Journal of Geodynamics 47 (2009) 39–46
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Journal of Geodynamics
journal homepage: http://www.elsevier.com/locate/jog
Methods of determining weight scaling factors for geodetic–geophysical
joint inversion
Caijun Xu a,b,∗ , Kaihua Ding a , Jianqing Cai b , Erik W. Grafarend b
a School of Geodesy and Geomatics, Key Laboratory of Geo-space Environment and Geodesy, Ministry of Education, Wuhan University,
129 Luoyu Road, Wuhan 430079, China
b Institute of Geodesy, University of Stuttgart, Geschwister-Scholl-Str. 24D, D-70174 Stuttgart, Germany
a r t i c l e
i n f o
Article history:
Received 18 January 2008
Received in revised form 13 June 2008
Accepted 14 June 2008
Keywords:
Geodetic–geophysical joint inversion
Weight scaling factor
Helmert method of variance components
estimation
Cross validation test method
a b s t r a c t
Geodetic–geophysical joint inversion is a hybrid inversion of different types of geodetic data, together with
geophysical or seismic, geological data. In the joint inversion, weight scaling factors of different datasets
are of vital importance and should be fixed properly. This paper aims to analyze the general weight scaling
factor fixing methods and to study their impacts on joint inversion. The result, validated and evaluated
by the cross validation test method, showed that it is not prudent to fix the inversion parameter only by
considering the objective function to be a minimum and that the parameter should be determined by the
actual circumstances. At last, a more reliable inversion result was obtained by using the Helmert method
of variance components estimation (VCE) for the fixing of weight scaling factor.
© 2008 Published by Elsevier Ltd.
1. Introduction
Geodetic–geophysical joint inversion makes use of various data
to extract common information, including geodetic, seismic, geological and geophysical data. The theory of geodetic–geophysical
joint inversion is based on Backus–Gilbert theory (Backus and
Gilbert, 1967, 1968, 1970). In 1977, Matsu’ura inverted for fault
parameters by using geodetic data, and brought forward the concept of geodetic inversion (Matsu’ura, 1977a,b). Since then, a rapid
and great development has occurred, with the continuum, linear form and single dataset replaced by the discrete, non-linear
form and various datasets. Examples include the joint inversion
of electronic distance-measuring instrument (EDM), global positioning system (GPS) and very long baseline interferometry (VLBI)
(Lisowski et al., 1990), the joint inversion of leveling, GPS and gravity (Zhao and Sjöberg, 1993; Zhao, 1995; Li et al., 2002; Li and Xu,
2005), the joint inversion of geodetic and seismic data, and the
joint inversion of geodetic, seismic and geologic data (Holt and
Haines, 1993, 1995; Holt et al., 2000; Williams et al., 1993; Tinnon
et al., 1995; Shen-Tu et al., 1998; Shen-Tu and Holt, 1999; Xu et
al., 2000, 2003, 2005; Segall and Matthews, 1997; England and
∗ Corresponding author at: School of Geodesy and Geomatics, Wuhan University,
129 Luoyu Road, Wuhan 430079, China. Tel.: +86 27 68778805; fax: +86 27 68778371.
E-mail address: [email protected] (C. Xu).
0264-3707/$ – see front matter © 2008 Published by Elsevier Ltd.
doi:10.1016/j.jog.2008.06.005
Molnar, 1997; Wu et al., 2001; Kreemer et al., 2000; Wan et al.,
2004).
However, a problem in the geodetic–geophysical joint inversion remains: how should we fix weight scaling factors of different
datasets so as to achieve a reliable result? As we know, different weight scaling factors represent different contributions
to the inversion result, so only careful fixing of weight scaling factors can lead to a correct/sensible result. This paper
firstly introduces the general methods to fix weight scaling factors in the geodetic–geophysical joint inversion, then discusses
and analyzes them in detail on the basis of a case study in
China.
2. Methods
This section introduces four methods that are usually used for
fixing weight scaling factors. Without the loss of generality, the
joint inversion of geodetic and seismic data is taken as a case to
introduce the geodetic–geophysical joint inversion.
GPS data and seismic moment tensor data have been used to
invert for the crustal motion velocity field and strain rates through
bicubic Bessel interpolation (Shen-Tu et al., 1998; Holt et al., 2000).
The relation between the horizontal velocity field u(x̂) and the rotation vector function W(x̂) can be described as (Haines and Holt,
1993)
u(x̂) = r[W(x̂) × x̂]
(1)
40
C. Xu et al. / Journal of Geodynamics 47 (2009) 39–46
where x̂ is the unit radial position vector of points on the surface
of the Earth and r is the Earth radius (6.371 × 109 mm). W(x̂) is
expanded as a bicubic Bessel spline function of the longitude and
latitude of the point. It can be constrained to simulate rigid plates
and to accommodate rapid spatial variations in some regions where
deforming is occurring. Eq. (1) can also be written as
ū(x̂) ≡
u(x̂)
= [W(x̂) × x̂]
r
(2)
ε̇¯ ˆ
∂W
=
·
cos ∂
(3a)
ε̇¯ ˆ · ∂W
= −
∂
(3b)
1
ε̇¯ =
2
ˆ
ˆ · ∂W − · ∂W
∂
cos ∂
N
(3c)
1
M
obs
here V1 = ε̇¯ − ε̇¯ , V2 = ufit − uobs , are the vectors of residual
errors. P1 = Q1−1 , P2 = Q2−1 , are the weight matrices.
The observation equations are
V1 = B1 X̂ − L1
(6a)
V2 = B2 X̂ − L2
(6b)
where X̂ is the unknown rotation vector, B1 and B2 are the coefficient matrices, and L1 and L2 are the observation vectors, which are
obs
ε̇¯
and uobs , respectively.
In order to make the objective function y minimum, we set its
first derivative with respect to X̂ to zero:
∂y
fit
obs
¯ fit ¯ obs
(ε̇¯ ij − ε̇¯ ij )(Q1 )−1
ij,pq (ε̇pq − ε̇pq )
+ (1 − )
(5)
ij
where the derivatives of W(x̂) are determined from the spline interˆ and
polation of the values of W(x̂) between the grid points and ˆ
are unit vectors in the east and north directions, respectively.
The average seismic strain rates in Eqs. (3a)–(3c) can be obtained
through a moment tensor summation (Kostrov, 1974). Often, the
location of earthquakes is centralized in some regions, not uniformly distributed in the whole study area. So, in order to obtain
the continuous strain rate filed in the whole study area, smoothing
of each strain rate is needed (Haines and Holt, 1993).
The corresponding objective function in the inversion is
y=
y = V1T P1 V1 + (1 − )V2T P2 V2
fit
where the unit of ū(x̂) in the equation is 1.0 × 10−9 a−1 , which is the
same as that of W(x̂). The sets of equations that relate the rotation
vector W(x̂) to the horizontal strain rates on the surface of a sphere
are (Haines and Holt, 1993)
ent weights. It is necessary to rescale Q1 and Q2 by adjusting in
the geodetic–geophysical joint inversion.
Because the relation between the velocity field and the rotation
vector is linear (Eq. (1)), and the strain rate can be linearly expressed
by the first derivative of the rotation vector (Eqs. (3a)–(3c)). Eq. (4)
can be denoted by the vectors:
fit
obs
(ufit
− uobs
)(Q2 )−1
i,j (uj − uj )
i
i
(4)
1
where subscripts ij, pq denote the tensor components of the strain
rate tensor, Q1 is the a priori variance–covariance matrix of the
average horizontal strain rate tensor from the seismic moment
tensor data, Q2 is the a priori variance–covariance matrix of the
obs
velocity vectors, ε̇¯
is the observed average strain rate tensor
ij
fit
component from the seismic moment tensor data, ε̇¯ ij is the fit-
is
ted average strain tensor by bicubic Bessel interpolation, uobs
i
the observed velocity field from GPS data, and ufit
is the fitted
i
velocity field through bicubic Bessel interpolation, M, N are the
total numbers of GPS data and grid areas, respectively, and is the weight scaling factor of the seismic moment tensor data,
and satisfies ∈ [0, 1]. Here, it should be noted that Q1 can be
obs
obtained in the process of calculating ε̇¯
(Haines and Holt, 1993),
ij
and Q2 can be obtained in the process of solving for the velocity field. Q1 and Q2 not only make each of the terms in Eq. (4)
dimensionless, but also reflect initial weights of each observed
value in the geodetic–geophysical joint inversion. But these initial
weights are not accurate for the geodetic–geophysical joint inversion, because these datasets involve their own systematic errors
separately. This leads to that the a priori variances of unit weight of
the datasets cannot be fixed exactly. On the other hand, mis-scaling
of variance–covariance matrices may make the joint inversion to be
underdetermined or mixed-determined. In addition, the weights
are related to the inversion norms, different norms require differ-
∂X̂
= 2V1T P1 B1 + 2(1 − )V2T P2 B2 = 0
(7)
It also can be written as
B1T P1 V1 + (1 − )B2T P2 V2 = 0
(8)
From Eqs. (6a), (6b) and (8), the normal equation in the
geodetic–geophysical joint inversion can be obtained:
[B1T P1 B1 + (1 − )B2T P2 B2 ]X̂ = B1T P1 L1 + (1 − )B2T P2 L2
(9)
It shows that X̂ is related to , and the relation between them is
non-linear. So, if is given a value in its domain, which is from 0 to
1, X̂ can be solved correspondingly.
It is very crucial to fix the value of properly in the
geodetic–geophysical joint inversion. Here, we discuss four methods for fixing .
2.1. Method I: taking and X̂ as unknown parameters to invert
together
Due to the fact that the relation between X̂ and is non-linear
(Eq. (9)), and y is related to and X̂ (Eqs.(5), (6a) and (6b)), the
relation between and y should also be non-linear and y can be
decided by . In a practical sense, a series of values of the objective
function can be obtained when traverses its domain, and then
the minimum of these values can be chosen as the final value of y.
Accordingly, the unknown rotation vector X̂ and the scaling factor
are taken as the final results.
2.2. Method II: fixing directly according to the a priori
information
In order to scale the amplitudes of each data set, a ratio of the L2
norms of the observation vectors was used as the weight factor for
joint inversion of body- and surface-wave data (Honda and Seno,
1989), tsunami and leveling data (Satake, 1993) and gravity and GPS
data (Zhao, 1995). The weight scaling factor can be solved by
1−
ε̇¯ = ob
uob or
02
1−
= 21
0
(10)
2
where 02 and 02 are the squares of the standard errors of the
1
2
observed average strain rate data, which is related to the seismic
moment tensor, and the GPS velocity field data, respectively, ε̇¯ ob C. Xu et al. / Journal of Geodynamics 47 (2009) 39–46
41
Fig. 1. Distribution of plotted grids (blue lines), focal mechanisms of all the earthquakes (1903–2003), GPS sites (black stars) and geological background (white lines) used
in the study.
Table 1
Inverted scale factors, values of objective function and standard errors of unit weighta
Method
y
ˆ 01 (×10−9 a−1 )
ˆ 02 (×10−9 a−1 )
ˆ 0 (×10−9 a−1 )
I
II
III
IV
1.000
0.016
0.671
N/A
251.849
7299.655
3052.550
4301.620
2.204
0.691
1.162
1.111
0.250
1.991
1.167
1.425
0.335
1.803
1.166
1.384
a In this table, is the scaling factor of the seismic moment tensor data; y is the value of the objective function; ˆ 01 , ˆ 02 and ˆ 0 are the a posteriori standard errors of the
unit weight of seismic data, the GPS data and the joint data, respectively; (I) represents the results when the value of the objective function is minimum; (II) represents the
results when the scaling factor is fixed to be the ratio of standard errors of seismic data and GPS data; (III) represents the results by using VCE; (IV) represents the results
when the scaling factors of the two datasets are equal.
and uob denote the L2 norm of the observed average strain rate
data and the GPS velocity field data, respectively.
According to Eq. (10), can be solved, and then the unknown
rotation vector X̂ can be got by Eq. (9).
2.3. Method III: solving by Helmert method of variance
components estimation
mation (VCE) to fix the scaling factor. In this paper, we assume the
initial variances of unit weight of seismic moment tensor data and
GPS data to be 02 and 02 , respectively. The relation between their
1
2
estimates and the corresponding sums of squared misfit is (Cui et
al., 2001; Grafarend, 2006):
S ˆ = W
(11)
and
The uniform error equation can be written as Eqs. (6a) and (6b),
regardless of the linearity between the geodetic observations L and
the unknown parameter X̂. A joint inversion problem, shown as
Eqs.(6a) and (6b), can be solved by using variance components esti-
S =
n1 − 2tr(N −1 N1 ) + tr(N −1 N1 )
2
tr(N −1 N1 N −1 N2 )
tr(N −1 N1 N −1 N2 )
n2 − 2tr(N −1 N2 ) + tr(N −1 N2 )
2
,
Table 2
Comparisons of inverted velocity fielda
Method
I
II
III
IV
Ave. (cm/a)
Ave. (cm/a)
Vi
V1i
V2i
1.09
1.23
1.15
1.16
0.74
0.75
0.73
0.73
0.52
0.57
0.55
0.55
Max. (cm/a)
dV
dV1
dV2
−0.07
0.07
−0.01
–
0.01
0.02
−0.00
–
−0.03
0.02
−0.01
–
dV
0.88
1.33
0.23
–
dV1
dV2
0.85
0.26
−0.10
–
0.24
1.31
−0.20
–
a In this table, Ave. and Max. represent the average and maximum; V i , V i and V i (i = I, II, III, IV) represent the velocity of the 286 grids, and the component in the north
1
2
and east directions, inverted by the i th method, respectively; dV, dV1 and dV2 denote the corresponding differences between the above values of the 286 grids inverted by
using Methods I–III, relative to Method IV, separately.
,
.
ˆ = S −1 W
dE1 (×10−9 a−1 )
where n1 and n2 are the numbers of the observed average strain rate
tensor data and the GPS velocity field data, respectively. Because
there are three average strain rate tensor components in each grid,
n1 is three times N in Eq. (4). Similarly, because there are two horizontal velocity components at each GPS site, n2 is two times M in
Eq. (4). The solution of Eq. (11) can be expressed as
Max.
(12)
(i = 1, 2)
(13)
(×10−9
i
a−1 )
dE2
where c is an arbitrary positive constant. The iteration does not stop
until the variances of unit weight are identical or their ratio can be
regarded as 1.0 through some necessary tests, such as hypothesis
testing, and so on. The weight scaling factor can be solved by
(c/ˆ 02 ) )
(i = 1, 2; j = 1, 2, 3 . . .)
(14)
2
dE1
2.4. Method IV: without considering the weight scaling factors
(×10−9
a−1 )
Ave.
Often, the weight scaling factors are regarded equal, which
means that we do not rescale the a priori variance–covariance
matrices of the different datasets, so is taken as 0.5 in Eq. (4)(e.g.
Haines and Holt, 1993; Holt and Haines, 1995; Shen-Tu et al., 1998;
Shen-Tu and Holt, 1999; Holt et al., 2000). In this paper, this method
is compared with the other three methods to judge their applicability in the geodetic–geophysical joint inversion.
a−1 )
(×10−9
E1i
Ave.
Method
Table 3
Comparisons of inverted strain ratesa
E2i
3. Results
While fixing the weight scaling factors by using the above Methods (I–IV), we do the geodetic–geophysical joint inversion from
GPS data and seismic moment tensor data, and obtain the crustal
motion velocity field and the strain rates in the study area (mainland China and its neighborhood). The spatial distributions of focal
mechanisms of earthquakes, the geological background, the irregular grids and GPS sites are concerned in the study area (Fig. 1). In
this paper, we plot 286 irregular grids, collect 911 seismic moment
tensors from the Harvard centroid moment tensor (CMT) Catalog
from 1903 to 2003, and select 1139 GPS velocity observations from
the China Earthquake Administration, referred to Eurasian Plate, as
the original data for the geodetic–geophysical joint inversion. Fig. 1
−3.41
7.57
−0.70
–
(×10−9
a−1 )
dE2
(×10−9
ratio of ˆ 02 to ˆ 02 is smaller than 1.0, increases. Only when the
1
2
ratio is nearly 1.0 does become stable (the numerator and the
denominator in Eq. (14) vary with the same proportion), and all
the inversion parameters can be acquired. Eq. (14) expresses the
solution of through two datasets, and it also can be expanded to
more datasets to solve , only expanding the domain of i to vary
from 1 to the number of types of datasets.
5.19
−9.26
1.25
–
a−1 )
1
−161.13
−171.75
−173.11
–
j
i th data in the j th VCE iteration. It controls the variance of . If
the ratio of ˆ 02 to ˆ 02 is larger than 1.0, decreases, and if the
6.46
2.09
−2.14
–
i
Min.
where (c/ˆ 02 ) denotes the adjustment factor of the weight of the
−7.28
−21.73
−11.23
−12.47
j
,
j
5.61
16.58
8.31
9.01
i
i
107.25
102.88
98.65
100.79
(
I
II
III
IV
=
j
dE1
1
j
dA
(c/ˆ 02 )
(×10−9
dA
Pik−1 ,
(0 a−1 )
c
ˆ 02
(0 a−1 )
Pik =
a−1 )
In practical application, iteration is needed to obtain the solution
by adjusting the weight as follows:
a In this table, Ave., Min. and Max. represent the average, minimum and maximum; Ai , E i and E i (i = I, II, III, IV) denote the values of the direction of principal axis, the first principal strain rate and the second principal strain
1
2
rate of the 286 grids, inverted by the i th method, respectively; dA, dE1 and dE2 denote the differences between the above values of the 286 grids inverted by Methods I–III, relative to Method IV, separately.
T
39.19
43.78
12.70
–
2
13.60
166.31
6.91
–
ˆ 02
167.29
174.21
175.62
–
1
−15.26
−142.02
−5.44
–
V2T P2 V2
ˆ 02
(0 a−1 )
V1T P1 V1
T
Ai
W =
ˆ =
−37.82
−39.28
−12.65
–
N = BT PB = B1T P1 B1 + B2T P2 B2 = N1 + N2 ,
dE2 (×10−9 a−1 )
C. Xu et al. / Journal of Geodynamics 47 (2009) 39–46
dA (0 a−1 )
42
C. Xu et al. / Journal of Geodynamics 47 (2009) 39–46
also shows that the distribution of GPS sites is heavily nonuniform,
dense to the east and sparse to the west.
Table 1 summarizes the results inverted for by these four methods. Included are the weight scaling factor, the value of the objective
function, and the a posteriori standard error of unit weight. Then
the velocity field and the strain rates in the irregular grids inverted
using the weights given by Methods I–IV (Tables 2 and 3) were
compared and the corresponding differences in the regular grids
(20 × 20 ) were shown visually (Figs. 2 and 3).
Table 1 shows that the weight scaling factor is 1.0 for Method I,
which indicates that GPS data is useless in the inversion, and only
the seismic data is used. The reason for occurring such circumstances is that the inversion of seismic moment tensor data alone
leads to the inversion that is barely overdetermined, because of the
smoothing for strain rates, and if there is not the smoothing, the
inversion would be underdetermined. In contrast, it is common for
several GPS velocities to contribute to the estimation of the rotation vector, making the inversion problem mostly overdetermined,
except for a few places where data are sparse. As a result, the use
of an objective function based purely on fit to the data is inappropriate, and leads to a choice of weights that favor the dataset
that is closest to an exactly determined inversion or underdetermined inversion. It is therefore clear that it is not reasonable to
do the inversion just by confining the objective function to be a
minimum, and that some constraints should be given to inversion
parameters according to the actual situation. The scaling factor is
0.016 when it is fixed directly by the a prior standard errors of
the seismic data and the GPS data, which are 12.5 × 10−9 a−1 and
1.6 mm/a (1.6/6.371 = 0.25 × 10−9 a−1 ), respectively. These values
can be obtained by calculating the averages of the original errors
43
of these data. It means that the seismic data play a very tiny role
in the joint inversion. It is due to the low accuracy of the seismic
data and the much higher accuracy of the GPS data. But the good
results are not only related to high accuracy, but also the distribution of data. So, it is better to use this method according to the actual
problem. Table 1 also shows that the ratios of the a posteriori standard error of the unit weight from each dataset obtained by these
methods are 8.816(2.204/0.25), 0.347, 0.996 and 0.780 separately. It
means that these two data are merged at last by using VCE because
they have the same a posteriori variances. And the weight scaling
factor got by using VCE is 0.671, which is closest to 0.5 when compared with the other weight scaling factors. Not only the scaling
factor, but also the other items in the table by Methods III and IV
are very close. Furthermore, 0.671 means that the seismic data play
a slightly more important role in the geodetic–geophysical joint
inversion than the GPS data. This is because: (1) the strain rates in
each grid are obtained by averaging neighboring grids, so there are
enough strain rates in each grid to invert for the unknown parameters, while GPS data is applied as constraints to the velocity; (2)
although there are a great number of GPS sites, their spatial distribution is not uniform: more than one GPS site for some grids,
but none for others. Thus, the nonuniform distribution of the GPS
data reduces their constraints to the velocity, and then reduces the
weight scaling factors in the geodetic–geophysical joint inversion.
Table 2 shows that the velocity field obtained by fixing the scaling factor when the objective is minimum (Method I), through the
a priori information (Method II) and by VCE (Method III) are different from that without considering the scaling factor (Method
IV). The maximums of the difference are 0.88 cm/a, 1.33 cm/a and
0.23 cm/a, respectively. It is clearly shown that various differences
Fig. 2. Inverted velocity field in regular grids (20 × 20 ) by Methods I–IV. (A)–(C) represent the differences between the former three results and the last one, separately.
44
C. Xu et al. / Journal of Geodynamics 47 (2009) 39–46
Fig. 3. Inverted strain rates in regular grids (20 × 20 ) by Methods I–IV. (A)–(C) represent the differences between the former three results and the last one, separately.
Fig. 4. The statistical values of differences about strain rates. A, B and C represent
the difference got by Method I, II or III, relative to Method IV, separately (according
to Table 3).
between Methods III and IV are minimum. Table 3 shows that the
strain rates, obtained by Methods I–III are also different from that
by Method IV. But various differences between Methods III and IV
are also minimum(Fig. 4).
geodetic–geophysical inversion set to do the joint inversion. Secondly, we test the methods from strain rate estimates, choosing a
mixed test data set, which includes GPS velocity and seismic strain
rate tensors. From the total 279 strain rate tensors, we randomly
select 24 and 48 strain rate tensors, mixed with 139 and 249 GPS
velocity observations, as the mixed test sets: test set (mix1) and test
set (mix2), respectively, and use the rest to do the joint inversion
separately. Lastly, we compare the modeled and observed values of
the test set through the internal fitting precision and the “external
fitting” precision (Table 4).
The internal fitting precision can be expressed by RMS misfit
between the modeled and observed velocity (strain rate) components of the inversion set.
IG =
4. Cross validation
In order to test model reliability and appraise the inversion
results, we apply a cross validation method (Efron and Tibshirani,
1993, 1997), which is the statistical practice of partitioning a sample data into subsets such that the analysis is initially performed
on a single test, while the other subset(s) are retained for subsequent use in confirming and validating the initial analysis. Firstly,
from the total 1139 GPS velocity observations, we randomly select
139 and 249 observations as the test sets, which are denoted test
set (139) and test set (249), respectively, and use the rest as the
IS =
M1
[(Vem − Veo )2i + (Vnm − Vno )2i ]
1
2M1
N1
m
o 2 m
o 2
m
o 2
[(ε̇¯ − ε̇¯ )i (ε̇¯ − ε̇¯ )i + (ε̇¯ − ε̇¯ )i ]
1
3N1
(15a)
(15b)
where IG and IS are the internal fitting precisions of GPS data
and strain rate data, respectively, subscripts m and o denote the
modeled value and observed value separately, Ve , Vn are the east
and north components of velocity, respectively, ε̇¯ , ε̇¯ and ε̇¯ are
the horizontal strain rate components, M1 and N1 are the number
of inversion set for GPS velocity and strain rates separately.
C. Xu et al. / Journal of Geodynamics 47 (2009) 39–46
45
Table 4
The internal fitting precision and the “external fitting” precisiona
Method
139 (cm/a)
IG
139 (cm/a)
EG
249 (cm/a)
IG
249 (cm/a)
EG
mix1 (×10−9 a−1 )
IS
mix1 (×10−9 a−1 )
ES
mix2 (×10−9 a−1 )
IS
mix2 (×10−9 a−1 )
ES
I
II
III
IV
0.265
0.222
0.233
0.229
0.288
0.298
0.270
0.270
0.271
0.218
0.230
0.227
0.289
0.301
0.268
0.269
4.208
21.399
7.329
8.312
4.337
22.009
7.579
8.713
3.838
23.131
7.809
8.560
3.754
22.576
8.075
9.033
a
139 / 249 and 139 / 249 represent the internal fitting precision and the “external fitting” precision of GPS for test set (139/249), respectively; mix1 / mix2
In this table, IG
IG
EG
EG
IS
IS
mix1 / mix2 represent the internal fitting precision and the “external fitting” precision of strain rates for test set (mix1/mix2), respectively.
and ES
ES
The “external fitting” precision can be expressed by the RMS
misfit between the modeled and observed velocity (strain rate)
components of the test set.
EG =
ES =
M2
[(Vem − Veo )2i + (Vnm − Vno )2i ]
1
2M2
N2
m
o 2
m
o 2
m
o 2
[(ε̇¯ − ε̇¯ )i + (ε̇¯ − ε̇¯ )i + (ε̇¯ − ε̇¯ )i ]
1
3N2
(16a)
(16b)
where EG and ES are the “external fitting” precisions of GPS data
and strain rate data, respectively, M2 and N2 are the number of the
test set for GPS velocity and strain rates separately.
The internal and external fitting precisions are given in Table 4,
and the stability of the method can judge from the difference
between these two quantities. The difference is smaller, the method
is more stable.
Table 4 shows that the internal fitting precision is slightly higher
than the “external fitting” precision in general. Method I gives the
best match between the internal and external fitting precision, but
it does not mean that this method is the best. As mentioned above,
the scaling factor solved by Method I is 1.0, which indicates that
the joint inversion can be done by seismic moment tensor data
alone, and it has little effect on GPS data, whether GPS data is in the
inversion set or the test set. So, the difference between these two
quantities is naturally the smallest. While among the differences
between them by other methods, it is the smallest by using VCE,
which shows that VCE is the most stable method discussed here,
and also the best for fixing the weight scaling factor. Besides, Methods III and IV give very similar results, mainly because the scaling
factor solved by using VCE is 0.671, not far from 0.5. And it shows
that the proper weight has been checked and confirmed by using
VCE rather than simply being assumed. For test set (139) and test set
(249), or test set (mix1) and test set (mix2), the differences between
their internal fitting precisions and “external fitting” precisions are
tiny, implying that the inversion results are reliable.
5. Conclusions
Although the geodetic–geophysical joint inversion relies on the
spatial distribution of data to some extents in a joint inversion with
different kinds of datasets, the weight of these datasets should be
considered in the joint inversion model; the weight scaling factor
should be fixed properly.
The Helmert method has been used for many problems, but in
the geodetic–geophysical joint inversion it was first introduced to
determine weight scaling factors for different datasets. Compared
to other current methods for fixing the weight scaling factor, as
discussed in the paper, more reasonable results can be obtained by
using the Helmert method of variance components estimation.
Acknowledgements
We thank China Earthquake Administration for providing us the
data. This work was supported by the National Natural Science
Foundation of China (No. 40574006, No. 40721001), the Program
for New Century Excellent Talents in University (NCET-04-0681),
the Programme of Introducing Talents of Discipline to Universities
(No. B07037), and Key Laboratory of Dynamic Geodesy, Chinese
Academy of Sciences (L04-02). The authors thank Prof. Randell
Stephenson, Prof. Paul Cross, Dr. Zhenhong Li and two anonymous
for their detailed, constructive comments. The first author would
like to thank the Ministry of Science, Research and the Arts Baden
Württemberg (Germany) for the fellowship grant.
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