The elastic property is usually represented by several independent parameters called elastic moduli—like Young’s, shear, bulk, etc. In practice, however, it is more commonly represented by more easily measurable physical properties such as seismic P- and S-wave velocities (Vp and Vs) and density (rho), in which Vp and Vs are theoretically derivable in combination from the elastic moduli and density. It therefore usually refers to these Vp, Vs, and rho when terms like “earth’s property” or “a (layered) earth model” are used in a seismic method. Among these properties, it is usually the S-wave velocity (Vs) that is estimated from the inversion of surface wave data. This is because the type of data used for the inversion has been the fundamental-mode (M0) dispersion curve, whose shape is determined mostly by the Vs structure of the earth (Fig. 3). A more accurate expression should be that it has been historically assumed that the fundamental mode dominates in the recorded field data. In consequence, surface wave data processing has been following this procedure of constructing the M0 curve and then finding the earth’s Vs structure whose theoretical M0 curve most closely matches the constructed (experimental or measured) M0 curve. S-Velocity (Vs) is one of the critical parameters in geotechnical engineering because it can be a direct indicator of shear and Young’s moduli, which in turn are often associated with materials’ stiffness and load-bearing capacity of the ground.
It is the 1D (depth) variation that is found for the Vs structure because the measured curve is most sensitive to the vertical variation of Vs and lateral variation is largely averaged off during data processing. This type of 1D Vs model is often referred as a layered earth model, which may include Vs, Vp, and rho all together or simply Vs alone with others fixed. A layered earth model inevitably carries no lateral variation as the case with an arbitrary earth model (Fig. 4). This means the surface wave method gives the 1D Vs structure (also called profile) most representative of the subsurface materials below the receiver spread by approximating them as a layered earth model despite the fact that, in reality, a certain degree of lateral variation always exists. Because of the nature of multichannel processing, it is usually the center location of the receiver spread to which this 1D Vs profile is assigned as the most representative surface location if the coordinate information is necessary. With accumulation of multiple number of this 1D Vs profile assigned with a unique surface coordinate, a 2D Vs profile is constructed by using an appropriate spatial interpolation scheme.
This type of M0-curve based inversion used to find the 1D Vs profile has been the most typical approach historically adopted, and this is currently the case with the MASW method as well. This, however, does not necessarily mean it is the only viable approach with the MASW method. In fact, there are several other approaches currently being researched or already under routine use that use other types of surface wave data rather than the M0 curve either separately or in combination. They include
The special edition in 2005 on the surface wave method published by the Journal of Environmental and Engineering Geophysics (JEEG) presents a summary table of typical inversion approaches adopted in both the two-receiver (SASW) and the multichannel (MASW) methods. Regardless of any specific type used, the surface wave inversion is non-unique and non-linear problem to varying extent under different circumstances. In addition, it is not guaranteed if a final solution indeed represents the global minimum instead of one of the local minima.
Historically, the inversion of surface waves has meant an estimation of the earth’s properties from the measured surface wave data (Fig. 1). It is the earth’s elastic property that is estimated among many properties—electric and magnetic to name a few. Various types of surface wave data can be used for the estimation that may include raw field record, dispersion image, dispersion curve(s), and a preliminary 2D shear(S)-wave velocity (Vs) map (see different inversion approaches listed below). Regardless of the type of data used, the surface wave inversion cannot be solved directly, requiring an optimization technique to find the most probable solution in a pool of infinite candidates. This technique can be a deterministic approach, random, or combination of both. On the other hand, the forward modeling (Fig. 2) of surface wave phenomena is usually a unique problem.