Abstract Linear inversion is defined as the linear approximation of a direct-inverse solution. This definition leads to data requirements and specific direct-inverse algorithms, which differ with all current linear and nonlinear approaches, and is immediately relevant for target identification and inversion in an elastic earth. Common practice typically starts with a direct forward or modeling expression and seeks to solve a forward equation in an inverse sense. Attempting to solve a direct forward problem in an inverse sense is not the same as solving an inverse problem directly. Distinctions include differences in algorithms, in the need for a priori information, and in data requirements. The simplest and most accessible examples are the direct-inversion tasks, derived from the inverse scattering series (ISS), for the removal of free-surface and internal multiples. The ISS multiple-removal algorithms require no subsurface information, and they are independent of earth model type. A direct forward method solved in an inverse sense, for modeling and subtracting multiples, would require accurate knowledge of every detail of the subsurface the multiple has experienced. In addition, it requires a different modeling and subtraction algorithm for each different earth-model type. The ISS methods for direct removal of multiples are not a forward problem solved in an inverse sense. Similarly, the direct elastic inversion provided by the ISS is not a modeling formula for PP data solved in an inverse sense. Direct elastic inversion calls for PP, PS, SS, … data, for direct linear and nonlinear estimates of changes in mechanical properties. In practice, a judicious combination of direct and indirect methods are called upon for effective field data application.

Abstract We develop a new way to remove free-surface multiples from teleseismic P- transmission and constructed reflection responses. We consider two types of teleseismic waves with the presence of the free surface: One is the recorded waves under the real transmission geometry; the other is the constructed waves under a virtual reflection geometry. The theory presented is limited to 1D plane wave acoustic media, but this approximation is reasonable for the teleseismic P-wave problem resulting from the steep emergence angle of the wavefield. Using one-way wavefield reciprocity, we show how the teleseismic reflection responses can be reconstructed from the teleseismic transmission responses. We use the inverse scattering series to remove free-surface multiples from the original transmission data and from the reconstructed reflection response. We derive an alternative algorithm for reconstructing the reflection response from the transmission data that is obtained by taking the difference between the teleseismic transmission waves before and after free-surface multiple removal. Numerical tests with 1D acoustic layered earth models demonstrate the validity of the theory we develop. Noise test shows that the algorithm can work with S/N ratio as low as 5 compared to actual data with S/N ratio from 30 to 50. Testing with elastic synthetic data indicates that the acoustic algorithm is still effective for small incidence angles of typical teleseismic wavefields.