To establish the horizontal crustal movement velocity field of the Chinese
mainland, a Hardy multi-quadric fitting model and collocation are usually
used. However, the kernel function, nodes, and smoothing factor are difficult
to determine in the Hardy function interpolation. Furthermore, the
covariance function of the stochastic signal must be carefully constructed
in the collocation model, which is not trivial. In this paper, a new
combined estimation method for establishing the velocity field, based on
collocation and multi-quadric equation interpolation, is presented. The
crustal movement estimation simultaneously takes into consideration an Euler
vector as the crustal movement trend and the local distortions as the
stochastic signals, and a kernel function of the multi-quadric fitting model
substitutes for the covariance function of collocation. The velocities of a
set of 1070 reference stations were obtained from the Crustal Movement
Observation Network of China, and the corresponding velocity field
was established using the new combined estimation method. A total of 85
reference stations were used as checkpoints, and the precision in the north
and east component was 1.25 and 0.80 mm yr

Horizontal movement velocity fields provide the main basic data for Earth science research. Because measuring a station's velocity by repeated observation is always limited, with many points that cannot be directly measured, a mathematical velocity field model is always used to obtain the velocity field. How to obtain reliable horizontal velocity fields from measured points has been the focus of a lot of research all over the world (Argus and Gordon, 1991; Huang et al., 1993; Liu et al., 2001, 2002; Chai et al., 2009; Jiang and Liu, 2010; Hu and Wang, 2012; Zeng et al., 2012, 2013). As we know, the horizontal velocity of a ground point mainly consists of two aspects: the overall movement and the local deformation reflected ground motion. The motion model for horizontal velocity is often used in geophysical models and the statistical fitting method. The geophysical model is often implemented using the Euler vector method (Argus and Gordon, 1991). Generally, the condition for using the Euler vector method is to divide the block reliably and treat each division as a rigid body, but many areas do not satisfy the requirements of a rigid body, which limits the usefulness of the method. The statistical fitting method mainly includes multi-quadric functions (Huang et al., 1993; Liu et al., 2001; Zeng et al., 2013) and collocation (Liu et al., 2002; Chai et al., 2009; Jiang and Liu, 2010; Zeng et al., 2012). The multi-quadric functions (Hardy, 1978) can be used for fitting the parameters of measured points and estimating the parameters of unmeasured points. The mathematical methods are used in this case, while their physical meaning is not clearly considered. The key issues and difficult problems in their application are the choice of kernel function, smoothing factor, and node. Some scholars have systematically researched their application in horizontal velocity field models in the Chinese mainland (Huang et al., 1993; Nie et al., 2007; Zeng et al., 2013). In order to consider the changing information of velocity fields in different local regions, the least squares collocation method, whose key problem is to determine the covariance function of signal vectors, can be adopted (Yang, 1992). To balance the contribution of the estimation results from the signal covariance matrix and observation noise, we can use the variance components to estimate the collocation solutions (Yang and Liu, 2002; Yang et al., 2008) or adaptive collocation solutions (Yang and Zeng, 2009; Yang et al., 2009, 2011), but both of them require iterative calculations. Because the collocation covariance function is very difficult to build, some scholars have analyzed and compared the relationship between the multi-quadric function and covariance collocation (Tao et al., 2002); some scholars, using the compensation principle of least squares, have pointed out that choosing the appropriate regularization parameters after the semi-parametric model may include the collocation model (Tao and Yao, 2003); and some scholars have established a parameter estimation model combining the multi-quadric function and configuration models and used the distance function as the signal covariance when estimating the gravity field to obtain results close to the multi-quadric function model (Wang and Ou, 2004).

In view of the above-mentioned facts, in this paper, we take Euler vectors as the long-term overall movement trend of the function model and regard the local variations of horizontal movements as stochastic parameters. At the same time, the stochastic characters are taken into account. We utilize a multi-quadric kernel function to obtain its normalized matrix and combine the characters of the collocation and multi-quadric functions, which not only avoids having to establish the covariance function but also achieves an integrated effect using both collocation and multi-quadric functions (Tao and Yao, 2003). On the basis of the method proposed above, we established a Chinese mainland horizontal movement velocity field by using the velocities of a set of 1041 reference stations, obtained from the Crustal Movement Observation Network of China. In this paper, a new combined estimation method using the kernel function of a multi-quadric fitting model to replace the covariance function of collocation was proposed and tested.

It is well known that the horizontal velocity of a ground point can be
described in two parts. One is the holistic long-term trend of horizontal
movement expressed by the Euler vector, which is related to the block where
the sites are. The other part is the local movement change, modeled as
stochastic signals, which is mainly affected by local surface displacement
(Freymueller et al., 2013). Therefore, the function model of point
horizontal velocity is

The relation between the long-term movement trend of horizontal velocity on
a ground station and the Euler vector of plate movement can be expressed as
follows (Jin et al., 2006; Yang and Zeng, 2009):

If we take the local movement variation signal

The Euler vector estimation

In practice, the most difficult aspect of adopting the collocation method is
that the signal's a priori variance cannot be determined accurately. The
signal covariance functions are generally built by adopting observed data
based on certain principles (Zeng et al., 2012). Thus, the priori covariance
matrix determined in this way has a strong correlation with the current
measured data. It not only affects the optimality of collocation, but also
reduces its range of application. Allowing for that, the requirements of
precision and density of observed data are relatively high when estimating
covariance functions. At the same time, the user should be well informed
about the physical field that it is used in. Thus, it is difficult to obtain
a general application and influence the optimality of the collocation model
(Tao and Yao, 2003). We noticed that the signal covariance function of
collocation is a kind of function about distance in general (Zeng et al.,
2012). Therefore, in this paper, we do not look upon the local deformation
parameters as random signals, like the collocation, but as non-random
variables, taking into account their random nature (Tao and Yao, 2003),
reflected by the distance function, to construct the covariance matrix of
the local signals. The covariance matrix of the local deformation is

We still use the Eq. (1) as the observation equation. Because of this, the
distance function does not explicitly consider the physical nature of the
local deformation. Using the compensation least squares method estimation
criterion,

Considering the collocation problem of the non-observed point's signal
parameter

Thus, the key to the above problem is still to determine the covariance matrix of the local deformation parameters; in this paper we will transform the local deformation parameter covariance matrix to the covariance function to determine it.

Velocity residuals statistics for different kernel functions (mm yr

External inspection accuracy of different kernel functions (mm yr

Multi-quadric functions were first proposed in 1978 by Hardy (Hardy, 1978).
The basic idea is that any smooth surface can be approached at any precision
by using a series of finite and regular mathematical functions, and the
non-observed points can be estimated by making use of the observed points.
Because it can be designed flexibly and the controllability is strong, the
method has been widely used in the interpolation problems involved in
geoscience since the approach was proposed. Tao and Yao (2003) discussed
the relationship between the multi-quadric function and collocation in detail and thought
that the multi-quadric function is a kind of special covariance function.
Based on this idea, we introduce a multi-quadric kernel function to
determine the covariance matrix of local deformation. Thus, the covariance
matrix (5) and (8) in Eq. (4) are completely determined by the multi-quadric
kernel function. Currently, the most commonly used multi-quadric kernel
functions in addition to the conical surface are positive double curved
surface, inverted double curved surface, positive thrice curved surface, and
inverted thrice curved surface.

Positive double curved surface:

The symbol

Inverted double curved surface:

Again,

Positive thrice curved surface:

Inverted thrice curved surface:

The velocity residuals statistics of 85 external checkpoints. The
horizontal axes represent the horizontal velocity residuals (N: north; E:
east; m yr

1

We used the same observation data as in Yang and Zeng (2009), that is, 1041
repeat observation stations of the Crustal Movement Observation Network
project, whose precision of horizontal velocity is superior to fitting point
coordinates' velocity by 3 mm yr

In order to evaluate the internal precision of the model, the calculation
formula for the root-mean-square error (RMSE) of the horizontal residual
velocity of the 985 observation stations is calculated:

The RMSE of the external checkpoint horizontal velocity can be defined by

Since the multi-quadric function is used to determine the signal priori random information in the multi-quadric collocation model, different kernel functions have different results. We use inverted and positive, double curved, twice and thrice curved surface to obtain results. Table 1 shows statistics for velocity residuals for the different kernel functions and Table 2 shows statistics for the 85 points of external inspection accuracy of the different kernel functions. It can be seen that the different kernel functions have different effects on the accuracy of the results.

Euler vectors of Chinese mainland.

Internal precision of the Euler vectors (mm yr

1

From the 85 external checkpoint precision statistics, the RMSE for east
and north of the positive thrice curved surface function is 1.25 and 0.89 mm yr

From the internal and external statistical accuracy, the positive
surface function and the inverse surface function of the statistical results
have obvious differences. The 85-point accuracy of the inverted double
curved surface and inverted twice and thrice curved surface in the north
direction are, respectively, 1.18, 1.45, and 1.77 mm yr

As we all know, there are three classical and common methods for establishing velocity field, i.e., Euler vector method, the collocation method, and the multi-quadric function method. In order to examine the effective of the proposed multi-quadric collocation model in calculating the horizontal velocity field, the above-mentioned three methods were used for comparison.

Scheme 1: with the physical significance of the plate tectonics Euler vector model being exact, we utilize the least squares principle to solve the unified Euler vector and the local horizontal velocity changes of the GPS stations in the Chinese mainland.

Velocity residuals statistics (mm yr

External examination precision (mm yr

Scheme 2: we use the collocation method to estimate the long-term overall
movement trend (Euler vector) and the local velocity variance of the GPS
stations. In the process of calculation, we use the Hirvonen function as the
stochastic signal covariance function, and make use of the local velocities
of the 1041 stations, which are obtained by extracting the regional velocity
field, to fit the covariance function. The covariance functions (Yang et
al., 2011) of velocity signal in the north (

Scheme 4: the multi-quadric collocation model, being used to solve the points' the horizontal velocity, has a specific physical significance and is utilized to estimate the long-term overall movement trend (Euler vector) and the velocity changes of local points, with using Eqs. (7), (9), and the inverted twice curved surface, in process of the calculation. We determined the best value of smoothing factor in the inverted twice curved surface function based on the actual distribution of data.

Table 3 lists the Euler vectors of the Chinese mainland calculated by
different schemes. Table 4 lists the external precision of the Euler vectors
of the Chinese mainland calculated by different schemes. Table 5 lists the
velocity residuals statistics for different schemes. Table 6 shows the
precision statistics of 85 external checkpoints. Figure 1 shows the
velocity residual statistical results of 85 external checkpoints. It
reveals that for the multi-quadric collocation model proposed in this study,
(1) the numbers of external checkpoints whose velocity residuals were
between

A comprehensive analysis of the results shows the following.

As can be seen from Table 3, the Euler vector, calculated by Euler vector estimation method (Scheme 1), collocation (Scheme 2), and multi-quadric collocation models (Scheme 4) are equivalent in magnitude and trends, but as a result of the difference of the function model and the signal covariance function between collocation and the multi-quadric collocation model, they are slightly different in values and in the estimated remaining horizontal velocity.

The least squares model based on the Euler vector only takes into
account the entire horizontal movement of the Chinese mainland, so its
precision is poor. From the perspective of internal precision (Table 4),
horizontal residuals for the east and north directions are up to 30.17 and
27.69 mm yr

Solving using the collocation method not only takes into account the
overall orientation, but also determines the local area random effects. It
makes a more reasonable distribution of the residuals (Table 5). The
internal precision is 1.60 for east and 1.43 mm yr

Using a multi-quadric function method can obviously improve the
precision of the velocity model. The internal residual statistic is 2.79 for east and 2.25 mm yr

Synthesizing the collocation method and multi-quadric function method not
only obviously improved internal precision, with east and north precision
reaching 0.78 and 0.73 mm yr

The Euler vector, collocation, and multi-quadric collocation models have physical meaning, and can work out the Euler vectors of Chinese mainland blocks. However, the Euler vector method has larger differences from the other two methods, which, in particular, have smaller differences in latitude and longitude components and rotating angle speed aspects. The Euler vectors determined by the collocation method and the multi-quadric function configuration model are very close.

According to the comparison and analysis for the calculation, we have
established a

In Fig. 2, the Chinese continental crust shows eastward movement as a whole. Meanwhile, the China continental region possesses significantly clockwise rotation, but velocity values show difference in some parts of China, of the horizontal velocity field, removing the continental movement background in China, in the Fig. 3. There is an eastward–southeastward–southwestward movement model with the feature of higher velocity values in the west by 104 longitude degrees in the Sichuan–Yunnan region. In the northwest region, the continental crust displays an NNW motion model. In the Northeast Plain, there was westward movement. Conversely, South China represents an SSE motion model with small velocity values relative to the others in the China.

Multi-quadric functions and the collocation method are the most commonly used methods of fitting the observed points and estimating the non-observed points. However, it is difficult to select the kernel function, smoothing factor, and node when we use the multi-quadric function method. The key to applying the collocation model is to establish a reliable covariance function. Usually, it tends to adopt an empirical formula in a gravitational field, where the majority of applications are established, by using a measured data covariance model, which not only affects the optimality of the collocation method but also reduces its usable area.

Taking into account the establishment of the covariance function is a relatively high requirement for data, and generally, it is difficult to achieve. Bearing in mind that the covariance function is a function of the distance, the local deformations covariance matrix is often chosen as a simple function of distance in multi-quadric functions. The estimating effects of this integrated approach are similar to the collocation method. Because the solution is simple and reasonable, it can greatly expand the scope of application of the collocation model.

This paper was completed on the basis of the Crustal Movement Observation Network Engineering and other GPS works. Special thanks to all Chinese colleagues for working hard in the field to collect the GPS data. Thanks also go to the teams for long-term data management and analysis. We acknowledge financial support from National 863 Project (no. 2013AA122501) and National Natural Science Foundation of China (nos. 41274036, 41474015, 41374019, 41374003, 41274040, and 41020144004). Edited by: J. C. Afonso