INTRODUCTION

The role of global navigation satellite systems (GNSS) for assessment and prediction of earthquake hazard constantly increases. Unlike the traditional geodetic methods, repeated, multiple, and continuous GNSS measurements can provide fast estimates of changes in the spatial location of points at the Earth’s surface to highest accuracy, regularity over time, and measurement rate. There are several examples where data from GNSS monitoring were successfully used for the estimation and interpretation of coseismic and postseismic processes and for prediction of earthquake hazard in different areas worldwide (Kaftan, 2017, 2021; Kaftan et al., 2021). Strain data are obtained from geodetic measurement networks. As to scale, they are subdivided into three hierarchical levels: global, regional, and local networks. Each level has to deal both with general and with special problems. In the first of the above cases, the main process under study is the movement of global tectonic plates, in the second the goal is to monitor how sites are being displaced in zones of dynamic influence due to regional faults, and in the third, the problem is to study local geodynamic processes. The problem that is common to all three levels is the determination of accurate spatial coordinates of GNSS stations.

Depending on what is the level concerned, users select appropriate strategies for field observations, processing, analysis, and interpretation of data that differ significantly. The choice of a strategy affects the accuracy and reliability of the results, as well as the economic efficiency of the work.

The success achieved in dealing with recent crustal movement observations at the global level with baselines (vectors of coordinate increments between GNSS stations) extending for one thousand kilometers or longer helped foster the conviction that the global approach is universal for any problems in geodynamics. The methodology used for the processing and interpretation of GNSS observations began to be transferred en masse from the global to the local level. However, the problems to be dealt with at the local level include dynamical properties of objects at a quite different scale, energy, velocity, and temporal sequence (Kuzmin, 2019). In that case sharp fluctuations of movements do not constitute a statistical error (or background) as traditionally thought, but a parameter that characterizes an important property of the geologic medium, viz., its response to external stress fields and dynamical stability. Their identification requires special approaches to data processing and interpretation, as well as calls for configuration optimization in application to local GNSS observation networks.

In this connection, the goal of the present study is an analysis of opportunities offered by observation networks for local strain monitoring using GNSS techniques.

STRATEGIES EMPLOYED FOR PROCESSING OF GNSS OBSERVATIONS

At present, there are two main approaches to the processing of GNSS observations used for precision coordinate determinations. These are the method of Double Differenced GNSS determinations and the Precise Point Positioning (PPP) method. The first of these is usually referred to as the relative method, because it provides accurate relative positions (coordinate increments) for GNSS stations. The second is called the absolute method, because it provides coordinates (position) of an individual GNSS station relative to the accurate coordinates (ephemerides) of GNSS satellites, so that the relative positions of stations is a side result in this case.

Both of these approaches can provide a nearly one-centimeter accuracy for the desired increments of coordinates and for the coordinates themselves. It should be noted that we do not consider here other known options of GNSS determinations such as real time kinematics (RTK) or differential coded determinations, because these do not yield the accuracy required for geodynamical problems, but are designed for dealing with routine problems arising in topography, navigation, land planning, cadaster, design, construction and other work which does not require very high positioning accuracy.

The accuracies achievable for determination of positions and mutual positions using the РРР and DD techniques were compared by (Melnikov, 2018) for the variant of a local strain network, showing that the two accuracies are about equal. However, it should be borne in mind that the first strategy used accurate products provided by the International GNSS Service, IGS, while the second merely relied on measured differences between the carrier phases and the accurate IGS production was not required. This circumstance provides a substantial advantage to the DD difference method for obtaining accurate mutual positions two weeks before the accurate coordinates obtained using the РРР strategy. For this reason we will consider the difference method as being the preferred option for local GNSS monitoring.

COORDINATE REFERENCE SYSTEMS AND REFERENCE FRAMES

Time series of coordinates are obtained by the GNSS techniques in different coordinate reference systems. A reference system is the set of prescriptions, conventions, and models required for specifying the three axes of coordinates at a given time.

A reference frame is the implementation of a reference system represented by the observing network equipped with a variety of measuring instruments and the coordinates of its stations.

Global Reference Systems

All coordinate reference systems are conventionally divided into global, regional, and local systems. The chief difference between these consists in the rules for specifying the origin and axes of a spatial system of coordinates, as well as spatial orientation of the latter in the body of the Earth.

Since the satellite era began, researchers started devising global reference systems that are necessary for settling in the nearest extraterrestrial space. The appearance of the first global systems of coordinates connected to the Earth is due to the names of notable geodesists L.P. Pellinen (Kashin, 1999) and W. Kaula (1966). The first fully-fledged global reference systems for general use were the World Geodetic System 1984 (WGS84) and the 1990 Parametry Zemli (Earth Parameters) (PZ-90). These reference systems include spatial rectangular systems of coordinates, rules to specify the axes and their orientation in the body of the Earth, global models for the gravity field and for the figure of the Earth.

The best and very general system is the International Terrestrial Reference System developed by the International Earth Rotation and Reference System Service, IERS [https://www.iers.org/IERS/EN/Home/home_node.html]. The system was developed in cooperation with the International Astronomical Union (IAU) and the International Union of Geodesy and Geophysics (IUGG). The system is being used under the auspices of the International Association of Geodesy with participation of several services of space observation: the International DORIS service, the International GNSS Service, the International Laser Ranging Service, and the International Very-Long Baseline Interferometry Service. All of these services make measurements within the global geodetic network, and which are used to determine the coordinates and velocities of movement for its stations, thus forming the International Terrestrial Reference Frame, ITRF. In this process the International GNSS Service supports its own part of the united ITRF as an independent reference frame based on continuously operated GNSS stations. The frame is used by the service to find accurate ephemerides of satellites in global radio navigation systems.

Models of global physical processes are used for accurate specification of the origin and axis orientations for the spatial system of coordinates in the body of the Earth, as well as for kinematic operation of the global reference frame. The models include those of earth tides, oceanic loads, atmospheric loads, the position of the average conventional pole, pole tides, and the loading due to oceanic pole tide. Local effects are also modeled, viz., the effects of temperature and pressure on the position of a station, variations and position of the phase center for a GNSS antenna. The kinematic implementation of global coordinate reference systems consists in relevant geodetic networks with catalogs of sites for the corresponding initial epoch, and the components of velocities of motion for the sites.

At present, the orientation of the coordinate axes and the specification of the origin for the abovementioned reference systems (WGS84, PZ-90, and ITRS) in the body of the Earth are made consistent among themselves to the matter of a few centimeters. The national reference systems of global coverage, WGS84 and PZ-90, have the orientation and fixation in the body of the Earth that matches that of the ITRF as being the leading system and universally acknowledged to be the most accurate and stable.

Reginal Reference Systems

Examples of regional coordinate systems and their implementations to be considered here are the European, the Russian, and the North American system.

The European regional reference system (European Terrestrial Reference System) was set up in 1989 by the European Subcommission of the International. Association of Geodesy (EUREF) [http://www.euref.eu/euref_egrs.html].

At present this reference system is based on the most accurate global reference system (ITRS). However, the plate tectonics makes the coordinates of the European stations in the ITRS move slowly at an approximate rate of 2.5 cm/year. For this reason the EUREF set up the European terrestrial reference system ETRS89 in 1990 at the symposium in Florence so it would be based on the ITRS and strictly tied to the stable part of Europe, hence making the connections between the European stations constant (Altamimi, 2018).

The State Geodetic Coordinate System of the Russian Federation (GSK-2011 in Russian) is the implementation of a geocentric coordinate reference system. With regard to the principles of fixation in the body of the Earth, GSK-2011 is identical with the International Terrestrial Reference System (ITRS) developed in accordance with recommendations issued by the IERS. It has a kinematic catalog of coordinates and velocities of movement for geodetic sites (Kaftan et al., 2019).

Consider an example showing estimation of the shear rigidity in the lithosphere inferred from determinations of horizontal crustal movements by the GNSS techniques (Furst et al., 2018). Figure 1 shows the distribution of vectors representing movements of the ground surface based on GPS measurements for the San Andreas fault zone in the western U.S. where areas of slow movements (e.g., Sierra Nevada) are adjacent to areas of high velocities west of the San Andreas. The velocity vectors are shown in the North American reference frame. This representation characterizes unilateral movement of the entire extensive area, as well as preventing visualization of mutually oppositely directed, mirror-asymmetrical crustal movements on both sides of the boundary between the North American and the Pacific plate. Visually, the character of motion as expressed in the regional reference system is presented as unilateral toward the northwest. One cannot clearly discern the right lateral movements on the fault system at the boundary of the tectonic plates; it is best expressed in coseismic movements and less obviously in the accumulation of elastic deformation during the precursory periods of great seismic events.

Fig. 1.
figure 1

The distribution of geodetic measurements in the North American reference frame. Black arrows indicate the velocity field in the North American system of coordinates. The analysis area is enclosed in a rectangle. Red stars mark earthquake locations: Landers (L), Northridge (N), and the Hector mine (HM) (Furst et al., 2018).

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Local Reference Systems

Having considered the modern global and regional reference systems, we can state that these were created for dealing with concrete problems and for convenience in the study of a definite object. Global reference systems primarily provide solutions for problems such as the study of changes in the shape and size of the Earth as a whole, the motion of global tectonic plates along the underlying viscous substratum, and so on. Some implementations of the International Reference System ITRS made a straightforward use of a geological model for the motion of tectonic plates, viz., NUVEL-1 (Argus and Gordon, 1991) for specification and orientation of the vectors of motion at stations of the global network.

Application of global and regional reference systems is inconvenient and unreasonable to the study of several local geodynamic processes such as relative sliding of sides of earthquake-generating faults, crustal movements and deformations in areas of volcanic activity, in operated areas of mineral deposits, in locations of nuclear waste disposal, etc. In these cases the monitoring observation networks that implement a local reference system cover areas that are much smaller compared with what the global and regional systems do. Quite frequently, the background motions of tectonic plates that are home to local networks mask those motions which accompany the relevant local process. A typical example is displayed in Fig. 1 where the velocities of displacement as shown do not obviously reveal right lateral strike slip movements along the strike of earthquake-generating faults. The resulting picture does not provide visualization of local geodynamic effects such as the presence of asperities where the sides of earthquake-generating faults slide relative to each other, and where no oppositely directed movements should be observed.

A local reference system does not require complex procedures for establishment related to abovementioned global geophysical and astronomical processes because of its small geometric dimensions. Moreover, such a procedure when incorporated in data processing would burden the desired results with the additional uncertainties inherent in modern geophysical models.

Local reference systems are used in the technologies of geodynamic monitoring using the GNSS techniques at local test areas (Kaftan and Ustinov, 2015; Kaftan et al., 2017, 2019). One need not rely on accurate global and regional implementations of reference systems. It is sufficient to make use of “approximate” orbital implementations such as, e.g., WGS84 and PZ-90. The available broadcast ephemerides for satellites in the these radio navigation systems can be determined to within a few meters. This enables one to obtain highly accurate vectors of increments for the coordinates of nearby GNSS stations that are a few tens of kilometers distant from each other. It remains to fix the origin of the coordinate axes and their orientation, which is provided for by adjustment of the network of increment vectors as a free network, without having to rely on stations of global and regional implementations. Like all the other reference systems, a local system is conventional. It is in force only during the surveying period. These circumstances significantly simplify the procedures of mathematical data processing and favor the achievement of higher accuracies in the characteristics of recent crustal movements and deformations.

Reference systems such as the above are usually called “inner” coordinate systems. The theory of obtaining such “inner” coordinates can be found, e.g., in (Grafarend and Schaffrin, 1974; Welsch, 1979). We implement this approach in a method for adjusting repeated measurements in order to study movements and deformations of the ground surface (Kaftan et al., 2019).

At the same time, observations at local networks involve several features that ought to be borne in mind in the planning process. It is not always feasible to achieve the optimal network configuration for a local network owing to several factors (forests, absence of roads, relief, etc.). In this connection we consider below how network configuration affects the accuracy attainable in the determination of movements and deformations.

THE EFFECTS OF CONFIGURATION FOR A LOCAL GNSS NETWORK ON THE ACCURACY OF STRAIN ANALYSIS

We recall that the optimal shape of a network triangle should be close to equilateral when setting up geodetic networks using electron optical means of measurement. This was found out by geodetic studies as long ago as the time when governmental triangulation networks were set up. That requirement is not critical for GNSS networks when dealing with the determination of the coordinates of a site. In this situation one can use the maximum number of measured connections (baseline vectors) between GNSS stations in data processing. At first sight this data processing option can yield the highest accuracy in coordinate determination due to an increased number of redundant measurements. However, there is a certain restriction in that, as baselines get longer, the accuracy of their vectors becomes worse. The effect is invariably mentioned in documentation for GNSS instrumentation. An experimental proof of the effect based on measurements in a local strain network was demonstrated by Melnikov (2018). The issue of whether long baselines should be included in data processing calls for further study, since long baselines that connect stations situated at the opposite sides of the observing network may prove useless because of their inferior accuracy with respect to the short vectors that connect adjacent stations.

When global and regional reference frames are being developed by economically developed states, with a single measurement session involving some hundreds, not infrequently thousands, of GNSS stations, the number of measured baseline vectors reaches a few hundreds of thousands. The computation of a large number of such vectors is time consuming. For this reason, e.g., the strategy of GNSS measurement processing using the 2015 Bernese program package, which is primarily designed for developing global and regional geodetic networks, envisages an obligatory procedure, viz., preliminary configuring the GNSS network before computing the chosen baseline vectors and subsequent network adjustment.

Nevertheless, it is important for strain analysis to use the maximum number of baseline vectors for computing horizontal ground deformations (Zubovich and Mukhamediev, 2010). The researchers are prompted to adopt this solution by the approach (Shen et al., 1996) that was implemented, e.g., in the program package due to Teza et al. (2008). This approach hides the notion of a finite element, since deformations are computed from any number of displacement vectors for GNSS stations that fall in a certain prescribed area, around a grid point of a regular grid. We lose thereby the understanding of the object being deformed (a geometrical figure) which is a finite element in the classical approach, namely, the triangle of the geodetic network (Tsuboi, 1933; Frank, 1966). In this connection the advantage of the classical approach also consists in the procedure which is used to obtain the optimal configuration of a geodetic network using the Delone triangulation (Delone, 1934]. The work of (Wu et al., 2003; Dokukin et al., 2010) showed that the use of long triangles reduces the accuracy of the desired deformation characteristics. At the same time, the issue concerning the use of the greatest number of baseline vectors at GNSS stations in order to enhance their accuracy remains unresolved.

AN EXPERIMENTAL TEST OF THE EFFECT THE CONFIGURATION OF AN OBSERVING NETWORK HAS ON THE ACCURACY OF DETERMINATION FOR GROUND SURFACE DISPLACEMENT VECTORS

We developed a special local test network of GNSS observation to deal with the problem formulated above, viz., estimation of the contribution due to long baselines into the accuracy of determination for displacements (Fig. 2). The characteristics of the test network are presented in Table 1.

Fig. 2.
figure 2

The configuration of observing network. (a) complete network, (b) abbreviated optimized network. Black triangles are GNSS stations. Black lines represent modified baselines.

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Table 1.   Characteristics of variants of observing network
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The experiment used the procedure for the processing of a local strain network considered by Kaftan et al. (2019). The procedure of GNSS observation processing was as follows.

The first step used daily sessions of GNSS measurements in the “static” regime with the method of double phase differences being employed to find exact values of baseline vectors. Each daily solution was used to create two variants of the vector set: the complete and the shortened (see Fig. 2). Continuous measurements were performed at network stations during 196 days.

The second step involved adjustment of i-1 differences between spatial 3D baseline vectors obtained by subtracting i vectors of the initial epoch from the vectors of the current measurement epoch. One important feature of this processing scheme was that the network was adjusted as one free from initial data errors without external reference stations using the principal pseudo-inverse matrix of the normal equations and proceeding according to the method of obtaining a local “inner” reference system (Grafarend and Schaffrin, 1974; Welsch, 1979). This restriction in the processing of local GNSS networks is referred to as no net rotation/translation in international practice, or in other words, the desired displacement vectors for the stations do not experience a general systematic displacement and rotation. The displacement vectors therefore become free of the trend of global motions in the plate on which the local strain network sits.

Adjustment by the method of least squares as applied to the complete (variant 1) and to the shortened (variant 2) network resulted in displacement vectors from the initial epoch to the current one dx. We calculated estimates of the variance for observation differences with unit weight μ2. The means of the standards with unit weight μ for each variant are presented in the last column of Table 1.

Comparison involved generalized estimates for the accuracy of change in horizontal displacements and for the efficiency of their determination at the level of measurement errors.

The rms errors of unit weight were obtained from the relation

$${{\mu }^{2}} = \frac{{{{{v}}^{T}}Q_{l}^{{ + {v}}}}}{{n - m}},$$

where the v are corrections to measured differences between components of baseline vectors, Ql is the inverse weight matrix of the measurements, n is the total number of measurements, and m is the number of redundant measurements.

The efficiency of strain determination was estimated from the dispersion ratio

$$F = \frac{{d{{x}^{T}}Ndx}}{{{{{v}}^{T}}P{v}}},$$

where dx is the vector of desired displacements at GNSS stations, N is the matrix giving the coefficients of normal equations, and P is the weight matrix.

The results from this comparison of two processing variants (Fig. 3a) show that all standard errors of unit weight for the complete network are invariably greater than those for the shortened network. The values of μ1 exceed the values of μ2 by factors of 1.1 to 2.3 (see Fig. 3b). The values of the Fisher test for the complete network are always below those for the shortened network (see Figs. 3c, 3d).

Fig. 3.
figure 3

Comparing the estimates of accuracy in network processing for the complete and the shortened variant. (a) comparing the standard errors for unit weight in the complete (red) and the shortened (blue) network; (b) ratio of the standard error of unit weight μ1 to μ2; (c) values of the Fisher statistics for the complete (red) and the shortened (blue) network; (d) ratio of Fisher statistics, F1 to F2.

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CONCLUSIONS

The above analysis of the experience gained in the use of coordinate reference systems for geophysical problems, as well as the results of an experiment using a test strain network, suggest the following conclusions.

The use of local reference systems for the coordinates of GNSS stations and their variation over time provides definite advantages for many problems in earth sciences, including the assessment and prediction of earthquake hazard. This approach seems to be more simple and effective in enhancing the accuracy of determination for ground surface movements and deformations owing to the involvement of the shortest baselines whose accuracy of determination is the best. The features inherent to local monitoring enable users to do GNSS processing and strain analysis without having recourse to more sophisticated scientific program packages (Bernese, GAMIT/GLOBK, GIPSY-OASIS), which primarily aim at the processing of global networks, and make use for the purpose of the most popular software packages produced by commercial companies who manufacture GNSS instrumentation.

A special experiment based on a large volume of measured data showed that, contrary to the “laws” of measurement adjustment, the variant with twice as few redundant measurements where long baselines have been excluded, yields a higher accuracy (by a factor of nearly one and a half) in the determination of spatial displacements. We estimated the effectiveness for the determination of spatial displacements using the Fisher test to show that the variance of desired characteristics in the abbreviated network significantly exceeds the error variance. We thus see that this analysis of coordinate reference systems, of the composition of local GNSS observation networks, and test calculations, all showed that the use of special methodological devices for data processing substantially enhances the accuracy of determination for pre-, co-, and post-seismic ground surface movements and deformations in the epicentral zones of large earthquakes, providing a basis for developing novel approaches to the prediction of their locations.