Page - 1167 - in Book of Full Papers - Symposium Hydro Engineering

3.2. INVERSE ANALYSIS The relationship between the experimental/field data and model can initially be expressed as: 𝐹 (𝒎 ) = 𝒖 𝒆 𝒙 𝒑 [5] Where F is the forward operator, mapping the material properties/damage parameters m (i.e. 𝜆 , 𝜇 and ρ) to the measured signals. As the solution of Eq. [5] may not exist as the measured data 𝒖 𝒆 𝒙 𝒑 are not in the range of F, we follow a least-square solution approach in Eq. [6]. Thus, the main concept behind the inverse analysis is to minimize the data residuals 𝛿 𝒖 or misfit between experimental or field data 𝒖 𝒆 𝒙 𝒑 and the simulated data from our forward model 𝒖 𝑚 𝑜 𝑑 . The objective function 𝐶 𝑓 (𝒎 ) which is to be minimized is the residual elastic energy in the residuals. That is the L2-norm of the error between the model response and the field or experimental data. 𝐶 𝑓 (𝒎 ) = 0.5||𝒖 𝑚 𝑚 𝑜 𝑑 − 𝒖 𝑒 𝑥 𝑝 || 𝐿 2 [6] Various strategies are available for inverse analysis, these include both direct and random-search strategies such as Newton methods (Gauss-Newton, Full Newton, etc), Conjugate Gradient, Nelder-Mead simplex methods, Particle Swarm Optimization, Simulated Annealing, Genetic Algorithm and Regularizing Iterative Methods by [25]. 4. NUMERICAL SIMULATION AND DISCUSSION The numerical simulation is carried out for a dam (true model) with material properties defined in terms of seismic velocities summarized in Table 1 and obtainable in Fig. 2. To ascertain the effect of acquisition geometry on quality of results, the sources-sensors setups in Fig. 1 are used to acquire all the waveform data necessary for carrying on the inversion. The dam true model consists of anomalies both in the dam structure and in the foundation. The anomalies represent weaknesses in various regions and of varying magnitudes. In the dam structure, the topmost anomaly represents a 20% reduction in the parameter values (Vp, Vs, ρ), the second 10% and the third (i.e. crack zone) has a 30% reduction. All the anomalies in the foundation have an 70% reduction in the material parameter values. 1167