principles and applications of microearthquake networks

4.4. Conipu titig Tru vel Time and Derivu tives 101
of point A be (0, zJ, and that for point B be (A, 0). For simplicity, we will
label the points along the r) axis by their q coordinates; for example, point
Al has the coordinates ( Al, 0). We will also label the points along the line z
= h, by their +I coordinates: for example, point ql has the coordinates ( ql,
hl). We define 5 = zA - h,. Let the d’s be the angles measured from the
negative z direction to the lines joining point A with the appropriate points
along the line z = h, (see Fig. 24).
To find by an iterative procedure an angle 4 whose associated ray path
will reach the station, we first consider lower and upper bounds ($1 and
&) for the angle 4. If we join points A and B by a straight line, then
intersects the line z = h, at point ql such that its q coordinate is given by
q1 = h( ul, this ray A X will
emerge to the surface at a point with r) coordinate of
-
A1, which is always
less than A. Thus the angle associated with the ray Aql can be selected
as the lower bound of 4. To find the upper bound of 4, we let y2 be the
point directly below the station and - on the line z = h,. i.e., qz = A. Again
by Snell’s law and v2 > ul, the rayA q2 will emerge to the surface at a point
with coordinate of A*, - which is always greater than A. Thus the angle 42
associated with the ray Av2 can be selected as the upper bound
-
of 4.
To start the iterative procedure, we consider a trial ray A?* and its
associated trial angle & (see Fig. 24) such that q* is approximated by
(4.101) 6, = arctan(q,/ E, we select a new +* and repeat the
same procedure until the trial ray emerges close to the station. To perform
the iteration, we must consider the following two cases:
-
(1) If A* > A, the new & is chosen between the two rays Aql and
AX. To do this, in Eq. (4.100), the current values for A, and q*
replace A2 and q2 respectively, and a new value for q* is com-
(2)
puted. - -
If 3% < A, the new & is chosen between the rays A?* and AqZ. To
do this, in Eq. (4.100), the current values of A* and q, replace A1
and ql, respectively, and a new value for q* is computed. In both
cases, a new value for c$* can be obtained from Eq. (4.101).