Elements of Dynamic Ray tracing

Transcription

1 Elements of Dynamic Ray tracing ErSE360 Contents List of Figures 1 1 Introduction Coordinates systems 1 Cartesian coordinates 3 Ray coordinates γ 1 γ s 3 3 Ray-centered coordinate system q 1 q s 3 3 Coordinate transformations 6 31 Matrices 7 3 Jacobian 9 33 Hamiltonian in orthogonal curvilinear coordinates 10 4 Dynamic Ray Tracing 1 41 DRT in ray-centered Coordinates 1 4 DRT in Cartesian coordinates Paraxial traveltimes Algorithm for paraxial traveltime 15 5 Amplitude continuation Ray tubes 16 5 Geometrical spreading Coustics 18 6 Remarks 19 7 Exercise 19 References 19 List of Figures 1 Position vector in ray-centered coordinates 4 Difference between Frenet frame and ray-centered coordinates 7 3 Perpendicular planes and ray tube 16 4 Coustic points 18 1

2 ERSE360 1 Introduction Ray Tracing RT is an indispensable tool in applications which are concerned with the propagation of waves whether it is light or sound In seismic metho for geophysics we deal with sound waves propagating through the earth Apart from modeling these waves through wave equations elastic or acoustic there are many occasions when it is feasible to deal with traveltimes of the waves rather than the full wavefield This lea us to the evaluation of the rays which are lines connecting a source and any observation point such that they are always perpendicular to the wavefront Rays are a high frequency approximation to the wave equation In the last course equations for such an analysis which are eikonal and transport equations have been covered so we shall not into those derivations here Appropriate references are provided however Solution to the RT equations gives us kinematic part of the wave that is it tells us the path traced by a particular point on the wavefront and the time it took to reach from say point S to point R It gives us no information about the amplitude of the wave at the observation point For the amplitude at a particular point on the ray we solve the transport equation Solving each of these equations numerically for each point in the velocity model is expensive In these notes we shall consider a method of ray tracing called Dynamic Ray Tracing DRT which not only reduces the computation time for the traveltimes but also allows for the calculation of amplitudes along rays Albeit this is an approximate method The basic idea is to trace rays using conventional RT but doing so sparsely that is not covering every point in the model These are the so called central rays Then we calculate the traveltimes away in the so called paraxial vicinity from these central rays and this is where the approximation comes in The farther we are away from the central rays the less accurate are the traveltimes These are called the paraxial traveltimes Their accuracy depen of course on the complexity of the velocity model In this course we shall assume the medium to be smoothly varying and with no viscosity In that case the decay of the amplitudes of the waves is only due to geometrical spreading As we shall see an analytical solution to the transport equation involves a quantity J called the ray jacobian and the geometrical spreading is given by J Ray jacobian can also be expressed in terms of a matrix Q which can be calculated by DRT therefore DRT can give us the decay in amplitude due to geometrical spreading Paraxial traveltimes can be expressed in terms of matrices P and Q which are explicitly calculated in DRT Therefore with DRT they can also be evaluated Paraxial traveltimes can be evaluated explicitly using the Paraxial Ray Tracing PRT but we don t need to do it since DRT can take care of both traveltimes and amplitudes We shall also see that both PRT and DRT are identical as far as equations of motion are concerned The only difference is in the physical interpretation of the quantities being calculated In case of PRT the quantities calculated are the position and slowness components along rays in the paraxial vicinity of the central ray called paraxial rays In the case of DRT the calculated quantities are P and Q along the paraxial rays Coordinates systems In everyday life we face problems and our approach towar solving them depen on our frame of mind No matter what frame of mind we are in the nature of the problem is the same but our approach has a profound impact on the solution Same is true in physics We can choose any reference frame coordinates for a theory without altering the physics behind it but different coordinates will facilitate different features Therefore in our endeavor to study the DRT it would

3 ELEMENTS OF DYNAMIC RAY TRACING 3 be of prime importance to understand different coordinate systems which we can use to formulate the theory 1 Cartesian coordinates Let us start with the obvious Cartesian coordinates The RT equations with arclength s as a parameter are given by dx = V p dp 1 1 = V = 1 V where p = τ τ is the traveltime and V is the velocity Rays are the curves defined by vector x s which is function of the arc length Then from the first of equations 1 that the tangent to these curves is parallel to p therefore p gives the direction of travel of the ray First and second of equations 1 are a set of coupled first order ODEs To solve them we would need initial conditions that is three components of x 0 and three components of p 0 From eikonal equation we know that p = 1 V therefore only two of the the three components of p 0 are independent Once x and p have been solved for third one of equations 1 can be directly integrated to give the traveltime τ along the ray For spherical polar coordinates see [Červený 001] Ray coordinates γ 1 γ s Since we are tracing individual rays and in the end we want to be able to calculate the traveltimes in the neighborhoo of each of these rays it is important that we have a way of distinguishing one ray from the other Therefore we need some parameter or parameters which would stay constant along the ray Also they should be both necessary and sufficient to define a ray uniquely From the Cartesian coordinates above we can see that staying constant requirement is met by the initial conditions But since two different rays can have the same x 0 for a point source every ray will have the same starting point the three initial position component are not required We are left with the three components of initial direction p 0 but we saw above that only two of these are independent Therefore we conclude that in 3D only two independent parameters are required to uniquely identify a ray These two can be the angles specifying the initial direction of the ray also called the emergence angles It is obvious that indeed the initial direction is unique to every ray These two parameters for D only one is required can be any functions of the initial conditions but they have to be independent Let us call these two independent parameters γ 1 and γ Then consider a coordinate system with first two axes as γ 1 and γ and the third as s that is the arclength This is called the ray coordinate system In the ray coordinate system every point on s = constant plane would represent a single ray therefore is would indispensable if we have to calculate for example a rate of change of some quantity with respect to the rays themselves As we shall see this is precisely what we have to do in order to calculate geometrical spreading 3 Ray-centered coordinate system q 1 q s The two coordinate systems discussed above are such that they define every ray that it a point on every ray is a point in either one of those coordinates systems Now we shall consider a coordinate system which is attached to one particular ray Since orthogonal coordinate systems are easier to deal with it is natural that we look for coordinates which form and orthogonal system For a start we take the unit vector along one of the axes to be t s dx / dx = V p tangent to the ray itself and the other two unit vectors n and b to be perpendicular to t Coordinate along t is represented by s along n by q 1 and along b by q therefore a point is represented by q 1 q s One of the non-tangent unit vectors say n is defined as n s = dt / dt = 1 dts K K is the curvature of the ray therefore it is perpendicular to t s Then we define b as b s = t s n s The coordinate system

4 4 ERSE360 defined by t s n s and b s is called the Frenet frame We would also need their derivatives with respect to s which are a b c dt s = K s n s dn s = T s b s K s t s db s = T s n s These are the well known Frenet formulae [Thomas and Finney 1996] Now we want to calculate the distance from a point R on the ray to R in the Frenet frame According to Figure 1 the e R q q 1 R e 1 Plane perpendicular to the ray r q q s 1 Ray r00 s O 1 Figure 1 Position vector in ray-centered coordinates radius vector to the origin of the new coordinate system which is at R is r 0 0 s therefore the radius vector to R is 1 3 r q 1 q s = r 0 0 s + q 1 n + q b Length squared of a differential element of a vector in Cartesian coordinates is given as l = dx dx = dx + dy + dz In any other coordinate system say q 1 q q 3 square of the length dx becomes dx dx = dq i dx dx dq j dq i dq j g ij dq i dq j dq i is a unitary vector in the direction of q i and g ij is the so called metric tensor This means that if the new coordinate system is orthogonal we would have i = j in the new formula for square of the length Therefore in general when the new coordinate system is orthogonal square of the length becomes l = g ii dqi g 11dq1+g dq+g 33 dq3 that is the metric tensor does not have off-diagonal terms Now the differential of the vector in 1 For now consider the unit vectors e1 and e in Figure 1 to be n and b respectively The difference between these two is illustrated in Figure

5 ELEMENTS OF DYNAMIC RAY TRACING 5 equation 3 is 4 dr = dr dq 1 dq 1 {}}{ ndq 1 + dr dq dq {}}{ bdq + dr dq 3 dq 3 {[ }}{ t + dn q 1 + db ] q Length of dr can be calculated by taking the dot product of equation 4 by itself but let us first check whether our unitary vectors are orthogonal or not For equation 4 the cross terms dr dq 1 dr dq are zero because the vectors n and b are orthogonal But the off-diagonal terms involving q 3 are not zero for example dr dq 1 dr = T q which would in general be non-zero To see what went wrong we note that the third unitary vector is not in the direction of s This can be seen from the fact that it is a sum of t and other vectors which are not in the direction of t as is evident from equations Therefore when calculate the off-diagonal element of g ij with either i or j equal to three the dot product is not zero The problem is not t because it is orthogonal to dr n and dr dq b it is the other two vectors which are given by equations Therefore the Frenet frame is not a suitable choice for our purpose we require some other coordinates The solution is simple you might have guessed already We just need to make the derivatives of n and b parallel to t while keeping all three of them mutually perpendicular To distinguish from the Frenet frame notation we call our new vectors e 1 e and t Since we do not have to change t we keep the same notation We also set the new vectors according to 5 e 1 e = t e t = e 1 t e 1 = e This will ensure that we have a right-handed system of orthogonal coordinates Now to make de1 and de parallel to t we write de I s 6 = a I s t s I = 1 Where a s is some scalar function of s Throughout these notes indices in upper case for example I here run from 1 to To find a I we take the derivative of e t = 0 with respect to s This gives us de I t = e I dt By substituting this into the dot product of equation 6 with t we get a I = e I dt Then we use this value of a I in equation 6 again and get de I = e I dt 7 t We know that t = V p therefore dt = V dp + p dv Then since both e I are perpendicular to t hence also to p equation 7 becomes de I = V e I dp 8 pv Finally by using dp = 1 V = V V in equation 8 we get de I 9 = e I V p ei V = V A numerical solution of this differential equation will determine e I We need only to determine only of the e I the other one can be calculated by equation 5 p can be calculated by tracing a t dq 1

6 6 ERSE360 ray using conventional method this is the central ray The initial value for say e 1 at s = s 0 should be such that e 1 s 0 p s 0 = 0 and e 1 s 0 e 1 s 0 = 1 Now we shall write equation 4 in terms of our new coordinates that is 10 dr dq 3 dq 3 dr dr dq dq 1 1 dq dq {{}}{{}}{ [ }}{ dr = e 1 dq 1 + e dq + t + de 1 q 1 + de ] q [ e1 v e v = e 1 dq 1 + e dq q 1 + v v [ = e 1 dq 1 + e dq q 1 v + q v v q 1 v q q 1=0q =0 q ] t q 1=0q =0 The dot product of equation 10 does not have any off-diagonal term for g ij because all three vectors involved are orthogonal to each other Thus we have 11 where 1 dr dr = g 11 dq 1 + g dq + g 33 dq 3 = dq 1 + dq + h g ii h i and h 1 = 1 h = 1 h 3 = 1 + q I v v q I ] t I = 1 q 1=0q =0 h i are called the scale factors From this analysis we see that our new coordinate system is indeed orthogonal This new coordinate system with unit vectors e 1 e t and coordinates q 1 q s is called the ray-centered coordinate system There were two problems with the Frenet frame First one was apparent that it they were not orthogonal Second more subtle problem was that the unit vectors t n and b depended only on the local properties of the curve ray in our case This means that we would have had no way of distinguishing one ray from another if the tangent t happened to be the same for different rays at any point In the ray-centered coordinate system we overcome this problem by using the differential equation 9 to calculate the e I As we have to use the initial value p s 0 to determine e I and we know that the initial direction determines the ray uniquely the ray-centered coordinate system will depend on a particular ray rather than only the value of the tangent at a particular point The difference between Frenet frame and ray-centered coordinates as they move along the ray is illustrated in Figure As we saw above the unit vector t is common to both It is possible that at some point S on the ray e 1 e and n b are the same But as the ray moves they get rotated with respect to each other Note that e 1 e and n b would always lie in the same plane Of course we can apply simple rotation to get the ray centered coordinates from Frenet frame but we shall not go into that procedure here For details see [Červený 001] 3 Coordinate transformations Having studied the coordinate systems we would also need to study how to move back and forth between them As we shall see coordinate transformations play a very important role in not only simplifying the equations but also determining some important properties of the rays and their amplitudes We shall first look at the transformation matrices then the transformation of volume elements Jacobians of transformation and then the Hamiltonian systems in generalized coordinates

7 ELEMENTS OF DYNAMIC RAY TRACING 7 n e 1 e = n 1 t e b R t Ray S e = b Plenes perpendicular to the ray Figure Difference between Frenet frame and ray-centered coordinates 31 Matrices Throughout these notes the matrices with a hat example ˆQ would represent 3 3 matrices and without a hat example Q matrices matrices will represent the first two coordinates example q 1 and q of any system ˆQ Ray to Ray-centered Coordinates This transformation can be written as qi γ j By i and j can be regarded as rows and columns this transformation can be written as a matrix 13 ˆQ = q i γ j = q 1 γ 1 q 1 γ q 1 γ 3 q γ 1 q γ q γ 3 q 3 γ 1 q 3 γ q 3 γ 3 If we regard vectors as column matrices that is dˆγ = γ 1 γ γ 3 T and dˆq = q 1 q q 3 T equation 13 can be written as dˆq = ˆQdˆγ qi 14 or dq i = dˆγ j = Q ij dˆγ j γ j If we calculate ˆQ along a ray say R q 1 and q are zero This is because e I are perpendicular to t = vp which is tangent to the ray Moreover q 1 and q stay unchanged along the ray that is they are always zero Therefore we have q I γ 3 = 0 Then equation 13 becomes ˆQ R = Q 11 Q Q 1 Q 0 Q 31 Q 3 Q 33 Therefore we have R 16 det ˆQ R = Q 33 det Q R = q 3 γ 3 det Q R

8 8 ERSE360 where Q is the matrix which consists of first two rows and columns of rhs of equation 15 If we take the third coordinate to be the same in both ray and ray-centered coordinates we have det ˆQ R = det Q R Ĥ Ray-centered to Cartesian Coordinates This transformation can be written as dx i = H ij dq j = x i q j dq j or dˆx = Ĥdˆq For this matrix also we would need to determine it only along the ray R orthonormal along the ray that is The matrix Ĥ is 19 Ĥ 1 R = ĤT R det Ĥ R = 1 ˆQx Ray to Cartesian coordinates Again we shall consider the matrix only along the ray Like the other ones matrix ˆQ x is given by 0 dˆx = ˆQ x dˆγ or dx i = Q x ij dγ j = x i γ j dγ j By inserting the value of dˆq from equation 14 into equation 18 we get dˆx = Ĥ R ˆQ R dˆγ Then by comparing this with equation 0 we get 1 ˆQ x R = Ĥ R ˆQ R ˆP Ray coordinates to ray-centered components of slowness vector This transformation matrix transforms the ray coordinates to the components of slowness vector in the ray-centered coordinates The slowness vector p in ray-centered coordinates is given by p q = τ + τ + τ q 1 q q 3 where τ is the traveltime and the superscript q denotes the fact that the vector is in ray-centered coordinates The transformation along the ray R is written as dp q p q i τ 3 i = P ij dγ j = dγ j = dγ j or dˆp q = γ j q i γ ˆPdˆγ j R as 4 R ˆP x Ray to Cartesian components of slowness vector This matrix is defined along the ray R dp i = P x ij dγ j = pi γ j dγ j = R τ x i γ j dγ j or dˆp = ˆP x dˆγ R No superscript on the components p i means that they are in Cartesian coordinates The relation between ˆP x and ˆP is ˆP x R = Ĥ R ˆP R 5 M Second derivative of traveltime M is a matrix which can be represented in terms of P and Q as 6 M = τ x I x J = PQ 1 I J = 1 p q is not the physical slowness vector The components of the physical slowness vector p are related to the components of p q by p q h 1 i = p i

9 ELEMENTS OF DYNAMIC RAY TRACING 9 3 Jacobian After determining the transformation matrices we shall now turn our attention to the Jacobians of transformation Jacobians come across when we transform volume elements from one coordinate system to another Although details can be found in any text on calculus or classical differential geometry for example [Arfken and Weber 005] we shall briefly discuss the main features which will be of use to us A Jacobian is the determinant of the transformation For our purpose we look at the transformation from ray coordinates to Cartesian coordinates We already know the matrix for this transformation that is ˆQ x Then the Jacobian of transformation is det ˆQ x We set γ 3 u where u can be any monotonically increasing parameter along the ray like arclength s or traveltime τ Then we have x 1 x 1 x 1 J u = det ˆQ x γ 1 γ u x = x x 7 γ 1 γ u x 3 γ 1 γ u x 3 x 3 Since along the ray dx du = dx du = g 1 u dx dx and we know that = t equation 7 becomes x 1 x 1 γ 1 γ u t 1 u t 1 t t 3 J u = gu 1 x x γ 1 γ u t = g u 1 x 1 x x 3 8 u γ γ u γ u u x 3 x 3 x γ 1 γ u t 3 1 x x 3 γ 1 γ u u 1 γ u 1 u The last determinant in equation 8 is a vector triple product have 9 where Ω u = x γ 1 J u = 1 g u Ω u t = J g u x γ 1 x γ and J is called the ray Jacobian Since u x γ I x γ t Therefore we u are vectors in the directions of γ 1 and γ the direction of vector area Ω u is perpendicular to the plane u = constant It s magnitude Ω u is the area in u = constant plane Moreover the differential element of an area is given by 30 dω u = x x dγ 1 dγ = Ω u dγ 1 dγ γ 1 γ u Because wavefront is always perpendicular to the ray tangent we can get the projection of dω u on to the wavefront by 31 dω u t dω Then by substituting equation 30 in 9 we get 3 J = dω dγ 1 dγ Transformation of a volume element dxdydx is then given as [ x dxdydz = gu 1 x ] 33 t dγ 1 dγ du = gu 1 J dγ 1 dγ du γ 1 γ Before we leave this section note that 34 u J = g u J u

10 10 ERSE360 and especially J s = J 33 Hamiltonian in orthogonal curvilinear coordinates The Hamiltonian H x is a function of coordinates 3 say x i and momenta p i For our purpose it is the eikonal Since the eikonal equation remains constant along the rays H x x i p i is constant Rays can be parametrized by any monotonically increasing parameter u for example arclength s or traveltime τ H x x i p i will then be a function of u implicitly that is through x i and p i In the last course we derived RT systems for such parameters by calculating dh du Now we shall generalize that technique for curvilinear orthogonal coordinates say ξ 1 ξ ξ 3 These can be any coordinates as long as they are orthogonal to each other First we shall need to represent the eikonal equation in terms of ξ 1 ξ ξ 3 Since the eikonal equation is τ τ = V ξ 1 ξ ξ 3 we need to represent in terms of the curvilinear coordinates Let us write the expressions for length squared and in curvilinear coordinates which are 35a 35b = h 1dξ1 + h dξ + h 3dξ3 1 1 = e 1 + e h 1 ξ 1 h 1 + e 3 ξ h 3 ξ 3 where h i are the so called scale factors As we saw before h i = g ii remember that our curvilinear coordinates are orthogonal so g ij = 0 for i j Here e i represent the unitary vectors in the direction of ξ i Using equation 35b the eikonal in orthogonal curvilinear coordinates becomes 36 τ τ = 1 h τ h τ + 1 h τ 1 3 = 3 V ξ 1 ξ ξ 3 where τ i = τ ξ i Remember that the physical components of slowness vector in ξ i direction are 1 τ h i ξ i τ i is just an abbreviation Now we shall generalize the Hamiltonian As we know the eikonal can be expressed in different ways to be considered as a Hamiltonian and from these expression we can obtain RT equations for different parameters u [Slawinski 003] Here we shall write a general form for the Hamiltonian which is [ H ξ i τ i = 1 1 n h τ h τ + 1 ] n/ 37 h τ3 1 3 V n = 0 For practice you can prove that for n = 0 equation 37 becomes lim H ξ i τ i = 1 1 n=0 ln h τ h τ h τ3 + ln V 3 Before going any further with orthogonal curvilinear coordinates let us recall that for a Hamiltonian H x x i p i the ray RT system with a motonically increasing parameter u is given as [Slawinski 003 Červený 001] 39 dx i du = Hx p i dp Hx i du = x i du = p H x i p i 3 We will use the superscript x to denote Cartesian coordinates

11 ELEMENTS OF DYNAMIC RAY TRACING 11 If the Hamiltonian is written as H x x i p i = n 1 [ p k p k n/ 1/V n ] by inserting this into equations 39 the corresponding RT system is given by dx i du = p kp k n/ 1 dp i p i du = n x i V n For n = 0 it is 41 dx i = p kp k n/ 1 p i dp i = ln V x i du = p kp k n/ = V n = 1 As you can easily verify from equations 40 the case n = 1 correspon to u s and n = to u σ = τv For n = 0 equation 41 gives the RT system with u τ To derive a RT system for orthogonal curvilinear coordinates we just need to replace x i with ξ i p i with τ i note that the new components which replace p i are not the physical components of slowness vector and use equation 37 for H in equation 39 Then we get 4 dξ i du = τ An/ 1 i h i i du = 1 n ξ i n A n/ 1 τk h k V h k=1 k ξ i du = An/ = 1 V n If we are considering the Hamiltonian in orthogonal curvilinear coordinates for the case where n = 0 we have to use equation 38 in 39 then we get 43 dξ i = τ A 1 i h i i = ln V ξ i + A 1 3 τ k h k=1 k h k ξ i = 1 note that u has been changed to τ for n = 0 case So far we have dealt with RT systems in coordinates which were functions or some parameter u now we shall consider a situation where u is the third coordinates This is the case in ray and ray-centered coordinate systems where γ 3 u and q 3 u respectively An advantage of using such coordinate systems is that the conventional RT equations can be reduced from six to four To how this comes about consider equation 39 in Cartesian coordinates Set x 3 u so that the fist of these equations becomes dxi dx 3 = Hx p i When i = 3 in this equations we have dx3 dx 3 hence we conclude that H x must be of the form 44 H x x i p i = p 3 + H xr x i p I = 0 i = 1 3 and I = 1 = Hx p 3 = 1 and where H xr x i p I is the so called reduced Hamiltonian Since we can solve the eikonal equation for p 3 the reduced Hamiltonian can be evaluated by 45 H xr x i p I = p 3 To get the RT equations we note that the first of equations 39 has to be evaluated only for i = 1 because for i = 3 we have dx3 dx 3 = 1 which would be just 1 all along the ray The second one of equations 39 for i = 3 becomes dp3 dx 3 = Hx x 3 = 1 this is because of equation 44 Here also the derivative is just 1 all along the ray Therefore this one too has to be evaluated only for i = 1 Note however that p 3 is calculated by equation 45 Finally the third of equations 39 becomes H dx 3 = p 3 +p xr I p I this is because of Hx x 3 = 1 and equation 44 Thus we get our RT equations for a coordinate system where the third coordinate is a monotonously increasing parameter along

12 1 ERSE360 the ray These are 46 dx I dx 3 = HR p I dp I dx 3 = HxR x I H xr = p 3 + p I dx 3 p I As before to get the equation 46 in orthogonal curvilinear coordinates we first replace x i with ξ i and p i with τ i Equation for the reduced Hamiltonian corresponding to equation 44 then becomes 47 H ξ i τ i = τ 3 + H R ξ i τ I = 0 i = 1 3 and I = 1 and H R ξ i τ I is then given by 48 H R ξ i τ I = τ 3 Then the RT equations corresponding to equations 46 become 49 dξ I dξ 3 = HR τ I I dξ 3 = HR ξ I H R = τ 3 + τ I dξ 3 τ I Before leaving this section let us consider the general Hamiltonian from equation 37 for n = 1 in that case u ξ 3 s The reduced Hamiltonian becomes [ 1 H R ξ i τ I = τ 3 = h 3 V τ 1 h τ ] 1/ 50 1 h By inserting equation 50 in 49 we get the RT equations in orthogonal curvilinear coordinates that is 51 dξ I dξ 3 = 1 τ 3 h3 h I τ I I = τ 3 h 3 + h 3 dξ 3 h 3 ξ I τ 3 [ V h 1 τ1 h τ ξ I ξ I ξ I dξ 3 = h 3 V [ 1 V h 1 τ 1 + h τ ] ] 1/ = h 3 V τ Dynamic Ray Tracing Now we have gathered the tools necessary to understand DRT We shall look at the DRT first in ray-centered and then Cartesian coordinates 41 DRT in ray-centered Coordinates Calculation of the paraxial traveltimes are easier to do in the ray-centered coordinates Ray-centered coordinates are such that the third axis is the parameter which increases monotonously in the direction of the ray As we saw in the last section to derive a RT system for that case we need the reduced Hamiltonian Before going any further let us introduced some notation 5 v s = [V q 1 q s] q1=0q =0 v i s = [ V q 1 q s / q i ] q1=0q =0 v ij s = [ V q 1 q s / q i q j ]q 1=0q =0

13 ELEMENTS OF DYNAMIC RAY TRACING 13 Equations 5 are the velocity and its first and second partial derivatives along the ray that is q 1 q = 0 We already know the scale factors for ray-centered coordinates from equation 1 In our new notation h becomes 53 h = 1 + v 1 v I q I By using the scale factors for ray-centered coordinate system equation 1 and setting ξ 1 ξ ξ 3 to q 1 q s we get from equation 50 we get [ ] 1/ H R 1 54 q i τ I = τ 3 = h V q 1 q s τ 1 τ Similarly from equation 51 we get the RT equations in ray-centered coordinates that is 55a 55b dξ I = h τ I dξ 3 τ 3 [ I = h 1 dξ 3 τ 3 q I 1 V + τ 3 ] h h 3 q I Equations 55 are exact There is a problem however as far as numerical computation is concerned The rhs of equations 55b are zero along the ray that is when q 1 q = 0 but this is as a consequence of it being a difference of two terms which individually are not zero even when q 1 q = 0 Due to limited accuracy in numerical computations this difference might not always be zero as required theoretically This would introduce instability Since we are interested in the paraxial vicinity of the central ray we shall look for distances away from the central ray only up to the second order in q I that is in the so called quadratic vicinity Remember it was noted in the introduction that DRT is an approximate method this is where the approximation comes in First we do a Taylor series expansion of the velocity V about the central ray that is V = v + v I q I + 1 v IJq I q J With this expansion of V equation 53 for h and keeping only quadratic terms in q I our approximate reduced Hamiltonian becomes [ H R q i τ I = τ 3 = v v 1 v IJ q I q J 1 τ 1 + τ ] 56 Then from equations 49 the RT system becomes dq I = vτ I I = v v IJ q J and because τ i = h i p q i we get the final form for RT system in ray-centered coordinates that is 57 dq I = vpq I dp q I = v v IJ q J I J = 1 where the superscript q denotes the fact that the slowness components are in ray-centered coordinate system This is the Paraxial Ray Tracing PRT system in ray-centered coordinates We are a short way away from DRT now We need to calculate the rate of change of q I and p I given by the PRT equations 57 with respect to the rays themselves Since the rays can be identified by the first two axes of the ray coordinate system γ 1 γ We also note that since the third axis of ray coordinate system is monotonically increasing parameter which can be the arclength s This γ I and d means that to γ I this gives is d qi 58 γ K commute with each other Then we differentiate equations 57 with respect = v pq I γ K d p q I = v q J v IJ I J K = 1 γ K γ K

14 14 ERSE360 From equations 14 and 3 we know that q I γ K Q and pq I γ K P Then if we define another matrix V such that V IJ = v IJ = V q 1 q s / q I q J q 1=0q =0 equation 58 becomes 59 dq = vp dp = v VQ This the Dynamic Ray Tracing DRT system in ray-centered coordinates As we shall see the matrices P and Q calculated by DRT are essential in calculating paraxial traveltimes and geometrical spreading Geometrical spreading depen on the ray Jacobian which can be expressed in terms of Q according to equation 7 By differentiating the first of equations 59 with respect to s we can get a second order ODE for matrix Q only which is 60 v d Q v dq + VQ = 0 s Calculation of paraxial traveltimes in ray-centered coordinates involves a matrix M introduced in equation 6 A first order nonlinear ODE for M can be derived which is 61 dm + vm + v V = 0 4 DRT in Cartesian coordinates To get the DRT system in ray-centered coordinates we differentiated the PRT system also in ray-centered coordinates with respect to γ I along the ray Similarly to get DRT in Cartesian coordinates we differentiate equation 39 with respect to γ I H x depen on γ I implicitly that is through x i and p i For simplicity we shall take only one of the γ I say for I = 1 and write it as γ Then the derivative of the first of equations 39 x i γ for example is d x j du p x i x j γ + p x i p x j that slowness components are in Cartesian coordinates If we define 6 = H x xi Q x i γ u=const H x and we get the DRT in Cartesian coordinates which is p x j γ where the superscript x signifies the fact p x i γ u=cons P x i 63 where dq x i du dp x i du = Ax ij Qx j = Cx ij Qx j + B x ij P x j D x ij P x j 64 A x ij = H x p x i x j B x C x ij = H x x i x j ij = H x B x ij p x i p x j = H x x i p x j

15 Therefore for u σ we use H x = 1 coordinates that is 65 ELEMENTS OF DYNAMIC RAY TRACING 15 [ d dσ Qx i = P x i p x k px ] k 1/V and get the DRT in equations in Cartesian d dσ P x i = 1 x i x j 1 V Q x j 43 Paraxial traveltimes Now we shall look at traveltimes in the paraxial vicinity of the central ray These are called paraxial traveltimes As we can imagine ray-centered coordinates would give the simplest expression for paraxial traveltimes τ q 1 q s where q 1 and q are assumed to be small Of course on the central ray q 1 and q are zero Therefore the traveltime on the central ray is τ 0 0 s An equation for τ q 1 q s is obtained by its Taylor series expansion about a point say 0 0s on the ray only up to the second order in q I that is 66 τ q 1 q s = τ 0 0 s + qmq T q q 1 q T The matrix M can be written in terms of P and Q according to equation 6 It would be of practical importance to expand the traveltime in terms of Cartesian coordinates which is 67 τ x 1 x x 3 = τ x 10 x 0 x 30 + dxp x + dxm x dx T where dx x 1 x 10 x x 0 x 3 x 30 T and p x = τ The point x 10 x 0 x 30 is on the central ray and x 1 x x 3 in the paraxial vicinity Matrix can be evaluated in terms of P x and Q x with an equation analogous to 6 that is 68 M x = P x Q x 1 where matrices Q x and P x are given by equations 0 and 4 respectively P x and Q x can be calculated using equations Algorithm for paraxial traveltime Finally we are in a position to outline the steps required to calculate the paraxial traveltimes First in the ray-centered coordinates 1 Determine the central ray of the ray-centered coordinate system by solving RT equations 39 in Cartesian coordinates Determine the traveltimes and p along those rays 3 Solve for the vectors e I of the ray-centered coordinate system using equation 9 along the central-ray 4 Determine the propagation of matrices P and Q using equation 59 5 Calculate the paraxial traveltimes using equation 66 In Cartesian coordinates the steps are as follows: 1 Determine the central ray of the ray-centered coordinate system by solving RT equations 39 in Cartesian coordinates Determine the traveltimes 3 Determine the propagation of matrices P x and Q x using equation 65 4 Calculate the paraxial traveltimes using equation 67

16 16 ERSE360 D D C C Central Ray Paraxial Ray A=A N a Vectors N and t are the normals to the planes ABCD and A B C D respectively t B B dω τ τ+ b Ray tube dω Central Ray Figure 3 Perpendicular planes and ray tube 5 Amplitude continuation It was mentioned in the introduction that DRT can be used to calculated the geometrical spreading in this section we shall see how Geometrical spreading comes as a consequence of initial energy being distributed over the wavefront Since energy is conserved but the area of wavefront gets larger as the wave propagates same amount of energy gets spread over larger and larger area Energy at a point of the wavefront is related to the amplitude at that point therefore as energy per unit area is decreased so does the amplitude of the wave This suggests that geometrical spreading can be defined in terms of the area of wavefront To do that we should first understand the concept of ray tubes 51 Ray tubes Consider a central ray calculated by the standard RT equations in Cartesian coordinates We also take other rays in the paraxial vicinity of the central ray to make a bundle of rays as shown in Figure 3a This is called the ray tube The face of the ray tube represents area perpendicular to the direction of tangent to the central ray Therefore the face of the ray tube represents the plane of constant u where u is the monotonically increasing parameter along the central ray If u τ the face represents a plane tangent to the wavefront This is because wavefront is the surface of constant τ For u s or u σ the plane of constant u will in general be different from the wavefront and therefore will not be perpendicular to the ray-tangent direction p Figure 3b However we can take the dot product of the vector area element with the tangent to the central ray that is t = V p see equation 31 Since we are interested in the area of the wavefront only we shall take the u τ in the analysis below which would mean that the face of our ray tube will always be tangent to the wavefront As we shall see in order to calculate the amplitudes along a ray we need to evaluate the divergence of the slowness vector p that is p = τ We recall from elementary calculus that divergence is nothing but the net outward flux per unit volume Therefore we select a differential volume element from our ray tube Figure 3a and write the divergence of p as 69 1 p = lim D 0 D A surf p da surf where D = δxδyδz is the volume of the ray tube and A surf is the boundary of D which is the surface A surf is the vector surface area of the ray tube whose direction is always outward normal

17 ELEMENTS OF DYNAMIC RAY TRACING 17 to the surface and magnitude is the surface area Surface area of the ray tube can be divided into three parts areas of the two faces and area of the side of the ray tube Since the sides of the ray tube are the paraxial rays themselves the net outward flux of p from the sides is zero that is A sides p da sides = 0 Faces of our ray tube are tangent to the wavefront therefore the vector area elements of the face da face is nothing but dω t where dω is given by equation 3 Therefore equation 69 becomes 70 p = 1 gτ 1 Jdγ 1 dγ We know that du = g u therefore gτ 1 71 Therefore we have 7 p = 1 V J τ = p = 1 V J [ dω p t τ+ dω p t τ ] = V Also p t = V 1 Then equation 70 becomes [ JV 1 τ+ JV 1 ] τ d J = 1 V J d J V 5 Geometrical spreading Now let us see how the amplitude of the wave changes along a ray Obviously we will need the transport equation to do that The transport equation for an acoustic isotropic medium with variable density ρ is [Červený 001 ] 73 τ Ψ ρ + Ψ ρ τ = 0 where Ψ is the amplitude along the ray We know that p = τ = V 1 d Ψ t Also is nothing but the component of gradient in the direction tangent to the ray Therefore we can write d Ψ Ψ ρ = t ρ With these relations equation 73 can be written as d Ψ + V Ψ τ = 0 ρ ρ 74 By multiplying equation 73 throughout by Ψ ρ we get d Ψ Ψ 75 + V τ = 0 ρ ρ Ψ Let ρ Φ then the solution of equation 75 is 76 Φ s = Φ s 0 e R s s 0 V τs where τ depen on s implicitly that is through x y z equation 7 to get 77 Φ s = Φ s 0 e R s s 0 V J d J V ρ Then we substitute for τ from or by evaluating the integral in the exponent [ ] J s0 /V s 0 J s0 V s 78 Φ s = Φ s 0 exp ln = Φ s 0 J s /V s J s V s 0

18 18 ERSE360 Ψ Now we restore ρ Φ and get 79 Ψ s = ρ s Ψ s 0 ρ s0 Ψs Therefore the amplitude at a point s on a ray is ρs 80 J s0 V s J s V s 0 ρ s V s J s 0 Ψ s = Ψ s 0 ρ s 0 V s 0 J s Because the quantities which are functions of the initial point s 0 remain constant along the ray we can gather them into one constant that is C s 0 Ψ s 0 Js0 ρs 0V s 0 Then equation 80 becomes ρ s V s 81 Ψ s = C s 0 J s This is the equation of the amplitude variation along a ray and it is inversely proportional to the square root of ray Jacobian Therefore in analogy to the conventional geometrical spreading which is inversely proportional to the radial distance here we define it as J Finally we would like to express equation 81 in terms of quantities which can be directly evaluated by DRT To do that we note that from equations 7 and 9 J = J s = det Q ˆ x From equation 1 we have det ˆQ x s = det Ĥ s det ˆQ s along the ray But due to equations 17 and 19 det ˆQ x s = det Q s Therefore we can way that J = det Q s Thus our final expression for amplitude continuation along a ray is 8 ρ s V s Ψ s = C s 0 det Q s The central ray can be obtained by conventional RT Therefore ρ s and V s along the central ray are known Matrix Q s can be calculated along the ray by DRT equations 59 a Coustic point of the first order b Coustic point of the second order Figure 4 Coustic points 53 Coustics We do not intend to cover coustics in these notes but a few remarks would not be out of place In analyzing the ray tubes we assumed that the rays in the tube will never

19 ELEMENTS OF DYNAMIC RAY TRACING 19 cross each other This means that we assumed that the area of the face of our ray tube will never become zero But in reality the rays do cross each other When the rays intersect along a line the point on the rays which intersect that line is called coustic point of the first order Figure 4a and when they intersect each other at a point it is called the coustic point of second order Figure 4b It is clear from Figure 4 that as coustics are formed the area of our ray tube goes to zero that is Ω = Jdγ 1 dγ = 0 This means that at coustics J = det Q = 0 From equations 81 and 8 we can see that at coustics amplitude goes to infinity Therefore coustics are characterized by vanishing of the Jacobian For further reading see [Červený 001] 6 Remarks DRT is a much richer subject than what could be presented here Some of the tricks like rotation of the Frenet frame to fit with the ray-centered coordinate system were not discussed in these notes The purpose here was to give the most basic elements of DRT Most of the derivations in these notes have been taken from [Červený 001] 7 Exercise 1 Prove equation 19 Hint use equation 10 Prove equation 5 3 Prove equation 6 4 Derive equation 51 in ray-centered coordinates for n = 0 1 in equation 37 Is there any difference or not explain 5 Show that rhs of equations 55b are zero along the ray q 1 q = 0 6 Derive equation 61 7 Derive the DRT equations in Cartesian coordinates for u s and u τ 8 Why does equation 66 not contain the linear term in q References [Arfken and Weber 005] Arfken G B and Weber H J 005 Mathematical metho for physicists Sixth Edition [Slawinski 003] Slawinski M A 003 Seismic waves and rays in elastic media 34 [Thomas and Finney 1996] Thomas G B and Finney R L 1996 Calculus and analytic geometry Sixth Edition [Červený 001] Červený V 001 Seismic ray theory