We deduce well-posed formulae for wave ﬁeld extrapolation in depth-dependent mediums. 1 Introduction The basic starting point in the subject of wave ﬁeld extrapolati on and its application to seismic inverse procedures is the derivation of the acoustic p ressure ﬁeld U(/vector r,z,t), developing in a medium (upper half-space z >0) with a constant refraction index, from a known pressure ﬁeld U(/vector r,z= 0,t) at its surface. In section 2 of this short comment, we improve mathematically some of those results of ref. [1] by co nsidering now the well- posedeness formulation of the problem. 2 The Depth-Extrapolation Prblem for a Medium with a Constant Refraction Index Let us consider the acoustic wave ﬁeld equation for a pressure ﬁel d developing in a medium (upper half-space z >0) deﬁned by a constant refraction index and from a known pressur e ﬁeld data U(/vector r,z= 0,t) butadded with depth derivative at the surface, namely: /bracketleftbigg∂2 ∂z2−1 v2∂2 ∂t2−(k2 x+k2 y)/bracketrightbigg U/vectork(z;t) = 0 (1) 1 U/vectork(z,t)|z=0=f/vectork(t) (2) ∂U/vectork ∂z(z,t)|z=0=g/vectork(t) (3) where U(/vectork,z,t) =/integraldisplay+∞ −∞dkxdkyexp(i/vectork·/vector r)U/vectork(z;t). (4) Let us analyze ﬁrstly the case of non-evanescent waves deﬁned by the condition |/vectork|2< w2 v2(with w≥0). Here U/vectork(z;t) =/integraldisplay∞ 0dwexp(iwt)U/vectork(z;w). (5) The time-domain Fourier transformed ﬁeld solution of eq.(1)– eq.(3) is explicitly given by U/vectork(z;w) =F+(w;/vectork) exp/bracketleftBigg/parenleftBigg i/radicalbigg w2 v2−(/vectork)2/parenrightBigg z/bracketrightBigg +F−(w;/vectork) exp/bracketleftBigg/parenleftBigg i/radicalbigg w2 v2−(/vectork)2/parenrightBigg z/bracketrightBigg (6) with F+(w;/vectork) =1 2 ˆfk(w)−iˆgk(w)/radicalBig w2 v2−(/vectork)2  (7) F−(w;/vectork) =1 2 ˆfk(w) +iˆgk(w)/radicalBig w2 v2−(/vectork)2  (8) and ˆfk(w) =f∞ 0dzexp(iwt)fk(t)dt (9) ˆgk(w) =f∞ 0dzexp(iwt)gk(t)dt. (10) Note that only on this situation of non-evanescent case, one could claim to obtain well- posed extrapolating formulae with our new condition eq.(3) , opposite to those somewhat diﬀerents results presented in ref. [1] without this condition. Let us thus introduce the following variable change in the Fo urier-Integral eq.(4) for the acoustic pressure ﬁeld: w′=w;kx=pxw;ky=pyw([1]). We thus, have the following 2 result: U(t,z;/vector r) =/integraldisplay+∞ −∞d2/vectorkei/vectork·/vector r/integraldisplay∞ 0dw eiwt(U/vectork(z;w)) = /integraldisplay1/c −1/cdpx/integraldisplay1/c −1/cdpy/integraldisplay∞ 0dw·eiwt(w2)eiw(/vector p·/vector r) /braceleftbigg/parenleftbigg e+iw/parenleftBig/radicalBig 1 c2−(/vector p)2/parenrightBig z/parenrightbigg F+(w, /vector pw) +/parenleftbigg e−iw/parenleftBig/radicalBig 1 c2−(/vector p)2/parenrightBig z/parenrightbigg F−(w, /vector pw)/bracerightbigg . (11) The above expression by its turn is the sum of four Fourier integr als which are going to be analyzed. the ﬁrst one is exactly given by the following e xpression (here ˆf/vectork(w)≡ ˆf(w,/vectork))1 U(1)(t,z;/vector r) =/integraldisplay1/c −1/cdpx/integraldisplay1/c −1/cdpy/integraldisplay∞ 0dw·w2eiw/bracketleftBig t+/vector p·/vector r+/parenleftBig/radicalBig 1 v2−(/vector p)2/parenrightBig ·z/bracketrightBig1 2ˆf(w, /vector pw).(12) As in ref. [1], we can re-write eq.(12) as a depth-extrapolato r integral operator (by means of a t-convolution integral) U(1)(t,z;/vector r) =/integraldisplay1/c −1/cdpx/integraldisplay1/c −1/cdpyR(1)(z,t,/vector r;p)(∗)t˜U(z= 0,t,p) (13) where the surface observed pressure ﬁeld is given by ˜U(z= 0,t,p) =/integraldisplay∞ 0dwe+iwtˆf(w, /vector pw) (14) and the depth-extrapolation Kernel is written explicitly a s ([2]) R1(z,t,x,p ) =/integraldisplay∞ 0dweiwteiw/parenleftBig /vector p·/vector r+/parenleftBig/radicalBig 1 v2−(/vector p)2/parenrightBig z/parenrightBig w2 =−π/parenleftbiggd2 d2ξ/parenleftbigg δ(ξ) +i πξ/parenrightbigg/parenrightbigg/vextendsingle/vextendsingle/vextendsingle ξ=t+/vector p·/vector r+/parenleftBig/radicalBig 1 v2−(/vector p)2/parenrightBig z. (15) The other part of the pressure ﬁeld in eq.(11) corresponding to t he knowledgement of the depth-derivative surface data eq.(3) and needed to turn t he extrapolation problem a well-posed mathematical problem is given by an analogous Fou rier integrals formulae U(2)(t,z,/vector r) =−i/integraldisplay1/c −1/cdpx/integraldisplay1/c −1/cdpy/integraldisplay+∞ −∞dw·w2eiw/bracketleftBig t+/vector p·/vector r/parenleftBig/radicalBig 1 v2−(/vector p)2/parenrightBig z/bracketrightBig ˆg(w, /vector pw) w/radicalBig 1 v2−(/vector p)2  =/integraldisplay1/c −1/cdpx/integraldisplay1/c −1/cdpyR(2)(z,t,/vector r;p)(∗)t˜Ut(z= 0,t, /vector p) (16) 1/integraltext∞ 0e±ibtdt=πδ(b)±iP{1 b} 3 with ˜Ut(z= 0,t, /vector p) =/integraldisplay∞ 0dw eiwtˆg(w, /vector pw) and R(2)(z,t,/vector r, /vector p ) =1/radicalBig 1 v2−(/vector p)2π i/parenleftbiggd dξ/parenleftbigg δ(ξ) +i πξ/parenrightbigg/parenrightbigg/vextendsingle/vextendsingle/vextendsingle ξ=t+/vector p·/vector r+(/radicalBig 1 v2−(/vector p)2)z. (17) The other two integrals are obtained by just changing z→ −zin the above obtained formulae. In the case of evanescent waves ([1]) deﬁned by the condition k2 x+k2 y≥w2 v2, the associated well-posed problem is governed by the following ini tial and boundary conditions imposed on eq.(1) U/vectork(z,t)/vextendsingle/vextendsingle/vextendsingle z=0=f/vectork(t) and lim z→∞U/vectork(z,t) = 0. (18) The solution takes, now, the following form U(3)(z,t,/vector r./vector p ) =/integraldisplay |/vector p|≥1 cd2/vector p/integraldisplay∞ 0dw eiwtw2eiw(/vector p·/vector r)exp/parenleftBigg −w/parenleftBigg/radicalbigg (/vector p)2−1 v2/parenrightBigg z/parenrightBigg ×˜f(w, /vector pw) =/integraldisplay |/vector p|≥1 cd2R(3)(z,t,/vector r./vector p )(∗)t˜Ut(z= 0,t, /vector p) (19) with the evanescent depth-extrapolating Kernel ([2]), ([3] ). R(3)(z,t,/vector r./vector p ) =−π/parenleftbiggd2 dξ2/bracketleftbiggi πξ/bracketrightbigg/parenrightbigg ξ=t+/vector p·/vector r+i/parenleftBig/radicalBig (/vector p)2−1 v2/parenrightBig z(20)