Study on accuracy of GPS for its application in SAR interferometry

Study on accuracy of GPS for its application in SAR interferometry

Parag S. Narvekar, K. S. Rao
Center for Studies in Resources Engineering
Indian Institute of Technology Bombay
parag_narvekar@yahoo.com

Abstract
One of the main applications of GPS is to find out the exact location of any point of interest almost anywhere on the globe. For some of the application this information is needed very precisely like for SAR Interferometry. Space borne synthetic Aperture Radar (SAR) interferometry is an exciting and powerful technique that has gained prominence in recent years through the ERS-1 satellite mission. Its potential is illustrated by recent examples of application to detection of centimetre-level surface displacements and production of wide area digital elevation models (DEMs), studies are going on to improve the accuracy to millimeter level. The position information obtained from the GPS can be used for improving the accuracy of ground control points in SAR system. Also the heights at particular locations can be compared with the heights of same locations in DEMs and can be used to improve the accuracy of SAR Interferometry. Thus there is need for obtaining GPS measurements at very high accuracy. In the present work the various sources of inaccuracies in GPS are studied e.g. Multipath, Receivers Noise, Clock error etc. A special study is done on the Atmospheric effects on signals using data of atmospheric parameters (pressure, temperature, humidity etc.) obtained from Indian Meteorological Department Pune and an attempt is made to find out in what extent it can affect the results.

Introduction
GPS is a satellite navigation system designed to provide instantaneous position, velocity and time information almost anywhere on the globe at any time and in any weather condition. Position information i.e. latitude longitude and height obtained from GPS can be compared with the DEMs generated by SAR Interferometry. GPS can be used to provide the Ground Control Points for SAR system and also to study the atmospheric effects on SAR Interferometry by comparing the Atmospheric delay in both the systems. Thus GPS can be used for improving accuracy in SAR Interferometry and there is need to estimate these errors and to bring them at high level of accuracy so that it can be compared with SAR System. Also a recording of a position is not of much use if it is not accompanied by some form of error estimate. There are various sources of inaccuracies are involved in Global Positioning System measurement, some of them are as follows Satellite clocks, Orbit error, Receivers Noise, Multipath, Intentional degradation of the satellite signal — Selective Availability (SA) (The government turned off SA in May 2000), Anti-spoofing (AS), Satellite geometry/shading, Ionospheric Delay, Tropospheric Delay etc.

Theory
There are two types of service available to GPS users, the SPS and the PPS.

SPS – Standard Positioning Service is the positioning accuracy that is provided by the GPS measurements based on the single L1 frequency C/A code.

PPS – Precise Positioning Service is the highest level of dynamic positioning accuracy that is provided by GPS measurements based on the dual frequency P-code.

The signals, which are generated from a standard frequency of 10.23 MHz, are L1 at 1575.42 MHz and L2 at 1227.60 MHz and are often called as the carriers. These signals are effected by various errors, some of them are discussed below and methods to tackle this problem are given. Atmospheric errors are discussed separately in Results and Discussion.

Selective Availability (SA)
SA essentially consists of two different components, known as dither and epsilon. Dither is an intentional manipulation of the satellite clock frequency resulting in the generation of the carrier waves and the codes with varying wavelengths. In other words, under SA, the distance between each C/A code chip will be variable, and no longer the designed 293m (chip length). The replica code generated within the receiver will still assume the chip length to be 293m and pseudorange measurements are based on this. Typical pseudorange errors for satellites with SA imposed are +/-100m.

Anti-spoofing
AS further alters the GPS signal by changing the characteristics of the P code by mixing it with a so-called W code resulting in the Y code. It is the latter that is modulated onto the carriers and is thus designed to prevent the ability of the receiver to make P code measurements. Many receiver manufacturers have already developed techniques to still make P code measurements with only a small addition in added noise (cross correlation techniques), Talbot (1992) or Ashjaee and Lorenz (1992).

Satellite Cloak Error
One billionth of a second (one nanosecond) of inaccuracy in a satellite clock results in about 30 centimeters (one foot) of error in measuring the distance to that satellite. For this reason, the satellites are equipped with very accurate (Cesium) atomic clocks. Even these very accurate clocks accumulate an error of 1 billionth of a second every three hours. We are trying to liniarize the pesudorange equations for the four satellites given as

with i = 4
Where Dti term represent the satellite clock corrections. GPS works on the principal that all of the satellites are synchronized with GPS master time. The pseudorange measurements must share a common time basis otherwise the position solution will be highly inaccurate. The satellite clock, however, are not in synchronization with one another. The satellite clock correction can be given by the equation

A further correction is made for single frequency users because the af0 clock offset term is based on dual frequency observations. af0, af1, af2, etc. are calculated separately.

which is the quantity measured on the ground before satellite launch. Thus, V GD could have a magnitude of 18 ns or larger for a given satellite. We consider here four psuedoranges because that is the minimum required to solve for the four unknowns of the four dimensional position and receiver clock offset. A taylor series is used to expand these equations about an estimated position (X,Y,Z,tB) with the higher order terms ignored. For the pseudorange measurement i

where (PRi - Dti) is the pseudorange based on the estimated position. Thus, each pseudorange can be approximated as

The coefficients of ¶X, ¶Y, ¶Z are direction cosines. We can now write the linearized equations in matrix form

which can be written in the form

¶Y = H ¶ß

An iterative computation is carried out to converge on the position solution. In general, ¶ Y is an m x 1 vector where m is the number of satellite in view. Accordingly H is an m x 4 matrix. One method of solving this equation is to take the generalized inverse of H

¶ß = (H^TH)^-1H^T ¶Y

The result can be used to converge on the solution within four to five iterations.

It is not necessary for the initial estimated position and the receiver clock offset to be accurate. The basic algorithm is as follows:

Calculate the satellite positions
Apply satellite clock corrections to the pseudorange measurements
From initial position estimate Iterate until convergence.

Multipath Error
In measuring the distance to each satellite, we assume that the satellite signal travels directly from the satellite to the antenna of the receiver. But in addition to the direct signal, there are reflected signals, from the ground and the objects near the antenna, that also reach the antenna through indirect paths and interfere with the direct signal. This has a number of effects: it may cause signal interference between the direct and reflected signal (see Figure 2 below) leading to noisier measurement, or it may confuse the tracking electronics of the hardware resulting in a biased measurement that is the sum of the satellite-to-reflector distance and the reflector-to-antenna distance.

The magnitude of the multipath effect on a phase observation can be estimated from the following mathematical relation (HOFMANN-WELLENHOF et al, 1998):

where
Df_m is the shift in carrier phase of the combined signal received at the antenna due to multipath, Is a damping factor which varies between 0 (no reflection) and 1 (reflected signal as strong as direct signal).

Some options for reducing the effect of multipath are:

Make a careful selection of antenna site in order to avoid reflective environments.
Use a good quality antenna that is multipath-resistant.
Use an antenna groundplane or choke-ring assembly.
Use a receiver that can internally digitally filter out the effect of multipath signal disturbance.
Do not observe low elevation satellites (signals are more susceptible to multipath).
In the case of pseudo-range positioning (single point or differential), averaging the computed results over a period of time will reduce the contribution of multipath errors on the averaged pseudo-range solution.

In the case of carrier phase positioning, longer observation sessions will tend to diminish the impact of multipath on the final baseline results.

Selecting a GPS Receiver
There are two main groups of receivers; those designed to track multiple GPS satellites simultaneously and those that sequence between satellites.

Sequencing Receivers - use a single channel to measure the C/A code and move it (multiplex) from one satellite to the next to gather this data. They usually have less components and circuitry so are cheaper and consume less power. Unfortunately, the sequencing can interrupt signal measurement and timing resolution (noise errors) of the satellite signal detection and can limit their overall accuracy.

Parallel Receivers - also know as Continuous Receivers, can monitor several satellites simultaneously. These units are valuable in high dynamic or high accuracy applications, so they are often used for mapping, surveying and scientific purposes. Besides the obvious advantage of being able to continuously measure a position, these multi-channel receivers can also eliminate the 'noise measurement' problem and provide for 'all in view' satellite tracking. Another capability of these receiver types is that with modern signal processing techniques, by the comparison of channels to each other, the units can 'calibrate out' any interchannel biases that can affect timing measurement and signal detection accuracy resolution.

Other Considerations - commercial single frequency (L1 signal) GPS receivers are achieving greater accuracies by tracking both the pseudo-random code and the L1 carrier frequency. Called 'carrier-aided tracking', this technique makes it possible for the receiver to resolve, with good precision, the exact 'edge' of the psuedo-random code signal. Thus more precise timing measurements, which in turn, translate into a better calculated positions can be achieved. Of course, dual frequency receivers provide for double satellite signal range measurement, plus compute the atmospheric 'delay' errors and use the carrier-phase observation techniques to provide accurate positioning improvement.

Atmospheric Effects
Atmospheric effect is divided in two types i.e. Ionospheric effect and Tropospheric effect depending on layers of atmosphere. From ground to above around 40 Kms is the troposphere and from 40 Kms to around 1000 Kms it is considered to be Ionosphere. The effects of atmosphere on GPS are discussed in separate section in results and discussion.

Results and Discussion
The atmospheric effect on GPS signals are studied and meteorological data obtained at Indian Meteorological Department Pune is used to fined out at what extent it can effect the measurement.

Ionospheric delay
The magnitude of the effect of the ionosphere is much more during the day than during the night. The magnitude also has a cyclical period of 11 years that reaches a maximum and a minimum. For the current cycle, the ionosphere will reach its peak magnitude in 1998 and its minimum in 2004. The cycle will then be repeated. The effects of the ionosphere, if not mitigated, can introduce measurement errors greater than 10 meters. The impact of the ionosphere on electronic signals depends on the frequency of the signal. The higher the frequency, the less is the impact. So if we transmit the patterns simultaneously via two different frequencies, the ionosphere may delay the code on one frequency, for example, by 5 meters and on the other frequency, say, by 6 meters. We cannot measure the magnitude of these delays, but we can measure their difference by observing the difference on their arrival time, which in this case translates into 1 meter of effective distance between them. By measuring this difference and using known formula for frequency dependency of the ionosphere delay, ionosphere effect can be removed.

For the GPS system with single frequency there will be the ionospheric error involved in the measurements. depending on radiosonde data available at meteorological department in Ahmedabad, phase shift in the signal with increasing Total Electron Content (TEC) for Indian atmospheric conditions is plotted as shown in figure 3.

Tropospheric delay
Tropospheric delay is a function of the satellite elevation angle and the altitude of the receiver, and is dependent on the atmospheric pressure, temperature, and water vapour pressure (BRUNNER & WELSCH, 1993). Models such as Hopfield, modified Hopfield, Lanyi, Chao etc. are used to estimate the tropospheric delay. Figures 4, 5, 6 shows the phase change in the signals with increasing atmospheric parameters i.e. Temperature, Pressure and Humidity.

Conclusion
Errors affecting the GPS measurements are studied and some ways to tackle this problem are found out e.g. satellite clock, multipaths, etc. In the present work the effect of atmosphere on signals are studied and the range of there effects on signals are plotted. If these solutions are applied to the GPS measurements the accuracy can be improved. Thus the position and height information can be compared with the height in the DEM generated by SAR interferometry technique. Also the atmospheric delay in GPS can be compared with the delay in SAR system so that the effect of atmosphere in SAR can be minimized.

References

S. K. Khare, M. S. Rao and K. S. Rao IIT Bombay, Generation of DEM using SAR interferometry technique - a case study of Bankura test site.
Ramon F. Hanssen (2001), Radar Interferometry the thesis, www.geo.tudelft.nl
Linearizing the GPS Pseudorange Equation, Digital Library and Archives, Scholarly Publishing Services.
Multipath Mitigation of Continuous GPS Measurements Using an Adaptive Filter, et.al, School of Geomatic Engineering, The University of New South Wales Sydney, NSW 2052, Australia.
Experimental Results on Three Multipath Compensation Techniques for GPS-based Attitude Determination, Alessandro Pasetti and Luisella Giulicchi, European Space Agency, ESA-Eatec.
Global Positioning System Standard Positioning Service Performance Standard, October, 2001, Assistant Secretary of Defense for Command, Control, Communications, and Intelligence.