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
- Acknowledgements

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
inversion.

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)}*

T^{(i)} = T^{(i)}_{wave}
- T^{(i)}_{mod}
.
| (1) |

T = (T)^{(1)}, T^{(2)}, ... ,
T^{(N)}^{T}
,
| (2) |

The dependence of arrival-time residuals

R = (R)^{(1)}, R^{(2)}, ... ,
R^{(N)}^{T}
| (5) |

X = (dx^{(1)}, dx^{(2)},
dx^{(3)},
dx^{(4)}=dt)^{T}
| (6) |

R(X) = T - P X
,
| (7) |

| (8) |

Since the exact hypocentral position is unknown, an approximate
hypocentral position is determined by minimizing the objective
function

y(X) = ^{1}/_{2}
R(X)^{T} C^{-1}
R(X)
.
| (9) |

C = C_{wave} + C_{mod}
| (10) |

Objective function (9) achieves its minimum for

X = (P^{T}C^{-1}P)^{-1} P^{T}C^{-1} T
| (11) |

R = T -
P (P^{T}C^{-1}P)^{-1}
P^{T}C^{-1} T
.
| (12) |

Operator **E** is a * projection operator * if
**E****E**=**E**.
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

P_{S} = P (P^{T}C^{-1}P)^{-1} P^{T}C^{-1}
| (13) |

P_{R} = 1 - P_{S}
| (14) |

T = S + R
,
| (15) |

S = P_{S} T and
R = P_{R} T
.
| (16) |

Inserting (15), (16), (14), and (13) into (11),
we see that final mislocation (11) is fully caused by vector **S**,

X = (P^{T}C^{-1}P)^{-1} P^{T}C^{-1} S
,
| (17) |

Covariance matrix **C**_{S} =
**P**_{S} **C** **P**_{S}^{T}
corresponding to **S** is

C_{S} =
P (P^{T}C^{-1}P)^{-1}
P^{T}
,
| (18) |

C_{R} = C - C_{S}
.
| (19) |

C_{SR} = 0
,
| (20) |

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.

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

P'_{S} =
P (P^{T}C'^{-1}P)^{-1}
P^{T}C'^{-1}
| (21) |

P'_{S} =
P (P^{T}C'^{-1}P)^{-1}
P^{T}C'^{-1}
| (21) |

P'_{R} = 1 - P'_{S}
.
| (22) |

C'_{S}
= P'_{S} C P'_{S}^{T}
,
| (23) |

C'_{R}
= P'_{R} C P'_{R}^{T}
,
| (24) |

C'_{SR}
= P'_{S} C P'_{R}^{T}
.
| (25) |

Using identity

(P'_{S}-P_{S}) C
P_{S}^{T} = 0
,
| (26) |

C'_{S}-C_{S}
= (P'_{S}-P_{S}) C
(P'_{S}-P_{S})^{T}
,
| (27) |

C'_{R}-C_{R}
= (P'_{S}-P_{S}) C
(P'_{S}-P_{S})^{T}
- (P'_{S}-P_{S}) C
- C (P'_{S}-P_{S})^{T}
,
| (28) |

C'_{SR}-C_{SR}
= -(P'_{S}-P_{S}) C
(P'_{S}-P_{S})^{T}
+ (P'_{S}-P_{S}) C
.
| (29) |

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:

(a)
Operator **P'**_{S} projects on the same subspace as
**P**_{S},
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}-**P**_{S}. For reasonable approximations
**C'** of **C**, i.e., for small
**P'**_{S}-**P**_{S},
covariance matrix **C'**_{S} remains quite similar to **C**_{S},
whereas for large **P'**_{S}-**P**_{S}, 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**.
In particular:
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}-**P**_{S} and increases for
large **P'**_{S}-**P**_{S}, see (27) and (17).
The accuracy of the hypocentre determination can
no longer be estimated because of the unknown correct value of **C**.

(b)
The projection subspace of operator **P'**_{R}
is rotated with respect to the projection subspace of **P**_{R}.
Residuals **R** thus may have to be rotated from the unknown projection
subspace of **P**_{R} to the unknown projection subspace of
**P'**_{R}, see terms (**P'**_{S}-**P**_{S}) **C** and
**C** (**P'**_{S}-**P**_{S})^{T} in (28) and (29),
otherwise they are changed with a random orientation.
For small **P'**_{S}-**P**_{S}, the size of the residuals
remains approximately the same from the statistical point of view,
whereas for large **P'**_{S}-**P**_{S}, the
arrival-time residuals become larger from the statistical point of view,
see term
(**P'**_{S}-**P**_{S}) **C**
(**P'**_{S}-**P**_{S})^{T} in (28).

(c) 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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