## Stochastic Estimation of Spatial Variability

The procedure for the estimation of the stochastic spatial variation of seismic motions from recorded data considers that the motions are realizations of space-time random ﬁelds, i.e., multi-dimensional and multivariate random functions of location, expressed as the position vector with respect to a selected origin, r = {x, y, z}T , superscript T indicating transpose, and time, t . At each location, the acceleration time history in each direction (north-south, east-west or vertical), a(r , t), is a stochastic process of time t , i.e., at each speciﬁc time t , a(r , t) is the realization of a random variable. This chapter begins with some basic deﬁnitions for random variables in Section 2.1. The concept of stochastic processes is then presented in Section 2.2. This section describes the time and frequency domain characterization of the process through its mean, autocovariance and power spectral density functions, and introduces the assumptions of stationarity and ergodicity. The derivations are presented for ground motion components recorded in a single direction, i.e., it is indirectly assumed that the motions in the three orthogonal directions can be analyzed separately; this assumption will be revisited in Section 3.3.2. The phase properties of recorded accelerograms are also highlighted in this section. Section 2.3 describes the joint characteristics of the time histories at two discrete locations on the ground surface. The time histories are now considered to be realizations of a bivariate stochastic process, also, often, termed bivariate vector process. This section presents the joint descriptors of the bivariate process, namely the cross covariance function in the time domain and the cross spectral density in the frequency domain. Section 2.4 introduces the concept of the complex-valued coherency in the frequency domain, and deﬁnes the lagged coherency as its absolute value, and the phase spectrum as its

phase. The interpretation of the lagged coherency and the phase spectrum of seismic data is presented in this section. The correspondence between the cross correlation function and the coherency and the deﬁnition of the plane-wave coherency are also illustrated in Section 2.4. The derivations for the bivariate stochastic vector processes set the basis for the description of the characteristics of multivariate processes, which are highlighted in Section 2.5. The multivariate stochastic process describes the joint characteristics of the recorded data at all considered discrete locations (stations) on the ground surface. This section also presents the additional assumptions of homogeneity and isotropy, which will be utilized in Chapter 3 for the parameterization of the spatially variable ground motions. The section concludes with a brief description of the concept of the random ﬁeld viewed as the extension of the multivariate vector process to the continuous case, for which the stochastic characteristics of the motions are provided at all locations on the ground surface. The bias and variance characteristics of the stochastic estimators, as well as the necessity for their smoothing either in the time or the frequency domain are also presented in Sections 2.2-2.4. To illustrate the concepts and derivations of the stochastic estimators described in this chapter, the data recorded at the SMART-1 array (Fig. 1.2) during Event 5 are utilized in example applications.