We propose a new moment technique for the inverse problem of W ave Propaga- tion in an inhomogeneous (half – space) medium in the acousti cal Wave ﬁeld Born Approximation by using he Analytical Regularization Schem e of Bollini, Giambiagi and Dominguez. Key-words: Seismic; Inverse Problem. 1 One of the most chalenging problem in the inverse methods for th e Geophysical prob- lem of imaging an inhomogeneous medium described by a depth – v ariable refraction index M(z) (0≤z <∞) from the backscattered acoustical wave ﬁeld in the Born – appr oxima- tion is the problem of its ill – posedeness ([1]) – pag. 323). In t his Brief Report we will propose a solution of this diﬃculty by considering the Analytica l Regularization Scheme previously used in Quantum Field Theory ([2]) to reduce the ima ging of the medium to the well-know problem of determining a function by knowing i ts associated moments ([3]) – Theorem 15.26). Let us start our analysis by considering the corresponding Fredh olm integral equation relating the medium refraction index ( M(z) (0≤z <∞) to the Born approximated backscattered acoustical Wave Field US(ξ,ξ;w) on the medium boundary ( ξ= (x,y)εR2, −0< z < +∞and written in the frequency domain US(ξ,w) = (w)2·/integraldisplay+∞ −∞d2η/integraldisplay∞ 0dzM(z)·exp/bracketleftBig 2iw/radicalbig (η−ξ)2+z2/bracketrightBig/parenleftbig (η−ξ)2+ 2/parenrightbig−1.(1) Unfortunately, the direct solution of the Fredholm Integral E quation eq.(1) to evaluate M(z) is an ill - conditioned mathematical problem since its kerne l is not square integrable and thus, deﬁning a non – bounded operator in the Square Integr able functions which by its turn make this operator loses its inverse continuity in thi s function space. In order to overcome such mathematical problem, we follow an u sual procedure bor- rowed from similar distribution multiplication problem in qu antum Field Theory ([2]). We propose to consider an analytically continued kernel in eq. (1) where the problem of ill – posedeness is absent and at the ed of the calculation, we mak e an analytical contin- uation of the result to the Physical Problem as ﬁrst proposed by Bo llini, Giambiagi and Dominguez in 1964. Let us, then, study the analytically contin ued problem Uα S(ξ,ξ;w) = (w)2·/integraldisplay+∞ −∞d2η/integraldisplay∞ 0dz m(z)·exp/bracketleftBig 2iw/radicalbig (η−ξ)2+z2/bracketrightBig/parenleftbig (η−ξ)2+z2/parenrightbig−α/2 (2) where αis chosen in such way that it guarantee its square integrability . In this region ofαvalues, it is possible to interchange the η−zintegration order by the use Fubbini Theorem and evaluate explicitly the η-integration for z >0 ([2]). −U(α) S(ξ,ξ,w ) w2=/integraldisplay∞ 0dzm(z)·/bracketleftBigg∞/summationdisplay m=0(2i)m·(w)m m!×/parenleftBigg πΓ/parenleftbigα−m 2−1/parenrightbig Γ/parenleftbigα−m 2/parenrightbig·(z2)+(1−(α−m) 2)/parenrightBigg/bracketrightBigg (3) 2 where we have used the result /integraldisplay+∞ −∞d2p(p2+β2)m−α 2=πΓ/parenleftbigα−m−2 2/parenrightbig Γ/parenleftbigα−m 2/parenrightbig(β2)−(α−m−2 2). (4) By considering now the existence of power expansion associated to the analytically continued Back-Scatttered Acoustic Wave Field (which is math ematically the correct assumption at least for large time), since w→0+in low frequency regime ([4]) −U(α) S(ξ,ξ,w ) w2=∞/summationdisplay m=0U(α) m·wm m!(5) and comparing with the power expansion of eq.(3), we get the mo ment problem relation- ship U(α) m=(2i)m Γ/parenleftbigα−m 2/parenrightbig·π·Γ/parenleftbiggα−m−2 2/parenrightbigg/parenleftbigg/integraldisplay∞ 0dz·m(z)·zm+2−α/parenrightbigg . (6) At this point we implement the Physical Limit of α→2 (see eq.(1) and taking into account that U(α−2) 0has an inﬁnite piece U(α−2) 0=/integraldisplay∞ 0dz·m(z)·lim α→2/parenleftbigg2 α−2/parenrightbigg ·z2−α =−2/parenleftbigg/integraldisplay∞ 0dzm(z)/parenrightbigg lim α→2/parenleftbigg1 α−2/parenrightbigg −2/integraldisplay∞ 0dz·m(z)lg z (7) The inﬁnite piece is disregarded in our approach by suppossing th at the “medium ´ area”/integraltext∞ 0m(z)dzvanishes identically. Otherwese eq.(7) must be understood as a ﬁ nite- part prescription. By grouping together eq.(6)–eq.(7) we reduce the problem of imaging the refraction index m(z) to the solution of the well-know Moment Problem of determini ng a function from its moment of order p U(2) (p>0)= (2i)pπ:=m(p)/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright /integraldisplay∞ 0dz·m(z)·zp(8.a) U(2) (0)=−2·/integraldisplay∞ 0dz m(z)lg(z)dz (8.b) In the case that m(z) has a Fourier Transform which is an analytical function arou nd the origin in the Fourier Domain (for instance when m(z) has compact support), we can solve formally eq.(8.a). The validity of this claim is a result of the equation below ˜m(k) =1 2π/integraldisplay∞ 0dz·eikzm(z) =1 2π∞/summationdisplay p=0(i)pkp p!m(p)(9.a) 3 with U(2) (m>0)= (2i)mπ/integraldisplay∞ 0dz·zm/parenleftbigg/integraldisplay+∞ −∞dxeikz˜m(k)/parenrightbigg = (2i)mπ·1 (i)m·/parenleftbiggdm˜m(k) dkm/parenrightbigg k=0(9.b) As a consequence, we have the ﬁnal result connecting the coeﬃcie nts of eq.(9.a) and eq.(5). U(2) (m>0)= 2m·π·m(m). (10) Finally, we remark that the above mathematicall operations hold true at least if |m(m)| ≤e−m·Cm, where Cmare the Taylor coeﬃcients of an analytical real function with the property of Σ C2 m<∞.