ISRM Suggested Method for Monitoring Rock Displacements Using the Global Positioning System (GPS)

Appendix

1.1 Sources of Error

For any measurement device, it is important to know the source of the errors and the corresponding correction methods. Sources of errors in positioning by GPS are summarized in Table 2. The errors can be classified into three groups, namely, errors related to satellites, errors associated with the propagation medium of the signal from the satellites to the receivers, and errors occurring in the vicinity of the antenna by receiver noise and signal disturbances.

Table 2 Sources of errors

Errors related to satellites are satellite ephemeris and clock parameters, which are broadcast by satellites in the navigation message. The control segment, managed by the USA, is responsible for correcting these errors.

The GPS signals are affected by the ionosphere and the troposphere through which they travel from the satellites to a receiver (Fig. 14). Ionospheric delays occur as a result of refractive effects, due to the total electron content, when signals propagate through the ionosphere extending from a height of about 50 to 1,000 km above the earth. These errors can be substantially removed by using a dual-frequency (L1/L2) receiver in the relative positioning. When a baseline length between two antennas is less than a few kilometers, it will be mitigated by taking the difference in carrier phases at two points to eliminate their common-mode errors (Appendix “Fundamental Equations for Relative Positioning”). A single-frequency (L1) receiver is enough, therefore, for use with a short baseline length, as described in this Suggested Method.

Fig. 14

Refraction of GPS signals in the atmosphere

Tropospheric delays occur as a result of refractive effects, due to the air density, when signals propagate through the troposphere extending from a height of about 0 to 11 km above the earth. The air density is a function of the pressure of dry gases and water vapor. This means that the measurement results are subject to the influence of the meteorological conditions along the signal propagation path. When the difference in height between antenna points is more than a few tens of meters, the bias of the tropospheric delays may not be able to be ignored in precise monitoring with an accuracy at the millimeter level, even for short baseline lengths. It is recommended that users employ an appropriate model (modified Hopfield model, Saastamoinen model, etc., Hoffman-Wellenhof et al. 2001; Misra and Enge 2006) to correct such tropospheric delays in order to realize precise monitoring. Tropospheric models are usually installed in the baseline analysis software. Users can select a model with the measured meteorological data for reducing these errors.

Multipath and signal disturbances, caused by the obstruction around antennas, and receiver noise affect the measurement results in the vicinity of the antenna.

Multipath is the phenomenon of a signal reaching an antenna via two or more paths; it is mainly caused by signal reflections from objects (buildings, walls, fences, etc.) and from the ground surface in the vicinity of the antennas. GPS antennas are designed to reduce the effect of multipath, and antenna/receiver manufacturers have developed and implemented proprietary techniques for dealing with it. Naturally, the primary defense against multipath is to position the antennas away from any reflective objects and to set a mask angle for cutting multipath signals.

Obstructions above an antenna (trees, slopes, etc.) may block the antenna signal reception or may cause a disturbance to the signals, and then overhead obstructions become error sources in GPS displacement monitoring. Antennas should be located in areas with a sufficiently open sky. When there are unavoidable obstructions (mainly trees) above the antennas, it is a good practice to analyze the data without the signals transmitted from the satellites behind the obstructions.

Carrier phase measurements are affected by random measurement errors due to receiver noise. Generally, receivers can measure the carrier phases of signals with a precision of 0.5–1 % of a cycle. Since the wavelength of L1 is about 19 cm, measurement errors due to receiver noise will be estimated at 1–2 mm. Users can adopt an adequate method (e.g., statistical model) to reduce this type of random error.

1.2 Fundamental Equations for Relative Positioning

The fundamental equation for relative positioning is described as follows (Misra and Enge 2006). The carrier phase of the signals transmitted from satellite k at measurement point mi is expressed as follows (Fig. 15):

$$\phi_{mi}^{k} = \frac{{r_{mi}^{k} + I_{\phi } + T_{\phi } }}{\lambda } + \frac{{c\left( {\delta t_{mi} + \delta t^{k} } \right)}}{\lambda } + N_{mi}^{k} + \varepsilon_{{\phi_{mi} }}^{k}$$

(1)

where $r_{mi}^{k}$is the distance between measurement point mi and satellite k as follows:

$$r_{mi}^{k} = \sqrt {(x_{mi} - X_{k} )^{2} + (y_{mi} - Y_{k} )^{2} + (z_{mi} - Z_{k} )^{2} }$$

(2)

(x _mi , y _mi , z _mi ) and (X _k , Y _k , Z _k ) are the coordinates of measurement point mi and satellite k, respectively. $I_{\phi }$ is the ionospheric delay, $T_{\phi }$ is the tropospheric delay, and $\lambda$ is the wavelength of the signal. $\delta \,t_{mi}$ and $\delta \,t^{k}$ are the biases of the receiver clock and the satellite clock, respectively. c is the velocity of light, $N_{mi}^{k}$ is the unknown integer ambiguity of the carrier phase, and $\varepsilon_{{\phi_{mi} }}^{k}$ is an observation error.

Fig. 15

Satellites, measurement point, and reference point

The single-phase difference between measurement point mi and reference point mb for satellite $k,\phi_{mi}^{k} - \phi_{mb}^{k} ,$ is taken. In the same manner, another single-phase difference, $\phi_{mi}^{l} - \phi_{mb}^{l} ,$ is taken for satellite l. The double-phase difference, $\left( {\phi_{mi}^{k} - \phi_{mb}^{k} } \right) - \left( {\phi_{mi}^{l} - \phi_{mb}^{l} } \right) ,$ is obtained by using the above two single-phase differences as follows:

$$\phi_{mi - mb}^{k - l} = \frac{{r_{mi - mb}^{k - l} }}{\lambda } + N_{mi - mb}^{k - l} + \varepsilon_{mi - mb}^{k - l} + \frac{{T_{{\phi_{mi - mb} }}^{k - l} }}{\lambda }$$

(3)

Equation (3) is the fundamental equation for relative positioning. It is noted that the biases of the receiver clock and the satellite clock are eliminated as their common errors during the process of deriving the double-phase difference. In addition, the ionospheric delay is also eliminated when the baseline length between the measurement point and the reference point is short, as in the Suggested Method, less than a few km in length.

On the other hand, tropospheric delay $T_{{\phi_{mi - mb} }}^{k - l}$ remains when the difference in height between the measurement point and the reference point is more than a few tens of meters, even for such a short baseline length.

The observation equations for the relative positioning method are obtained from Eq. (3). The double-phase difference on the left side of Eq. (3) is observed by a GPS sensor. Then, the three-dimensional coordinates of the measurement point appearing in $r_{mi - mb}^{k - l} ,$ and integer ambiguity $N_{mi - mb}^{k - l} ,$ are determined by means of the least squares method for residual $\varepsilon_{mi - mb}^{k - l} .$

1.3 Trend Model: Model for Improving Measurement Results with Random Errors

The trend model is a smoothing technique for estimating the real values from scattered measurement data (Kitagawa and Gersch 1984). It is composed of a system equation and an observation equation, as follows:

$$\begin{aligned} & \Delta^{k} u_{n} = v_{n} \\ & y_{n} = u_{n} + w_{n} \\ \end{aligned}$$

(4)

where u _n represents the estimates for the exact values of the displacements and y _n is the measured displacement. The measurement interval is Δt and subscript n denotes progressing time t (t = nΔt). Δ is the operator for the finite difference (Δu _n  = u _n – u _{n − 1}) and Δ^k means the rank “k” difference.

Equation (4) is a kind of probability finite difference equation for rank k. v _n and w _n are white noises with an average value of 0, a standard deviation of τ, and an observation error with a standard deviation of σ.

The trend model can yield good estimates for exact displacements from scattered data obtained from the GPS monitoring system. Through experiments and practical applications, it was proven that the system can detect displacements of 1–2 mm and displacement velocities of 0.1 mm/day (Shimizu and Matsuda 2002).

1.4 Tropospheric Model

Modified Hopfield model (Hoffman-Wellenhof et al. 2001)

Tropospheric delays are defined by the following equation:

$$\Delta R^{\text{Trop}} = 10^{ - 6} \int {N^{\text{trop}} {\text{d}}s}$$

(5)

where $N^{\text{Trop}}$ is the refractivity.

Hopfield showed the possibility of separating $N^{\text{Trop}}$ into dry and wet components. The dry part results from the dry atmosphere, while the wet part results from water vapor. Equation (5) becomes:

$$\Delta R^{\text{Trop}} = 10^{ - 6} \int {N_{d}^{\text{Trop}} {\text{d}}s} + 10^{ - 6} \int {N_{w}^{\text{Trop}} {\text{d}}s}$$

(6)

Using real data covering the whole earth, Hopfield empirically found a presentation of the refractivity of the dry component as a function of height h above the surface:

$$N_{d}^{\text{Trop}} = N_{d,0}^{\text{Trop}} \left( {\frac{{h_{d} - h}}{{h_{d} }}} \right)^{4}$$

(7)

where the height of dry component h _d is assumed to be the following equation:

$$h_{d} = 40136 + 148.72\left( {T - 273.16} \right)\;({\text{m}})$$

(8)

where T is the absolute temperature (K). Similarly, the refractivity of the wet component is assumed to be:

$$N_{w}^{\text{Trop}} = N_{w,0}^{\text{Trop}} \left( {\frac{{h_{w} - h}}{{h_{w} }}} \right)^{4}$$

(9)

where the average value, $h_{w} = 11,000$ (m), is used as the height of the wet component.

Models for dry and wet refractivity at the earth’s surface have been used for some time. The corresponding dry and wet components are:

$$N_{d,0 }^{\text{Trop}} = c_{1} \frac{p}{T} \quad c_{1} = 77.64 \;{\text{(K}}/{\text{hPa)}}$$

(10)

$$N_{w,0}^{\text{Trop}} = c_{2} \frac{e}{T} + c_{2} \frac{e}{{T^{2} }} \quad c_{2} = - 12.96 \;({\text{K}}/{\text{hPa}}), \quad c_{3} = 3.718\, \times \,10^{5} \;({\text{K}}^{2} /{\text{hPa}})$$

(11)

where p is the atmospheric pressure and e is the partial pressure of the water vapor, namely:

$$e = 6.112\, \cdot \left( \frac{RH}{100} \right)\, \cdot \,\exp \left( {\frac{17.62T - 4813}{T - 30.03}} \right)$$

(12)

where RH is the relative humidity (%).

1.5 Terminology

1.5.1 Antenna Phase Center

The electronic center of the antenna often does not correspond to the physical center of the antenna. The radio signal is measured at the antenna phase center. The phase center cannot be physically measured. The offset of the physical phase center from an external point on the antenna can be known commonly by referring to the base/bottom of antenna.

1.5.2 Baseline

It is the length of the three-dimensional vector between a reference point and a measurement point (or between a pair of measurement points) for which simultaneous GPS data are collected.

1.5.3 Baseline Analysis (Post-Processing)

The act of using a computer program to compute baseline solutions from measured data (i.e., carrier phase and navigation data) by receivers at both a reference point and a measurement point is a baseline analysis. The three-dimensional relative coordinates (latitude, longitude, and height) of a measurement point from the reference point are provided in the 1984 World Geodetic System (WGS84).

1.5.4 Carrier

It is the radio frequency sine wave signal. In the case of GPS, there are two transmitted carrier waves, namely, L1 and L2. The L1 carrier frequency is 1,575.42 MHz and the L2 carrier frequency is 1,227.60 MHz.

1.5.5 Dual-Frequency (L1/L2) Receiver

A type of receiver that uses both L1 and L2 signals from the GPS satellites is a dual-frequency receiver. A dual-frequency receiver can compute more precise position fixes over longer distances and under more adverse conditions by compensating for ionospheric delays.

1.5.6 Elevation Mask (Mask Angle)

Satellites are tracked from above this angle. Users can avoid interference and multipath errors from under this angle. It is normally set to 15°.

1.5.7 Epoch

It is a measurement interval used by a receiver when measuring and recording the carrier phase.

1.5.8 Ionospheric Delay

Ionospheric delays occur as a result of refractive effects due to the total electron content when signals propagate through the ionosphere. The errors can be substantially removed by using a dual-frequency receiver. When a baseline length is less than a few kilometers, it will be mitigated by taking the difference in carrier phases at two points (Appendix “Fundamental Equations for Relative Positioning”).

1.5.9 L1/L2 Carriers

The frequencies of the L1 and the L2 carriers are transmitted by the GPS satellites.

1.5.10 Multipath

Interference, similar to ghosting on television, is called a multipath error. It occurs when the GPS signals traverse different paths before arriving at the antenna, typically as refracting from structures or other refractive surfaces (e.g., the ground, walls, fences, etc.) near the antenna.

1.5.11 Navigation Data

Data messages, containing the satellite’s broadcast ephemeris, the satellite clock (bias) correction parameters, constellation almanac information, and satellite health constitute the navigation data. They are transmitted from satellites.

1.5.12 Observing Session

A period of time over which GPS data are collected simultaneously by two or more receivers is called the observing session (length).

1.5.13 Point Positioning

Point positioning is a method of obtaining the absolute coordinates (longitude, latitude, and height in WGS84) of a point in an instant by one receiver. The receiver measures the transit time of the signal from satellites to the receiver and receives the navigation data. The standard accuracy is around 30 m.

1.5.14 Relative Positioning

The determination of the relative positions between two or more receivers, simultaneously tracking the GPS signals, is the relative positioning. One receiver is set on the reference point, while the second receiver is set on a measurement point. Three-dimensional relative coordinates of the measurement point are provided in WGS84.

1.5.15 RINEX

RINEX is the Receiver INdependent Exchange format. It is a set of standard definitions and formats to promote the free exchange of GPS.

1.5.16 Single-Frequency (L1) Receiver

A single frequency receiver is a device that can receive signals (L1 wave) and navigation data; it can measure the L1 carrier phase during a specific time period. It is used for the relative positioning of short length baselines.

1.5.17 Static Method

This belongs to relative positioning. It is commonly used due to its reliability and ease of data collection. It is performed by setting up antennas at two or more points for a predetermined observing session length.

1.5.18 Tropospheric Delay

Tropospheric delays occur as a result of refractive effects due to the air density when signals propagate through the troposphere. These errors can be reduced by using an appropriate model to correct tropospheric delays for a short baseline length.

1.5.19 WGS84

WGS84 is the World Geodetic System 1984. It is a global geodetic datum defined and maintained by the U.S. Department of Defense. A mathematical model (or reference ellipsoid) of the earth, whose dimensions were chosen to provide a “best fit” with the earth as a whole, is used. Descriptions of the GPS satellite orbits in the navigation message are referenced in WGS84.