- Ground-motion model is:
where y is in cm∕s2, θ

_{1}= -3.4712, θ_{2}= 2.2639, θ_{3}= -0.1546, θ_{4}= 0.0021, θ_{5}= -1.8011, θ_{6}= 0.0490, θ_{7}= 0.2295, σ_{r}= 0.2203 (intra-event) and σ_{e}= 0.2028 (inter-event). - All records from rock sites.
- Strong correlation between magnitude and distance in dataset.
- Use a derivative-free approach based on a hybrid genetic algorithm to derive the model. Use a simplex search algorithm to reduce the search domain to improve convergence speed. Then use a genetic algorithm to obtain the coefficients and uncertainties using one-stage maximum-likelihood estimation. Believe that approach is able to overcome shortcomings of previous methods in providing reliable and stable solutions although it is slower.
- In hybrid genetic algorithm an initial population of possible solutions is constructed in a random way and represented as vectors called strings or chromosomes of length determined by number of regression coefficients and variance components. Population size is usually more than twice string length. Each value of population array is encoded as binary string with known number of bits assigned according to level of accuracy or range of each variable. Use three operations (reproduction/selection, crossover and mutation) to conduct directed search. In reproduction phase each string assigned a fitness value derived from its raw performance measure given by objective function. Probabilities of choosing a string is related to its fitness value. Crossover or mating combines pairs of strings to create improved strings in next population. In mutation one or more bits of every string are altered randomly. The process is then repeated until a termination criterion is met. Demonstrate approach using test function and find small maximum bias in results. Conclude that method is reliable.
- Use Taiwanese dataset of Chen and Tsai (2002) to demonstrate method.
- Compare results with those obtained using methods of Brillinger and Preisler (1985), Joyner and Boore (1993) and Chen and Tsai (2002). Find differences in coefficients (although predictions are very similar except at edges of dataspace) and standard deviations (slightly lower for proposed method).
- Compare predicted motions for M
_{L}5.5 with observations for M_{L}5–6. Find good fit. - Plot total residuals against magnitude and distance and find no trends.
- Note that residuals show that model is satisfactory up to 100km but for larger distances assumption of geometric spreading of body waves in not appropriate due to presence of waves reflected off Moho.
- Note that near-source saturation should be included. Apply proposed method using a complex functional form with different equations for three distance ranges and compare results to those using simple functional form. Find differences at short and large distances.