Time-domain simulation of wave phenomena on a finite computational domain often requires a fictitious outer boundary. An important practical issue is the specification of appropriate boundary conditions on this boundary, often conditions of complete transparency. Attention to this issue has been paid elsewhere, and here we consider a different, although related, issue: far-field signal recovery. Namely, from smooth data recorded on the outer boundary we wish to recover the far-field signal which would reach arbitrarily large distances. These signals encode information about interior scatterers and often correspond to actual measurements. This article expresses far-field signal recovery in terms of time-domain convolutions, each between a solution multipole moment recorded at the boundary and a sum-of-exponentials kernel. Each exponential corresponds to a pole term in the Laplace transform of the kernel, a finite sum of simple poles. Greengard, Hagstrom, and Jiang have derived the large-$\ell$ (spherical-harmonic index) asymptotic expansion for the pole residues, and their analysis shows that, when expressed in terms of the exact sum-of-exponentials, large-$\ell$ signal recovery is plagued by cancellation errors. Nevertheless, through an alternative integral representation of the kernel and its subsequent approximation by a {\em smaller} number of exponential terms (kernel compression), we are able to alleviate these errors and achieve accurate signal recovery. We empirically examine scaling relations between the parameters which determine a compressed kernel, and perform numerical tests of signal "teleportation" from one radial value $r_1$ to another $r_2$, including the case $r_2=\infty$. We conclude with a brief discussion on application to other hyperbolic equations posed on non-flat geometries where waves undergo backscatter.

翻译：在有限计算域上对波现象进行时域模拟时，通常需要引入虚拟外边界。一个重要的实践问题是在该边界上指定适当的边界条件，通常是完全透明条件。此前已有研究关注该问题，而本文则考虑另一个虽相关但不同的问题：远场信号恢复。具体而言，我们希望从记录在外边界上的光滑数据中恢复能够到达任意远距离的远场信号。这些信号编码了内部散射体的信息，且通常对应于实际测量。本文将远场信号恢复表示为时域卷积形式，每个卷积涉及在边界上记录的解的多极矩与指数和核函数。每个指数对应核函数拉普拉斯变换中的极点项，即简单极点的有限和。Greengard、Hagstrom 和 Jiang 推导了极点残差的大ℓ（球谐指数）渐近展开，其分析表明，当用精确的指数和表示时，大ℓ信号恢复会受到抵消误差的困扰。然而，通过核函数的替代积分表示及其随后的较少指数项逼近（核压缩），我们能够缓解这些误差并实现精确的信号恢复。我们通过实验检验了决定压缩核的参数之间的标度关系，并对从径向值r₁到r₂（包括r₂=∞的情况）的信号“传送”进行了数值测试。最后，我们简要讨论了该方法在非平坦几何中涉及波后向散射的其他双曲型方程中的应用。