Propagating local circular wavefronts

A wavefront can be locally propagated by tracing local rays. Figure  shows a cell in a triangular grid. At two corners of the triangular cell, the attributes of the wavefront are known to be

$${\bf \eta}_a=\{t_a,(u_a,w_a),R_a,\gamma_a,J_a\}$$

and

$${\bf \eta}_b=\{t_b,(u_b,w_b),R_b,\gamma_b,J_b\},$$

respectively. The wavefront attributes at the third corner,

$${\bf \eta}_o=\{t_o,(u_o,w_o),R_o,\gamma_o,J_o\}$$

can be determined through a procedure of propagating a local wavefront that passes the line boundary between the points a and b.

 

vpwftprg1 Figure 4 Local wavefront propagation when the wavefront attributes at the points a and b are associated with a common wavefront. By tracing two local rays, the wavefront attributes at the point o is determined from the wavefront attributes at the points a and b jointly.

The procedure begins with local ray-tracing from two corners of the triangular cell, where the attributes of wavefronts are known. As shown in Figure , the ray a shoots from the point a and in the direction of the gradient vector (u_a,w_a), and the ray b shoots from the point b and in the direction of the gradient vector (u_b,w_b). These two rays form a ray-tube that guides the propagation of the local wavefront passing between the points a and b. If a point on the local wavefront follows the wavefront propagation and reaches the third corner of the triangle, it must cross the boundary ab and propagate through the triangular cell. Therefore, the two local rays are traced as if the straight-line boundary ab extends out at both ends and the velocity field within the triangular cell extends out with the velocity gradient being unchanged. The wavefront attributes at any point on the rays can be calculated by using the formulas presented in the last section.

As shown in Figure , the local rays a and b are traced to the points $a^\prime$ and $b^\prime$, respectively. At these two points, the normal distances from the third corner of the triangular cell to the rays are achieved. Let q_a and q_b denote the two normal distances, and define

where (x_o,z_o) are the coordinates of the third corner of the triangular cell, $(v_{a^\prime}u_{a^\prime},v_{a^\prime}w_{a^\prime})$ and $(v_{b^\prime}u_{b^\prime},v_{b^\prime}w_{b^\prime})$ are the unit vectors tangential to the rays at the normal points $(x_{a^\prime},z_{a^\prime})$ and $(x_{b^\prime},z_{b^\prime})$, respectively. With this definition, the normal distance is positive when a ray is on one side of the point o, and negative when a ray is on the other side. To ensure that the attributes of the wavefront at the point (x₀,z₀) is computed by propagating the local wavefront passing the line boundary between the points a and b, it is required that the point (x_o,z_o) is within the ray tube formed by the rays a and b. This condition is satisfied when $q_a\le 0$ and $q_b \ge 0$.

Because the normal points $a^\prime$ and $b^\prime$ are on the rays, the attributes of the wavefront at these points can be computed analytically. Let $R_{a^\prime}$ be the curvature radius of the wavefront at the normal point $a^\prime$.Then, the curvature center of the wavefront at the point is

$$\left\{ \begin{array} {lll} x_{a0} & = & x_{a^\prime}-v_{a^\... ...\prime}-v_{a^\prime}w_{a^\prime}R_{a^\prime}.\end{array}\right.$$ (12)

Similarly, the curvature center of the wavefront at the normal point $b^\prime$ is

$$\left\{ \begin{array} {lll} x_{b0} & = & x_{b^\prime}-v_{b^\... ...\prime}-v_{b^\prime}w_{b^\prime}R_{b^\prime},\end{array}\right.$$ (13)

where $R_{b^\prime}$ is the curvature radius of the wavefront at the point.

The local wavefront within the ray tube formed by the rays a and b is considered to be continuous if one of the following two sets of conditions are satisfied:

1.

$R_{a^\prime} \gt 0$, $R_{b^\prime} \gt 0$ and $w_{a^\prime}u_{b^\prime}-u_{a^\prime}w_{b^\prime} \gt 0$.

2.

$R_{a^\prime} < 0$, $R_{b^\prime} < 0$ and $w_{a^\prime}u_{b^\prime}-u_{a^\prime}w_{b^\prime} < 0$.

The first set of conditions define a divergent local wavefront and the second set a convergent local wavefront. Figure  displays examples of these two cases. If neither of the two sets of conditions hold, the local wavefront is considered to be discontinuous. Figure  displays two examples.

 

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Figure 5 Continuous local wavefronts. Left: a divergent local wavefront. Right: a convergent local wavefront.

 

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Figure 6 Two examples of discontinuous local wavefronts.

When the local wavefront is continuous, the attributes at the points a and b are associated with a common wavefront. Therefore, the curvature center of the wavefront at the point o is estimated by averaging the curvature centers of the wavefront at the points $a^\prime$ and $b^\prime$ as follows:

$$\left\{ \begin{array} {lll} x_{o0} & = & \alpha x_{a0}+\beta... ...\ \\ z_{o0} & = & \alpha z_{a0}+\beta z_{b0},\end{array}\right.$$ (14)

where $\alpha$ and $\beta$ are weighting coefficients determined by the normal distances, as follows:

 

$$\left\{ \begin{array} {lll} \alpha & = & \displaystyle{\vert... ...q_a\vert \over \vert q_b\vert+\vert q_a\vert}\end{array}\right.$$ (15)

The curvature radius of the wavefront at the point is then computed by

 

$$R_{o}=\hbox{sgn}(R_{a^\prime}+R_{b^\prime})\sqrt{(x_o-x_{o0})^2+ (z_o-z_{o0})^2},$$ (16)

and two components of the traveltime gradient by

 

$$\left\{ \begin{array} {lll} u_o & = & \displaystyle{(x_o-x_{... ... = & \displaystyle{(z_o-z_{o0}) \over v_oR_o}\end{array}\right.$$ (17)

where v_o is the velocity at the point o.

The wavefront at the point o is approximated by a local circular wavefront of radius R_o and centered at (x_o0,z_o0). As shown in Figure , the local circular wavefront at the point o intersects with the ray a at the point $a^{\prime\prime}$and the ray b at the point $b^{\prime\prime}$. Because these two points are on the rays, the attributes of wavefronts can be calculated analytically. Let $t_{a^{\prime\prime}}$ and $t_{b^{\prime\prime}}$ be the traveltimes at two intersections. Because both the point $a^{\prime\prime}$ and the point $b^{\prime\prime}$ are on the local circular wavefront defined at the point o, the weighted average of $t_{a^{\prime\prime}}$ and $t_{b^{\prime\prime}}$ is used to estimate the traveltime at the point o, that is

 

$$t_o = \alpha t_{a^{\prime\prime}}+\beta t_{b^{\prime\prime}}$$ (18)

where $\alpha$ and $\beta$ are defined in equation (15). It can be shown that in a linearly varying velocity field the geometrical spreading factor function J varies along a wavefront in proportion to the velocity. Therefore, the estimate of the geometrical spreading factor J_o from the attributes at the point $a^{\prime\prime}$ is $v_oJ_{a^{\prime\prime}}/v_{a^{\prime\prime}}$and from point $b^{\prime\prime}$ is $v_oJ_{b^{\prime\prime}}/v_{b^{\prime\prime}}$. The average of two gives

 

$$J_o = \alpha {v_o \over v_{a^{\prime\prime}}}J_{a^{\prime\prime}}+ \beta {v_o \over v_{b^{\prime\prime}}}J_{b^{\prime\prime}}$$ (19)

The last parameter to be estimated is the take-off angle of the ray that reaches the point o. It can be shown that, in two-dimension,

$$J={\partial \gamma \over \partial n},$$

where n is the distance normal to the ray. Therefore, we have

where $\bar{J}$ is the average geometrical spreading factor over the arc length $\Delta arc$.If the average of J_o and $J_{a^{\prime\prime}}$ is used to approximate the $\bar{J}$ over arc_a and the average of J_o and $J_{b^{\prime\prime}}$ over arc_b, then $\gamma_o$ is estimated by the weighted averaging as follows:

 

$$\gamma_o = \alpha(\gamma_{a^{\prime\prime}}-{2arc_a \over J_... ...a_{b^{\prime\prime}}-{2arc_b \over J_o+ J_{b^{\prime\prime}}}).$$ (20)

With equations (16) to (20), we can compute the attributes of the wavefront at the third corner of the triangular cell.

When the local wavefront is discontinuous, the attributes at the points a and b are associated with different wavefronts. Therefore the local wavefront at each point should be propagated individually. Figure  shows an example in which the local wavefront at the point b is propagated. In this case, the curvature center of the wavefront at the point o is assumed to coincide with the curvature center of the wavefront at the point $b^\prime$, that is, x_o0=x_b0 and z_o0=z_b0. The curvature radius is then

$$R_{o}=\hbox{sgn}(R_{b^\prime})\sqrt{(x_o-x_{o0})^2+(z_o-z_{o0})^2}.$$ (21)

The formula for computing the components of the traveltime gradient at the point o is identical to equation (17). The traveltime t_o, geometrical spreading factor J_o and take-off angle $\gamma_o$ are computed by simply setting $\alpha=0$ and $\beta=1$ in equations (18), (19) and (20).

 

vpwftprg2 Figure 7 The local wavefront propagation when the wavefront attributes at the points a and b are associated with different wavefronts. The wavefront attributes at the point o is determined from the wavefront attributes at each point (the point b as an example) separately.

There are two special cases in which the rays a and b do not form a ray-tube because of multiple arrivals. Figure  shows examples of two such cases. In both cases, the rays a and b converge with each other, that is

$$w_{a^\prime}u_{b^\prime}-u_{a^\prime}w_{b^\prime} < 0.$$

The first case has q_a < 0 and q_b < 0. The ray a guides the propagation of the local wavefront passing between the points a and b, whereas the ray b guides the propagation of another local wavefront. Therefore, only the ray a is used to compute the attributes of the wavefront at the point o. The second case has q_a > 0 and q_b > 0. The ray b, instead of the ray a, is used in the computation.

The wavefront may be diffracted at the points a and b because of the changes of the velocity gradients in the adjacent triangular cells. Figure  shows an example of diffraction at the point b. Two rays that are traced with the same initial conditions at the point b split towards two sides. The local wavefront corresponding to diffraction is defined between two rays. The attributes of the diffracted wavefront are computed by tracing the local ray connecting the points b and o.

 

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Figure 8 Two examples of local wavefronts of multiple arrivals.

 

vpwftprg3 Figure 9 An example of a diffracted local wavefront at the point b.