An Empirical Approach for the Estimation of GPS Covariance Matrix of Observations (2005)

The covariance matrix of observations plays an important role in GPS data processing. For example, the weighting of the observations are based on the covariance matrix of the observations. The covariance matrix of the observations also plays a role in the estimation of the covariance matrix of the parameters. In this paper we present an empirical stochastic approach to create covariance matrices for GPS. Our approach aims at a more realistic and complete information about the stochastic behavior of GPS observations. As a result, improvement in quality and quality control of estimated coordinates using GPS measurements is obtained. In the present case study, baselines are processed using pseudoranges. The observations are considered as a time series after double differencing. The advantages and ION 61st Annual Meeting The MITRE Corporation & Draper Laboratory, 27-29 June 2005, Cambridge, MA 1098 disadvantages of this empirical stochastic approach are explored, comparing it with more usual methods used for building observations covariance matrix. An analysis is made in terms of obtained coordinates where baselines between points with known coordinates were processed. Two baselines with length of approximately 2 km and 20 km were processed using three different weighting schemes for observations. In general, when the empirical approach was used a good improvement in the bias was obtained. The mean biases were at least 44 % smaller in the 2 km baseline case and at least 21 % in the 20 km baseline case. The model certainly brings improvement to the solution, with better results achieved for height determination, with improvements up to 72 % in mean bias for the shorter baseline. INTRODUCTION The covariance matrix of observations plays a fundamental role in GPS data adjustment. Usually the method of least squares is used to compute the coordinates of a given receiver. In the adjustment the covariance matrix of observations drives how each one of the observations will contribute for the update of the parameters. This matrix includes not just variances, but also a relation between all observations, the covariances. An usual approach to estimate the covariance matrix of observations is to set a weight for all observations and then, in case of baseline, propagate it using the double difference operator. It is not assumed any correlation between observations made in two different epochs, the so called autocorrelations. The weights can be set based on different types of information, such as elevation angle of the satellite or signal-to-noise ratio. Sometimes even the identity matrix (equally weighted observations) is used. In this work, an approach to build the covariance matrix of observations for GPS data processing is presented. The main objective of such technique is to populate the covariance matrix with realistic information, estimated by means of a stochastic analysis of the raw data. Trying to have as much as possible information inside the covariance matrix is a way to carry into the adjustment model a realistic picture of the quality and behavior of the observations. Eventually, the procured parameters will be adjusted using this matrix. The ultimate goal of this work is to improve the quality of GPS data processing. In order to analyze the data assuming a stochastic behavior a few requirements need to be satisfied, such as stationarity. Because of that, the GPS data need to undergo some modifications before it can be used. In our Empirical Stochastic model (which will be called herein with ESto model) all analysis is made based on raw data, without any external information, before the adjustment. In next Section a background explanation about stochastic processes and their analysis will be made. Next, the treatment given to GPS data in order to make its analysis as a stochastic process possible will be shown. A case study was carried out and its results are shown, as well as conclusions and recommendations for future work. STOCHASTIC PROCESSES A time series is a series of observations made through time. An important feature of a series is that, usually, the observations made at subsequent epochs are dependent on each other. The analysis of a time series is based on this dependency. In order to carry out the analysis, generally stochastic models are used. The utilization of mathematical models to describe physical phenomena is very common. Sometimes a model based on physical laws can be capable of determining the value of a given variable for a specified time. This is possible when the variable (in our case the GPS observable) is totally deterministic. A stochastic process is a statistical phenomena which occurs through time according to probability laws, [Box et. al., 1994]. A time series can be considered as a realization of a stochastic process. There is a particular process, called a stationary process, which has a particular statistic equilibrium state. A stochastic process is called widely stationary when its properties remain unaffected when the time origin is changed. This means that the joint probability function of a process with n observations n t t t z z z ,..., , 2 1 , observed at time instants n t t t ,..., , 2 1 is the same as a process with n observations k t k t k t n z z z + + + ,..., , 2 1 , observed at time instants k t k t k t n + + + ,..., , 2 1 for every integer k . If a process is stationary, we can say that: ION 61st Annual Meeting The MITRE Corporation & Draper Laboratory, 27-29 June 2005, Cambridge, MA 109 0 , 0 ) ( ) ( ≥ ≥ ∀ = + k t z z k t t ρ ρ , (1) where ) ( t z ρ is the probability density function, and because of that, we can conclude that the mean of a stationary process is constant: