Geoscience Reference
In-Depth Information
Appendix 4
The least-squares method
In geophysics it is often useful to be able to fit straight lines or curves to data (e.g., in
radioactive dating and seismology). Although the eye is a good judge of what is and is not a
good fit, it is unable to give any numerical estimates of errors. The method of least squares
fills this need.
Suppose that t 1 ,...,
t n are the measured values of t (e.g., for travel times in seismology)
corresponding to values x 1 ,...,
x n of quantity x (e.g., distance). Assume that the x values are
accurate but the t values are subject to error. Further assume that we want to find the particular
straight line
t
=
mx
+
c
(A4.1)
which fits the data best. If we substitute the value x = x i into Eq. (A4.1), the resulting value of
t might not equal t i . There may be some error e i :
(A4.2)
e i
=
mx i
+
c
t i
In the least-squares method, the values of m and c are chosen so that the sum of the squares of
the errors e i is least. In other words, i = 1 e i is minimized, where
n
n
e i
t i ) 2
(A4.3)
=
( mx i
+
c
i
=
1
i
=
1
To minimize this sum, it must be partially differentiated with respect to m , the result equated
to zero and the process repeated for c . The two equations are then solved for m and c :
n
e i
0
=
m
i = 1
t i ) 2
n
=
( mx i
+
c
m
i
=
1
n
=
2 x i ( mx i
+
c
t i )
i = 1
n
n
n
x i
(A4.4)
= 2 m
+ 2 c
x i 2
x i t i
i
=
1
i
=
1
i
=
1
636
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