Inverse scattering: asymptotic analysis

If an acoustic field is governed by the equation V 2 u + w 2 u + u2al (x)u + V (az(x)Vu) = -S(x-y) in R and U is measured on the surface of the earth, i.e. on the plane x3 = 0 for all positions of the source y and receiver x and for all frequencies, then: (i) we show that the low frequency portion of the data determines a 2 ( x ) while the high frequency portion of the data determines ( a , -a2)(l + a2)-' =p(x) . Here a2 = p ' A p is the relative variation of the density of the inhomogeneity and p ( x ) depends only on the relative variation of the velocity in the inhomogeneity; (ii) we show how to recover a2 from the low frequency data and how to recover p ( x ) from the high frequency data; (3) we give a formula for the high frequency asymptotics of U on the plane x3 = 0. In [ 11-[3] an exact solution to some three-dimensional inverse problems of geophysics is given. The problem we discuss here is the following one. Let U be the acoustic field generated in an inhomogeneous medium by a point source: Here al = K-IAK, a2 =p-'Ap are the relative variations of the bulk modulus and density, w is the frequency, 6(x) is the delta function. Assume that al(x) and a2(x) are compactly supported in R! = {x:x3 < 0}, and 1 + a2 > 0. This means that the inhomogeneity lies in the lower half-space. The data are the values of u(x,y, 0) measured on the plane P= (x:x3 =0} for all positions of the sourcey' =(yl,y2, 0) and the receiver x ' =(XI, x2,O) and for all w > 0. The problem is to recover aj(x), j= 1,2, from the data. We show that the low frequency portion of the data determines a2 and the high frequency portion determines p(x)=(l + a2)-'(a1 -4. Thus, by comparing two sets of data with the same high frequency portions of the data one concludes without calculations that the two sets of the data correspond to the media with different densities. One notes that p(x) = -( 1 + ~23)-~(2U3 + a i ) where a3 = v-lAv is the relative change in the velocity, v=(p-'K)''Z. If the low frequency portions of the data are the same, then one concludes that the difference in the data is due to the variation of the velocity. We give some methods for finding az(x) and p(x) separately from the low and high frequency data respectively and we give an asymptotic formula for U on P for large w. V2u+w2u+w2al(x)u+V . ( u ~ ( x ) v u ) = ~ ( x ~ ) ~ ~ R ~ . (1) If w + 0 then the limit equation (1) is of the form V2Go + V . ( u ~ ( x ) V G ~ ) = ~ ( X ~ ) (2) and u(x,y, w)+Go(x,y) as 0-0. The existence of this limit follows from the results in [3, 8 V.41 provided that 1 + a2 > 0. Let go =(4;n/x-yl)-'. Then (2) can be written as 0266-561 1/86/040043 + 04$02.50