Seismic Zone Map for India Based on Cluster Analysis of Uniform Hazard Response Spectra

Abstract

A novel methodology for obtaining a seismic zone map of India is demonstrated in this study, wherein a concrete theoretical framework is provided for deriving the zones and the respective zonal response spectra. The method involves time series clustering of uniform hazard response spectra (UHRS) that were obtained for the entire country on a 0.1° × 0.1° grid by performing probabilistic analysis corresponding to a 2475-year return period. The Euclidean distance between the UHRS values at all periods (27 data points between 0.01 s and 5 s) was taken as the similarity measure in an evolutionary particle swarm optimization algorithm. The analysis was conducted with a swarm population of 100 over 3000 iterations, and the mean UHRS of the resulting clusters was assumed as the cluster centre. Various quality/validity indices including the compactness measure, similarity measure, combined measure and Dunn Index were used to verify the results of the clustering. Based on these clusters, the entire country can be divided into seven zones, with a unique zonal spectrum for each zone.

Appendices

Appendix A

Particle Swarm Optimization

Particle swarm optimization (PSO), which is an evolutionary algorithm, makes use of the nature-bound swarm intelligence technique in solving an optimization problem (Kennedy & Eberhart, 1995). The logic in the algorithm is based on the natural behaviour or movement of birds/fishes in a swarm (flock/school). In the algorithm, a set of particles with memory are considered to fill the solution space. The particles have an adaptable velocity, which determines their movement in the search space towards the solution.

The PSO algorithm starts with initialization of the position vectors (x_i) of all the particles in the swarm. These position vectors are assumed to be the solutions of the optimization problem. Each of these particles also has velocity vectors (v_i), which determines its respective position in the subsequent time step, as given by Eq. 2. At every iteration, the objective function or cost function is evaluated based on the particle’s position x_i and the best position for the ith particle in the swarm is estimated. Since all the particles have memory, the current best position is compared against the personal best position (Pbest), which might have been observed in any of the previous iterations. Simultaneously, a global best position (Gbest) is obtained through comparison of all the particles’ positions in the swarm. Accordingly, based on the Pbest and Gbest values, the velocity of the particle i is updated (Eq. 1) and the procedure is repeated for all the n iterations to obtain new positions for all the particles (Eq. 2). It should be noted that in each of these iterations, the position of the particle is updated towards the global best position, mimicking the swarm phenomena.

$$v_{i}^{n + 1} = wv_{i}^{n} + c_{1} r_{1}^{n} \left( {{\text{Pbest}}_{i}^{n} - x_{i}^{n} } \right) + c_{2} r_{2}^{n} \left( {{\text{Gbest}}_{i}^{n} - x_{i}^{n} } \right)$$

(7)

$$x_{i}^{n + 1} = x_{i}^{n} + v_{i}^{n + 1}$$

(8)

Here, w is the inertia weight; c₁ is the cognitive acceleration parameter; c₂ is the social acceleration parameter; r₁ and r₂ are uniformly distributed random numbers bounded by [0 1] (Kennedy et al., 2001). The change in velocity and the bounds of the increased velocity at every time step is governed by the parameter w and the upper bound of the velocity vector (V_max). Consequently, the maximum step size of a particle in a single iteration is governed by parameters c₁ and c₂, whereas r₁ and r₂ maintain the population diversity. The bounds for the velocity vector are obtained using Eqs. 3 and 4.

$$V_{{{\text{max}}}} = 0.1 \left( {\max \left( x \right) - \min \left( x \right)} \right)$$

(9)

$$V_{{{\text{min}}}} = - V_{{{\text{max}}}}$$

(10)