principles and applications of microearthquake networks

6.1. Determination of Origin Time and Hypocenter 13 1
tion, and the z axis is along the vertical direction pointing downward. Any
position on earth may be specified by latitude 6, longitude A, and elevation
above sea level h. If the position is between 70"N and 70"s in latitude, it
may be converted to (x, y, z) with respect to the coordinate origin at
latitude bo, longitude A,, and elevation at sea level by a method described
by Richter (1958, pp. 701-703, as
(6.2)
x = 60*A(h - A,), y = 60-B($ - 40)
i = -0.001h
where x, y, and z are in kilometers, A. A,, 4, and 4, are given in degrees,
and h is given in meters. Values for A and B may be derived from Woodward
(1929)
(6.3)
A ~(1.8553654
+ 0.0062792 sin2@ + 0.0000319 sin4@)cos@
B = 1.8428071 + 0.0187098 sin2@ + 0.0001583 sin4@
where @ = $(+ + 4,) and is given in degrees.
In the earthquake location problem, we are concerned with a fourdimensional
space: the time coordinate t, and the spatial coordinates x, y,
and z. A vector in this space may be written as
(6.4)
x = (t, .Y, y. z)T
where the superscript T denotes the transpose. In Chapter 5, we used x to
denote a vector in an n-dimensional Euclidean space in which the coordinates
are x,. x2, . . . , x,. Since it is common to denote the spatial coordinates
by x, y, and i, we have changed our notation here. Furthermore, the
subscript k of the coordinates is now used to index the observation made
at the kth station.
To locate an earthquake using a set of arrival times 7k from stations at
positions (xk, yk, zk), k = 1, 19 7 . . . . rn. we must first assume an earth
model in which theoretical travel times Tk from a trial hypocenter at
position (x", y* , z* 1 to the stations can be computed. Let us consider a
given trial origin time and hypocenter as a trial vector x* in a fourdimensional
Euclidean space
(6.5) x* = (I*. .v*. ),* .-*)'I.
Theoretical arrival time fk from x* to the kth station is the theoretical
travel time Tk plus the trial origin time t", or
(6.6) tk(X*) = Tk(x*) + t*
Strictly speaking, Tk does not depend on t", but we express Tk as ak(x")
for convenience in notation.
for
k = 1, 2. . . . , rn