{
  "id": "1011.2527",
  "version": "v1",
  "published": "2010-11-10T23:18:02.000Z",
  "updated": "2010-11-10T23:18:02.000Z",
  "title": "An inverse problem for the wave equation with one measurement and the pseudorandom noise",
  "authors": [
    "Tapio Helin",
    "Matti Lassas",
    "Lauri Oksanen"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the wave equation $(\\p_t^2-\\Delta_g)u(t,x)=f(t,x)$, in $\\R^n$, $u|_{\\R_-\\times \\R^n}=0$, where the metric $g=(g_{jk}(x))_{j,k=1}^n$ is known outside an open and bounded set $M\\subset \\R^n$ with smooth boundary $\\p M$. We define a deterministic source $f(t,x)$ called the pseudorandom noise as a sum of point sources, $f(t,x)=\\sum_{j=1}^\\infty a_j\\delta_{x_j}(x)\\delta(t)$, where the points $x_j,\\ j\\in\\Z_+$, form a dense set on $\\p M$. We show that when the weights $a_j$ are chosen appropriately, $u|_{\\R\\times \\p M}$ determines the scattering relation on $\\p M$, that is, it determines for all geodesics which pass through $M$ the travel times together with the entering and exit points and directions. The wave $u(t,x)$ contains the singularities produced by all point sources, but when $a_j=\\lambda^{-\\lambda^{j}}$ for some $\\lambda>1$, we can trace back the point source that produced a given singularity in the data. This gives us the distance in $(\\R^n, g)$ between a source point $x_j$ and an arbitrary point $y \\in \\p M$. In particular, if $(\\bar M,g)$ is a simple Riemannian manifold and $g$ is conformally Euclidian in $\\bar M$, these distances are known to determine the metric $g$ in $M$. In the case when $(\\bar M,g)$ is non-simple we present a more detailed analysis of the wave fronts yielding the scattering relation on $\\p M$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-11-10T23:18:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R30"
    ],
    "keywords": [
      "wave equation",
      "pseudorandom noise",
      "inverse problem",
      "point source",
      "measurement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.2527H"
    }
  }
}