PHYSICAL CORRELATION?   IT MAY BE DANGEROUS!

Mikhail D.Gerasimenko, Nikolay V.Shestakov
Far East Federal University
Institute of Applied Mathematics FEBRAS, Radio st., 7, Vladivostok 690041, RUSSIA

Abstract. Unknown inestimable systematic errors that influence different observations give the reason for the physical correlations and not with algebraic ones. It is very important especially in the case of GPS measurements. Different investigators have obtained correlation coefficients regarding these reasons of up to 0.85 for identical satellite configuration and they are attempting to use these correlations or different correlation functions in the post processing of GPS measurements. As illustrated in our report, it may be very dangerous for practice and could lead to extraordinary wrong results. One of these unexpected effects may be artificial increasing the weight P of the final results. In limit case it may be that P→∞ if the weight even one observation p→0. This is absurd result clearly and it has no common physical sense. That's why the formal estimates of accuracy do not reflect the real situation with physical correlation of observations. It may be artificial improving of accuracy.  We have met this phenomenon when we developed the software of optimal GPS geodetic network design for fault- mechanics studies. Considering this undesirable phenomenon it should be noted that we do not exclude that it is needed to re-estimate different methods of improving of GPS results that are involving in post processing measurements different non-diagonal covariance matrix of observations as well.

1.	Introduction

Today GPS techniques are widely used in geodesy and geodynamics. In connection to it the interest to using the adjustment theory of correlated observations is significantly increased. The problem is that the GPS data from closely distributed stations are correlated because of practically identical atmosphere conditions, using the same satellite constellation and some other effects. Unknown inestimable systematic errors that influence different observations in an almost identical way give the reasons for the physical correlations. Georgiadou and Kleusberg (1988), for example, analyzed the multipath effect of one site on two consecutive days and they obtained correlation coefficients regarding this error source of up to 0.85 for identical satellite configurations. The influence of physical correlation on the accuracy estimates of GPS measurements is shown by El-Rabbany (1994) in an empirical way. These correlations have a considerable influence on the evaluation of GPS measurements (Schwieger, 1996). Sometimes this problem is solved by introducing into mathematical process a correlation matrix of GPS observations (Ananga et al., 1994; Johnson and Wyatt, 1994). The correlation coefficients of this matrix are processed by using some or another experimental or intuitive assumption which might don’t reflect real physical situation. 
		These correlation coefficients may have an extraordinary influence on processing results and in most complicated cases may lead to the absurd final result. But we can’t find any investigation connected with discussed here problem. This situation can be explained by great dimension of the correlation matrix, significant processing volume and its complication. In this case many details of computation process are hidden from researchers, especially if all computations are made by using a computer. Artificial accuracy increasing of the final processing results is interpreted by them as a correct choice of the modeled function that were used for correlation coefficients. But in reality this accuracy increasing may be only a result of formal operations and doesn’t reflect a real situation!
		We met with this phenomenon when we developed the software for optimal GPS network design for fault mechanics studies and earthquakes prediction. As a base conception we used an optimal design methodology by Gerasimenko et al. (2000). Correlation coefficients were processed by equation Johnson and Wyatt (1994)
		,				                                                          (1)
where L is a coherent length, () is a distance between geodetic points i and j. In some situations a quantity  may has a very big value. For example, if L=5 km as assumed by Johnson and Wyatt, 1994, and =1 km we have result. Those great values of correlation coefficients may have an extraordinary influence on processing results and in most complicated cases may lead to the absurd final result. But we can’t find any significant researches connected with discussed here problem. Mentioned above abnormal influence of correlation coefficients was displayed as increasing of accuracy of the fault parameters when we delete useless GPS stations (by weight decreasing to insignificant level) from observation scheme. This result doesn’t agree with a common sense. 
		We studied available for us scientific literature related with discussed here problem and have found there are no significant results for last ten years in correlation errors processing theory. Situation in this sphere is the same as fixed in a well-known Russia monograph of Kemnits (1970): “… this shortcoming of classic observation’s errors theory was removed by formal applying it to correlated observations through correlation theory”. But this “formal” using of mathematical methods for modeled physical correlation of observations may lead to extraordinary and unexpected results (as showing below on simple examples). 

2.	Weighted mean of the same correlated variables

Let n correlated observations of the same variable y be given as a vector of observations  with non identical weights. Weight coefficients matrix of the vector observations  is
,								           (2)
where  is the diagonal weight matrix, R is the correlation matrix. The matrix R has units on the main diagonal. Its non diagonal elements  are correlation coefficients of observations  and .
		Solution by least square method a system of equations   
          								           (3)
give the estimate for  unknown quantity
              ,	                          					(4)
where matrix  consists of units only,  is the weight matrix, V is the vector of residuals. The value 
										(5)
is a weight of weighted mean of correlated observations , where  are the elements of the weight matrix . Therefore estimate for the variance  is 
.									(6)
		Let’s consider more detailed a case when n=2, i.e. only two correlated observations  and  with correlation coefficient   were made. In these circumstances we will have

,						(7)
,							(8)

.						(9)
		Finally we make a preliminary inspection a few particular examples of using equations (7)-(9) in the next section.

3.	A few particular examples of extraordinary absurd results

		Example 1. When all observations have identical weights, from equation (9) we have the least squares estimate. It is obvious and has no any objections. But with its weight 
								                     (10)
a situation isn’t so simple. As correlation coefficient can changes in limits  thus when r increases from 0 to +1, the weight  decreases from 2p to p. It agrees with common sense. But if r changes from 0 to –1, the quantity. In limit when  we obtain the weight ! This is a clearly absurd situation.
		Example 2. Not identical weights. Let’s assume, , where c is some constant. In that case we obtain
,			           				          (11)
.						          (12)
From equation (11) it follows that if  then weight =min  when ! More than that when  (second observation is absent), the  weight
								          (13)
and if  we are obtaining the absurd result . We are having this situation with only one observation that have weight !
		Example 3. From equation (12) as a particular case we could obtain that if  or  then the estimate. And its weight , i.e. the final results are fully independent from a first observation which cannot happen for independent observations.
		Example 4. Let’s assume that we observed some distance twice with correlation coefficient . The value of observations are m and m. The weights are equal to  and . According to common sense it obvious that the result  must be in limits . Using the equation (12) we obtain ≈102.57m! The result seems to be obviously absurd because it is more than numerical value of any observations.  For independent observations this phenomena is not observed. 

4. Conclusion

At first glance the above discussed effects relate with only some extreme cases and they have no significant interest for practice. But here we showed the case of only two correlated observations. And even in this simplest case we met unexpected peculiarities and absurd results which can not be left without attention. And what effects could we wait in enough spacious geodetic network? By another words, a problem of investigation of correlated observations and their using in practice needs in solving. Considering these undesirable phenomena it should be noted that we do not exclude that it is needed to re-estimate different methods of improving of GPS results that are involving in post processing measurements different non-diagonal covariance matrix of observations as well. The formal estimates of accuracy do not reflect the real situation with physical correlation of observations. It may be artificial improving of accuracy.  

  References

1.	Ananga N., Coleman R., Rizos C. Variance-covariance estimation of GPS networks, Bulletin Geodesique, Vol.68, 77-87, 1994.
2.	El-Rabbany A.E.-S. The effect of physical correlations on the ambiguity resolution and accuracy estimation in GPS differential positioning,  Ph.D. dissertation, University of New Brunswick, Department of Geodesy and Geomatics Engineering, Technical Report No.170, Frederictin, 1994.
3.	Georgiadou Y., Kleusberg A. On Carrier Signal Multipath Effects in Relative GPS Positioning, Manuscripta Geodaetica, 172-179, 1988.
4.	Gerasimenko M.D., Shestakov N.V., Teruyuki Kato. On optimal geodetic network design for fault-mechanics studies. Earth, Planets and Space, Vol.52, 985-987, 2000.
5.	Johnson H.O., Wyatt F.K. Geodetic network design for fault-mechanics studies. Manuscripta geodaetica, Vol.19, 309-323, 1994.
6.	Kemnits U. V. Mathematical processing of the results of correlated observations, Moskow, Nedra, 1970. 
7.	Schwieger V. An Approach to Determine Correlations between GPS Monitored Deformation Epochs. Proceedings of the 8th FIG International Symposium on Deformation Measurements, 25-28 June 1996, Hong Kong, Hong Kong Polytechnic University, 17-26, 1996.