,~ mT(xr)D,.. Then, this location is considered as if it were a new 'recorded' one, to simulate the next unrecorded location based on N+I 'recorded' locations. By a similar step-by-step procedure, G(x,) ( r = 1, 2, ..., M) can be simulated at all desired locations.

Conditional simulation of non-Gaussian random field Suppose N(x) is a univariate and time-independent nonGaussian random field with zero-mean and targeted autocorrelation function "N ' NOCT(X~, ) X2) = E[N(xON(x2)]. Let B be the random variable representing N(x) at a certain location x, with zero mean and variance a2 = RNN(X, X) and a given non-Gaussian distribution function F~(b), which is also the (one-dimensional) distribution function of the nonGaussian random field N(x) at location x. Assume that

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Conditional simulation of random fields: I. Elishakoff et al. G(x) is a Gaussian random field having the same first and second moments as the non-Gaussian field N(x), namely zero-mean and autocorrelation function R,;~;(x~, x2) = ~x,
(i = 1, 2, ..., N)

(9)

By means of the simulation technique of Gaussian random fields reviewed in the previous section, a set of simulated values g(x~) of G(x) at unknown locations x~ (r = 1, 2, ..., M) can be obtained and then are mapped back into nonGaussian ones n(x~) of N(x~) in terms of following inverse mapping

n(x,) : F~' {F,[g(xr)l}

(r:

1, 2, ..., M)

(10)

n(Xr) (r = 1, 2, ..., M) consist of a set of approximate simulated sample values of the non-Gaussian random field N(x) under the condition of N(x;) = n(x;) (i = 1, 2, ..., N). Because of the nonlinearity of the mapping, M estimates n(xr) and N realizations n(x~) may not match, in the general case, the targeted autocorrelation function Re, one should update the assumed autocovariance function of the Gaussian random field as follows R~G(Xl, X2) = RGG(Xl, X2) + RGG(Xl, X2) [R~(Xl, X2) - RNN(X=, X2)]

560

(ll)

Engng Struct. 1994, Volume 16, Number 7

The procedure proceeds to equation (9) to start the next iteration until the condition ]eM(x~, X2)l<--e is satisfied. If N(x) is a non-Gaussian random field with nonzero mathematical expectation m(x), probability distribution function FNcx)(n) and autocorrelation function RNN(X~, X2), one can define a new field N~(x) = N(x) - re(x) so that N~(x) is a random field with zero mean, probability distribution function FN,(_,)(n~) = FN(~)(n~ + m) and autocorrelation function RN,N, (X~, X2) = RNN(Xl, X2) -- m(xl)m(x2). The conditional simulation of N(x) can be achieved through conditional simulation of N~(x) and then putting N(x) = Nt(x) + m(x). Numerical examples To verify the effectiveness and accuracy of the proposed procedure for conditional simulation of non-Gaussian random fields, we apply this procedure to simulate several types of non-Gaussian random fields such as the exponential random field, the truncated Gaussian random field, the 'contaminated' random field and the beta-distributed random field. The conditional simulation procedure can be also utilized to simulate the unconditional autocorrelation function and the probability distribution function through conditionally generating a number of samples of the random field; each sample is obtained by generating a realization of the random field at a certain location randomly, and then generating realizations at other locations based on the realization at that location and given correlation function by using the present conditional simulation procedure, and the results are compared with the targeted ones. The conditional mathematical expectation and variances are also calculated when realizations of the random fields are specified at several locations.

Example 1 Assume that N(x) is a one-dimensional random field and its N realizations n(x;) are given at locations x~, x2, ..., and x,,. N(x) possesses a non-Gaussian distribution, namely truncated Gaussian fields. The probability density function is assumed to be

Fu(~)(b) =

0,

for

-~c < b <

eft(b) - erf(-bt ) eft( b~ ) - e f t ( - b , )'

for

-b~ <- x < b=

1,

for

b~ < - x < w

-b 1

(12) where bj is a constant and eft(b) is the error function Ib

eft(b) =

/2

1 e - ~ dt \,2~v Jo

(13)

The random field has zero-mean and following variance 9.

Var[N(x)] = 1 -

7f'rr [erf(b,) - erf(-bl)]

exp( ) (14)

The autocorrelation function of N(x) is assumed to be exponential

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Three different values of the constant b~ are considered to be 1, 2 and 4. Figure 1 gives a line divided into 12 segments with length 0.5 to show spatial locations where samples are either recorded or to be simulated. Hence, the coefficient of correlation of the field between any two neighbouring locations is e ~ 5 -- 0.61, which is greater than 0.6. To verify the accuracy of the proposed conditional simulation procedure, we first simulate the unconditional probability function and the autocorrelation function of the random field by the present conditional simulation technique. Figure 2 illustrates the probability distribution functions of both the truncated Gaussian field and the associated Gaussian field with its mean and variance equal to those of the truncated Gaussian, for two different values of bj. Two probability distribution functions coincide practically for b~ = 4 and differ considerably from each other when bj = 1. Figure 2 also illustrates simulated truncated Gaussian probability distribution functions. The simulated results exhibit quite good agreement with the targeted functions for both two cases of b~. Figures 3 and 4 give the simulated conditional means and standard deviations, respectively, when realizations at x = 0, x = 3 and x = 6 are specified.

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Example 2 Consider now a 'contaminated' distribution, obtained by convex mixing of two distributions, F = aFi + 13/'2, a+13 = 1. Other examples of 'contaminated' distributions and their effects on human errors in reliability of structures can be found elsewhere 7. In the present investigation, the probability distribution F is composed of the convex combination of two distributions. In the numerical case under consideration, two-thirds of the standardized Gaussian and

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It is seen that in the vicinity of recorded locations, the conditional mean approaches the value of the realization itself, whereas the conditional variance tends to zero. Far apart from recorded locations, the conditional mean and variance converge to, respectively, unconditional mean and variance, as expected. The conditional mean depends on values of given realizations since all simulated samples should pass through the realization values. The conditional variances are independent of the realization values. Figure 5 portrays the simulated and targeted unconditional correlation functions. To assess the simulated results, the results for the corresponding Gaussian field for bl = 4 are also presented in Figures 3-5, which are obtained by the Hoshiya simulation technique 5 for Gaussian fields. Two solutions, for Gaussian and truncated Gaussian fields, in the case bl = 4 are indistinguishable as they should be, since in that case probability distribution functions of Gaussian and truncated Gaussian fields practically coincide.

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Figure 4 Conditional standard deviations f o r truncated Gaussian fields and Gaussian field corresponding to bl = 4 ' ( T = truncated)

Figure 2 Probability distribution functions for truncated Gaussian and corresponding Gaussian fields ('1-= truncated)

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Engng Struct. 1994, Volume 16, Number 7 561

Conditional simulation of random fields: I. Elishakoff et al. lated conditional means and conditional standard deviations when realization of the field at x = 3 is fixed at 1.25. As one can deduce from the previous example, the simulated conditional mean approaches the realization itself, while the conditional variance approaches zero, when the unrecorded location x,. tends to the recorded location -L. The conditional means and conditional standard deviations approach corresponding unconditional values 'far away' from the recorded location, where the edge effect diminishes (note some analogy with the edge effect in the theory of shells).

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Figure 6 Target and simulated probability distribution functions of contaminated field 1

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As the last example, we consider a two-dimensional nonGaussian random field. The field now has exponential distribution with nonzero mean. The probability distribution function of N(x, y) has the form of

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FN(,,,.~(b) =

0.8 0.6 0.4

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(19)

b --> 0

The mean and variance of N(x, y) are equal to oL and od, respectively. The correlation structure is still assumed exponential, i.e.

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- e -/'/''

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(20)

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Figure 7 Correlation function of contaminated field (3000 samples)

one-third of the unit-mean exponential are mixed. Figure 6 shows its probability distribution function graphically. The autocovariance function of the contaminated field is assumed to be exponential. Figure 6 also illustrates the simulated probability distribution function with 3000 samples. Figure 7 portrays the simulated and targeted unconditional autocovariance functions. The simulated unconditional probability function and autocovariance function agree well with the corresponding target probability function and with the autocovariance function, by using the proposed conditional simulation procedure of non-Gaussian random fields. Figure 8 presents the simu0 "M

where ~ is the distance between two points, ~ = ((x,-x2)2) ~/= + (y~-y2)2) I/2. Let N,(x, y) = N(x, y) - ~, so Ndx, y) is a zero-mean non-Gaussian random field with correlation function RN,N,(~) ---- R N i ~ ) -- ~X2, and probability distribution function FN,(b) = FN(b+eO. The conditional simulation of the random field N(x, y) is achieved through conditional simulation of Nj(x, y), and then letting N(x, y) = Ndx, y) + e~. Figure 9 gives the simulated and targeted autocorrelation functions of N(x, y) for o~ = 1. Fixing the allowed error at e = 0.02, the simulated correlation function converges to the target correlation function after five iterations. This indicates that the simulated results agree well with targeted values. Figure 10 portrays the simulated conditional means and conditional standard deviations of N(x, y) on two axes, x = 0 and y = 0, when the realization at x = y = 0 is specified at 1.01. The realization at x = y = 0 mainly affects its neighbouring domain. The conditional mean and variance tend to the unconditional mean and variance, respectively, at locations far away from the realization location.

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Engng Struct. 1994, Volume 16, Number 7

0.2 0 -1

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Conditional simulation of random fields: I. Elishakoff et al. Only univariate time-independent non-Gaussian random fields are considered in this paper. The generalization to multivariate and time-dependent fields is underway and will appear elsewhere.

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Acknowledgment

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This work has been supported by the National Center for Earthquake Engineering Research (NCEER), through grant 1-3211A.

References 0.5

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Figure 10 Simulated conditional means and standard deviations (s.d.) of 2D exponential field along x and y axes (3000 samples)

Conclusions This study combines the unconditional simulation technique of non-Gaussian random fields and the conditional simulation technique of Gaussian random fields to conditionally simulate a non-Gaussian random field. The error between autocorrelation functions of the simulated field and the targeted field is decomposed into a simulation error and a mapping error. The magnitude of the latter is used as the convergence criterion to continue or stop the iterative procedure. Several examples are given to illustrate the effectiveness of the proposed simulation procedure. Numerical experience showed that only two to four iterations are required to obtain convergent numerical results.

1 Kameda, H. and Morikawa, H. 'Simulation of conditional random fields - a basis for regional seismic monitoring for urban earthquake hazards mitigation', US-Italy-Japan Workshop~Seminar on Intelligent Systems, University of Perugia, Italy, 27-29 June 1991 2 Kameda, H. and Morikawa, H. 'Interpolating stochastic process for simulation of conditional random fields', Probabilistic Engng Mech. 1992, 7(4), 242-254 3 Vanmarcke, E. H. and Fenton, G. A. 'Conditioned simulation of local fields of earthquake ground motion', Stract. Safety, 1991, 10, 247-274 4 Kameda, H. and Morikawa, H. 'Conditioned stochastic processes for conditional random fields', J. Engng Mech 1993, 120(4), 855-875 5 Hoshiya, M. 'Conditional simulation of a stochastic field', in Structural safety' and reliabiliO, G. I. Schu~ller, M. Shinozuka and J. T. P. Yao (Eds), Balkema, Rotterdam, The Netherlands, 1994, pp 349-353 6 Hoshiya, M. and Maruyama, O. 'Stochastic interpolation of earthquake wave propagation', in Structural safety and reliability G. I. Schu~ller, M. Shinozuka and J. T. P. Yao (Eds), Balkema, Rotterdam, The Netherlands, 1994, pp 2119-2124 7 Yamazaki, F. and Shinozuka, M. 'Digital generation of non-Gaussian stochastic fields', J. Engng Mech., 1988, 14(7), 1183-1197 8 Journel, A. J. and Huijbregts, Ch. J. 'Mining geostatistics', Academic Press, 1988 9 Elishakoff, I. 'Probabilistic methods" in the theo~' of structures', John Wiley, New York, 1983 l0 Elishakoff, I. and Hasofer, A. M. 'Effect of human error on reliability of structures', in: Proc. AIAA Structures, Structural Dynamics and Materials Conf, Dallas, Texas, 1992, 3233-3236

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