A new analytical approach to reconstruct the acceleration time history at the bedrock base from the free surface signal records

Acceleration time histories of earthquake events are typically measured in seismic stations that are placed close to the soil top surface. These acceleration records are often used as input data for seismic analysis. It may be used for base excitation in seismic analysis of above ground structures with shallow foundations.. However it may not be used for seismic analysis of underground structures, or even for above ground buildings with deep foundations and several underground stories. The required base excitation data of the latter should have been measured below the top surface, at a level that may be determined according to the specific analyzed building geometry or at the bedrock below. If the acceleration time history at the bedrock would have been known, the seismic wave propagation through the soil medium, from the bedrock towards the top surface, could have been carried out and the base excitation of the buried structure could be determined. Since there is no data on the acceleration time history at the bedrock, and the only given data is the acceleration records at the top surface, the goal of this paper is to provide an exact reverse analysis procedure to determine the unknown acceleration time history at the bedrock that would exactly produce the measured acceleration time history at the top surface. Once this goal is achieved, seismic analysis of buried structures may be carried out with the determined acceleration record at the bedrock as input. This paper presents an analytical exact solution of the inverse problem for determination of the acceleration, velocity and displacement time histories at the bedrock base of a layered geological medium that are compatible with the given acceleration record at the soil top surface. This new proposed method is based on analytical solutions of the initial-boundary value problems of the linear wave equation in the case of a layered medium. The relationship between waves in one layer and waves in another adjacent layer is derived considering the continuity of stresses and displacements at the common interface between the layers. The efficiency and accuracy of the proposed method is demonstrated through several examples involving the nonstationary response of the free surface. The case of the San Fernando Earthquake is studied. Excellent agreement is achieved between the recorded free surface time history and the reconstructed signal. This excellent agreement is obtained due to the exact analytical method used in deriving the inverse problem solution. This exact analytical method allows one to obtain an acceleration (velocity/displacement) distribution along all the layers at any time.