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