The Efficient Method for Calculating the Physical Optics Scattered Fields from the Concave Surfaces

In this work, the numerical steepest descent path (NSDP) method is proposed to compute the highly oscillatory physical optics (PO) scattered fields from the concave surfaces, including both the monostatic and the bistatic cases. Quadratic variations are adopted to approximate the integrands of the PO type integral into the canonical form. Then, on involving the NSDP method, we deform the integration paths of the integrals into several NSDPs on the complex plain, through which the highly oscillatory integrands are converted to exponentially decay integrands. The RCS results of the PO scattered field are calculated and are compared with the high frequency asymptotic (HFA) method and the brute force (BF) method. The results demonstrate that the proposed NSDP method for calculating PO scattered fields from concave surfaces is frequency-independent and error-controllable. Numerical examples are provided to verify the efficiencies of the NSDP method.

Steepest descent integration: A novel method for computing wavefields radiated from borehole sources

Borehole sources, including chemical explosives, air gun, water gun, and piezoelectric transducers in the borehole, generate seismic waves inside and outside the borehole. Modeling the wavefield is of key importance in acoustic logging, crosshole tomography, mining geophysics, and deep sounding seismic for interpretation of amplitude information of real data and prediction of energy-radiation patterns. Classic methods for modeling the wavefield inside a borehole, such as real-axis integration, are challenged by highly oscillatory integrals encountered when modeling the wavefield outside the borehole. We have developed a novel method, called steepest descent integration (SDI), which evaluates the oscillatory wavenumber integration by numerically integrating along the steepest descent path. The oscillation along the new integration path is significantly reduced. The contributions of poles and branch cuts are added if they are located between the steepest descent path and the real axis. The SDI is applicable to arbitrary frequency and source-receiver distance. Comparison with real-axis integration shows that the method can compute highly oscillatory integrals with better efficiency and accuracy. In addition, the SDI is more numerically robust because it generates no spurious arrivals, which are evident in the real-axis integration. Analysis of numerical examples at different source-receiver distance shows that SDI is more efficient when computing far-field seismograms. This SDI can also be used to compute highly oscillatory integral in other wave-propagation problems.