{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T15:34:00Z","timestamp":1787240040670,"version":"build-2736575974"},"reference-count":94,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Rev."],"published-print":{"date-parts":[[1991,12]]},"abstract":"<jats:p>The statistical properties of acoustic signals reflected by a randomly layered medium are analyzed when a pulsed spherical wave issuing from a point source is incident upon it. The asymptotic analysis of stochastic equations and geometrical acoustics is used to arrive at a set of transport equations that characterize multiply scattered signals observed at the surface of the layered medium. The results of extensive numerical simulations are presented, illustrating the scope of the theory. A number of inverse problems for randomly layered media are also formulated where we recover large-scale properties of the sound speed profile from the statistics of reflected signals.<\/jats:p>","DOI":"10.1137\/1033136","type":"journal-article","created":{"date-parts":[[2005,3,7]],"date-time":"2005-03-07T02:21:47Z","timestamp":1110162107000},"page":"519-625","source":"Crossref","is-referenced-by-count":134,"title":["Frequency Content of Randomly Scattered Signals"],"prefix":"10.1137","volume":"33","author":[{"given":"M.","family":"Asch","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"W.","family":"Kohler","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"G.","family":"Papanicolaou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M.","family":"Postel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"B.","family":"White","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,18]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.109.1492"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-9394-2"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01209475"},{"key":"R4","volume-title":"Introduction to the theory of disordered systems","author":"Lifshits Ilya M.","year":"1988"},{"key":"R5","volume-title":"Scattering and Localization of Classical Waves in Random Media","author":"Sheng P.","year":"1989"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.62.575"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/BF01135526"},{"key":"R8","volume-title":"Stochastic Equations and Waves in Randomly Inhomogeneous Media","author":"Klyatskin V. I.","year":"1980"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/0147009"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0149033"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/0165-2125(90)90010-2"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/0165-2125(90)90023-W"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/0165-2125(91)90005-9"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.57.1000"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.59.1918"},{"key":"R16","unstructured":"P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou,    Ping Sheng,  Wave localization and multiple scattering in randomly-layered media,  Scattering and Localization of Classical Waves in Random Media, World Scientific,  1990,  563\u2013619"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.63.2228"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1016\/0165-2125(88)90004-2"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1364\/JOSAA.8.001109"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112092002052"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.60.108"},{"key":"R22","first-page":"1005","volume":"73","author":"Richards P. G.","year":"1983","journal-title":"Bull. Seismol. Soc. Amer."},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1111\/j.1365-246X.1983.tb01904.x"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1029\/JB080i023p03322"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1029\/JB091iB06p06465"},{"key":"R26","first-page":"1223","volume":"77","author":"Frankel A.","year":"1987","journal-title":"Bull. Seismol. Soc. Amer."},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1190\/1.1442932"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1121\/1.385105"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1016\/0165-2125(88)90046-7"},{"key":"R30","volume-title":"Topics in the theory of random noise. Vol. I: General theory of random processes. Nonlinear transformations of signals and noise","author":"Stratonovich R. L.","year":"1963"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1137\/1111038"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160270503"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1137\/0121002"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(72)90222-3"},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1063\/1.1666247"},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1063\/1.1666600"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1137\/0132043"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1016\/0165-2125(82)90022-1"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1137\/0146032"},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.1063\/1.863654"},{"key":"R40","doi-asserted-by":"publisher","DOI":"10.1137\/0144010"},{"key":"R41","doi-asserted-by":"publisher","DOI":"10.1016\/0165-2125(85)90007-1"},{"key":"R42","doi-asserted-by":"publisher","DOI":"10.1190\/1.1442527"},{"key":"R43","doi-asserted-by":"publisher","DOI":"10.1137\/0151019"},{"key":"R44","doi-asserted-by":"publisher","DOI":"10.1137\/0134036"},{"key":"R45","series-title":"MAA Stud. Math.","first-page":"111","volume-title":"Studies in probability theory","volume":"18","author":"Papanicolaou G.","year":"1978"},{"key":"R46","volume-title":"Approximation and weak convergence methods for random processes, with applications to stochastic systems theory","author":"Kushner Harold J.","year":"1984"},{"key":"R47","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160290202"},{"key":"R48","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160320203"},{"key":"R49","doi-asserted-by":"publisher","DOI":"10.1007\/BF01225144"},{"key":"R50","volume-title":"Asymptotic analysis for periodic structures","author":"Bensoussan A.","year":"1978"},{"key":"R51","doi-asserted-by":"publisher","DOI":"10.1007\/BF01183015"},{"key":"R52","volume-title":"Waves in layered media","author":"Brekhovskikh L. M.","year":"1980"},{"key":"R53","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-52369-4"},{"key":"R54","volume-title":"Electromagnetic Waves in Stratified Media","author":"Wait J. R.","year":"1970"},{"key":"R55","volume-title":"Seismic wave propagation in stratified media","author":"Kennett B. L. N.","year":"1983"},{"key":"R56","volume-title":"Radiation and scattering of waves","author":"Felsen L.","year":"1973"},{"key":"R57","volume-title":"The Embedding Method in the Theory of Wave Propagation","author":"Klyatskin V. I.","year":"1986"},{"key":"R58","unstructured":"S. A. Grekeskul, V. D. Freilikher, Waveguiding properties of randomly stratified media, Vol. 31, Radiofiz, 1988, 1210\u20131217, (pp. 861\u2013867 in English)"},{"key":"R59","doi-asserted-by":"publisher","DOI":"10.1070\/PU1990v033n02ABEH002545"},{"key":"R60","unstructured":"V. I. Klyatskin,  Localization and waves in layered randomly inhomogeneous media,  1991, Wave Motion, to appear"},{"key":"R61","series-title":"Lectures in Appl. Math.","first-page":"417","volume-title":"Mathematics of random media (Blacksburg, VA, 1989)","volume":"27","author":"Pastur L. A.","year":"1991"},{"key":"R62","unstructured":"B. M. Shevtsov, Three-dimensional problem of inverse scattering in stratified randomly inhomogeneous media, Vol. 25, Radiofiz, 1982, 1032\u20131040, (pp. 742\u2013749 in English.)"},{"key":"R63","unstructured":"B. M. Shevtsov, Statistical characteristics for wave packet scattering in a layered randomly inhomogeneous medium above a reflecting surface, Vol. 30, Radiofiz., 1987, 1007\u20131012, (pp. 750\u2013754 in English.)"},{"key":"R64","unstructured":"B. M. Shevtsov, Backscattering of a wave in layered regularly and randomly inhomogeneous media, Vol. 32, Radiofiz., 1989, 1079\u20131083, (pp. 798\u2013801 in English.)"},{"key":"R65","doi-asserted-by":"publisher","DOI":"10.1364\/JOSAA.7.000868"},{"key":"R66","doi-asserted-by":"publisher","DOI":"10.1051\/jphys:01987004804050900"},{"key":"R67","doi-asserted-by":"publisher","DOI":"10.1051\/jphys:01987004804052700"},{"key":"R68","volume-title":"Asymptotic expansions of integrals","author":"Bleistein N.","year":"1986"},{"key":"R69","doi-asserted-by":"publisher","DOI":"10.1016\/0165-2125(89)90003-6"},{"key":"R70","doi-asserted-by":"publisher","DOI":"10.1111\/j.1365-2478.1971.tb00610.x"},{"key":"R71","volume-title":"An introduction to probability theory and its applications. Vol. II","author":"Feller William","year":"1966"},{"key":"R72","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-88264-7"},{"key":"R73","doi-asserted-by":"publisher","DOI":"10.1121\/1.383794"},{"key":"R74","doi-asserted-by":"publisher","DOI":"10.1029\/JB079i014p02077"},{"key":"R75","volume-title":"SAFARI: Seismo-acoustic fast field algorithm for range independent environments","author":"Schmidt H.","year":"1987"},{"key":"R76","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(81)90050-4"},{"key":"R77","volume-title":"Methods of Mathematical Physics, Volume II","author":"Courant R.","year":"1957"},{"key":"R78","unstructured":"M. Asch, Masters Thesis,  Courant Institute, Ph.D. dissertation, New York University, New York,  1990"},{"key":"R79","volume-title":"Quantitative Seismology-Theory and Methods","author":"Aki K.","year":"1980"},{"key":"R80","doi-asserted-by":"publisher","DOI":"10.1137\/1025122"},{"key":"R81","doi-asserted-by":"publisher","DOI":"10.1137\/1029075"},{"key":"R82","volume-title":"Time series","author":"Brillinger D.","year":"1981"},{"key":"R83","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160260405"},{"key":"R84","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160410705"},{"key":"R85","volume-title":"Ondelettes et op\u00e9rateurs. I","author":"Meyer Yves","year":"1990"},{"key":"R86","volume-title":"Methods of mathematical physics. Vol. I","author":"Courant R.","year":"1953"},{"key":"R87","volume-title":"Non-linear and non-stationary time series analysis","author":"Priestley M. B.","year":"1988"},{"key":"R88","doi-asserted-by":"publisher","DOI":"10.1121\/1.388174"},{"key":"R89","unstructured":"Yu N. Barabanenkov, A. G. Vinogradov, Yu A. Kravtsov, V. I. Tatarskii, Application of the theory of multiple scattering to the derivation of the radiation transfer equation for a statistically inhomogeneous medium, Vol. 15, Radiofiz, 1972, 1852\u20131860, (pp. 1420\u20131425 in English.)"},{"key":"R90","doi-asserted-by":"publisher","DOI":"10.1111\/j.1365-246X.1985.tb05128.x"},{"key":"R91","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.63.2837"},{"key":"R92","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.42.4613"},{"key":"R93","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.40.2825"}],"container-title":["SIAM Review"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/1033136","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T15:14:31Z","timestamp":1787238871000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/1033136"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,12]]},"references-count":94,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1991,12]]}},"alternative-id":["10.1137\/1033136"],"URL":"https:\/\/doi.org\/10.1137\/1033136","relation":{},"ISSN":["0036-1445","1095-7200"],"issn-type":[{"value":"0036-1445","type":"print"},{"value":"1095-7200","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,12]]}}}