Arrival-time residuals and hypocentre mislocation
It is shown that
the arrival-time residuals and hypocentre mislocation are two mutually
independent consequences of the inaccurate seismic model and inaccurately
measured arrival times. The minimum residuals resulting from the
kinematic hypocentre determination contain no information on
the accuracy in determining the hypocentre position.
Hypocentre determination, seismic model, accuracy, arrival-time residuals.
- 1 Introduction
- 2 Exact hypocentral position and time
- 3 Approximate hypocentral position and time
- 4 Independence between arrival-time residuals
and hypocentre mislocation
- 5 Consequences of incorrect data covariance matrix
- 6 Conclusions
Iterative linearized procedures of kinematic hypocentre determination
are, as a rule, based on minimization of residuals of measured arrival
times, with respect to unknown hypocentral time plus calculated travel
times. This minimization is, of course,
correct (if the linearization leads to the correct minimum of the objective
function), but tempts seismologists to erroneously consider the error
of the final hypocentral position to be proportional to the resulting
minimum arrival-time residuals.
The aim of this paper is to demonstrate that the resulting minimum
arrival-time residuals carry information pertinent to the accuracy
of the model,
but no information on the accuracy of the hypocentre determination.
As a consequence, if the residuals are decreased by adjusting the model
(without additional information, using the arrival times not only for
hypocentre determination but also to update the model), the accuracy
of the corresponding hypocentral position cannot not be improved,
but is often made worse.
Assume, for the kinematic hypocentre determination,
that the arrival times of some elementary waves at each of
several receivers are measured. Let us denote
the total number of measured arrival times N.
For the sake of simplicity, we shall refer to the corresponding N
arrivals as individual "waves", although some of the arrivals may
correspond to the same elementary wave recorded at different receivers.
For the sake of simplicity, we shall consider only a Gaussian
error distribution, in agreement with the conventional linearized
2 Exact hypocentral position and time
Let us assume, in this section, that the hypocentre is located
at its exact position and at exact hypocentral time.
Let us emphasize that the exact hypocentral position
and time are unknown and cannot be determined.
Denote by T(i)mod the difference between
synthetic and exact travel times caused by the inaccurate seismic model
of the medium, and by T(i)wave
the difference between inaccurately measured
and exact arrival times of the ith "wave" used to determine
the hypocentre. Then the difference between measured and calculated
arrival times of the ith "wave" is
These differences form an N-dimensional vector
T(i) = T(i)wave
where N is the number of arrival times used for hypocentre determination.
Superscript T denotes transpose.
T = (T(1), T(2), ... ,
3 Approximate hypocentral position and time
The dependence of arrival-time residuals
on hypocentre mislocation
R = (R(1), R(2), ... ,
is, in a linear approximation,
X = (dx(1), dx(2),
pa(i), a=1,2,3,4 being the derivatives
of the ith synthetic arrival time with respect to the hypocentral
coordinates and hypocentral time (i.e., the ith space-time
slowness vector at the hypocentre). The use of linear approximation
(7) is correct here, because expansion (7) is centred at the exact
hypocentral position, X representing the error. The possible
quadratic term in Taylor expansion (7) would then be proportional
to the square of the error and should thus be negligible for reasonable
| P =
Since the exact hypocentral position is unknown, an approximate
hypocentral position is determined by minimizing the objective
y(X) = 1/2
is the data covariance matrix describing the standard deviations
of both the measured arrival times and synthetic travel times.
Objective function (9) achieves its minimum for
and the resulting residuals are
X = (PTC-1P)-1 PTC-1 T
Let us now remind the reader that both final mislocation (11)
and resulting minimum residuals
(12) are due to the unknown error T in measured arrival times
and synthetic travel times.
R = T -
4 Independence between
arrival-time residuals and hypocentre mislocation
Operator E is a projection operator if
As a consequence, if E is a projection operator, 1-E is
a projection operator, too. Here 1 is an identity operator.
Introducing two complementary projection operators
PS = P (PTC-1P)-1 PTC-1
[see (12)], the unknown error vector T may be decomposed as
with the two components
Vector S contains those parts of arrival-time errors (2)
which were eliminated by misplacing the hypocentre in space and time.
S = PS T and
R = PR T
Inserting (15), (16), (14), and (13) into (11),
we see that final mislocation (11) is fully caused by vector S,
and is independent of the resulting minimum arrival-time residuals R.
Similarly, the arrival-time residuals R are independent
of vector S.
X = (PTC-1P)-1 PTC-1 S
Covariance matrix CS =
PS C PST
corresponding to S is
see (13), and covariance matrix CR =
PR C PRT
corresponding to R is
The cross-variance CSR =
PS C PRT
between S and R is zero,
if the data covariance matrix C in (9) is chosen correctly.
The cross-variance between X and R is then zero too.
Only if an incorrect matrix C in (9) were chosen, would there be
a statistical relation between the residuals and mislocation:
the statistical expectation of both of them would be larger than
for the correct value of C in (9).
The incorrect choice of data covariance matrix C is discussed
in more detail in the next section which serves as an example and
may be skipped by readers not detail-oriented.
5 Consequences of incorrect data covariance matrix
Assume that the correct data covariance matrix C in (9) is not known
and is replaced by incorrect estimate C' during the hypocentre
determination procedure, and thus equations (11) to (17)
are also affected in this way.
For instance, projection matrices (13) and (14) are replaced by
Variances (18) to (20) are then replaced by
Let us emphasize that if data covariance matrix C is known, variances
CS, CR and CSR
may be determined
using (18) to (20), whereas if the correct data covariance matrix C
is not known and is replaced by C', variances (23) to (25)
cannot be determined.
the differences of variances (23) to (25)
with respect to variances (18) to (20) may be expressed as
PST = 0
= (P'S-PS) C
= (P'S-PS) C
- (P'S-PS) C
- C (P'S-PS)T
= -(P'S-PS) C
+ (P'S-PS) C
Thus, if the correct data covariance matrix C
is replaced by its incorrect
approximation C', the results of the kinematic
hypocentre determination are changed in the following way:
Operator P'S projects on the same subspace as
generated by columns of matrix P defined by (8).
Vector S thus still belongs to the same 4-D subspace.
Vector S may be decreased or increased with the same probability, depending
on difference P'S-PS. For reasonable approximations
C' of C, i.e., for small
covariance matrix C'S remains quite similar to CS,
whereas for large P'S-PS, covariance matrix C'S
increases, see (27).
The same applies to mislocation vector X because
mapping (17) of S on X is independent of the choice of C.
The resulting hypocentral position and time are shifted in an unknown
direction and with the same probability of both orientations,
i.e., the mislocation
may be decreased or increased with the same probability. The mislocation
thus remains nearly the same from the statistical point of view for small
difference P'S-PS and increases for
large P'S-PS, see (27) and (17).
The accuracy of the hypocentre determination can
no longer be estimated because of the unknown correct value of C.
The projection subspace of operator P'R
is rotated with respect to the projection subspace of PR.
Residuals R thus may have to be rotated from the unknown projection
subspace of PR to the unknown projection subspace of
P'R, see terms (P'S-PS) C and
C (P'S-PS)T in (28) and (29),
otherwise they are changed with a random orientation.
For small P'S-PS, the size of the residuals
remains approximately the same from the statistical point of view,
whereas for large P'S-PS, the
arrival-time residuals become larger from the statistical point of view,
(P'S-PS)T in (28).
A statistical dependence arises between the residuals and hypocentre
mislocation, see (29), however it is unknown.
The arrival-time residuals and hypocentre mislocation are two mutually
independent consequences of the inaccurate seismic model and inaccurately
measured arrival times. The resulting minimum residuals contain
no information regarding the accuracy in determining the hypocentre position.
This conclusion has been derived for reasonably small mislocation
vectors X, but is even more valid in cases
of larger mislocations or non-unique solutions.
This study has been motivated by discussions with
Tomas Fiser of the Institute of Rock Structure and Mechanics,
and Frantisek Hampl, Josef Horalek, Ivan Psencik,
and Jan Sileny of the Geophysical Institute.
The research has been partially supported
by the Grant Agency of the Czech Republic under Contract 205/95/1465,
and by the Grant Agency of the Academy of Sciences of the Czech
Republic under Contract 346110.
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PAGEOPH, 148 (1996), 337-342.
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