Biased Estimator And Its Application In Surveying

Ill-conditioning problems extensively exist in data processing of surveying and other various engineering fields, in GPS fast stationary positioning surveying, for example ,because of the tiny time intervals when collecting raw datas, the coefficient matrix of normal equation are often ill-conditioned ,as a result ,the deviation of the solution of the parameters by the Least Square estimator are too large ,and the process to acquire these solutions become very unstable. In fact, analyzing the essence, overcoming the effect of ill-conditioning and obtaining more accurate and stable parameters estimator is a new task in GPS surveying data processing, which have been determined as an important studying field in contemporary surveying error theory and engineering data processing by the International Association of Geodesy (IAG).Based on the surveying practice,the author researched the biased estimator and its theory and methods on solving the ill-conditioning problems. The following are the main results :1. According the physical situation of surveying and the different mechanization of ill-condition, the author analyses the causes of ill-conditioning in surveying field by sorting them and explored the methods which can resolve the ill-conditioned problems both with empirical knowledge and mathematical methods.2. The comparisons between several most important biased estimators, which are used widely in almost all engineering fields nowadays, that is ridge estimator , principal components estimator and LS estimator etc. are conducted by using the criterion of mean squared error; and the conditions to show the superiority of these estimators over the LS estimator have been obtained by comparing the results computed with these biased estimators methods with LS ,the author researched the effect the solutions can be improved. The results show that these methods all can improve the LS estimation .3. Several common ridge parameter determation methods and their theory have been discussed;Base on some experiment,the result that these ridge parameters'determation methods can improve LS is researched, these experiment results under these methods show that they all can improve the LS estimate solutions .Then the author generalizes and summarizes the applicability range and these methods'merits and faults.4.A new partial biased estimate method which base on the Singular value decomposition is put forward, for the convenience to discuss ,the new method is abbreviated as"SVDRE". This method is presented because the current part biased estimate method only divide the coefficient matrix of normal equation into two parts:"ill-conditioning"and"non-ill-conditioning",but not emphaze the fraction of the"ill-conditioning part". The new method divides the Singular values into different parts according the condition of coefficient matrix and gives different constants to revise the small singular values and the results show that this method can improve the accuracy of the solutions of parameters, besides, while the"partial ill-conditioning"effect is notable ,this method can acquires more accurate solution than other methods.5.As is known, the biased estimator can solve the ill-conditioned problems, but they must calculate the coefficient matrix of normal equation, however, the condition number of the coefficient matrix of normal equation is twice of the design matrix, this is very bad for number stability .Aiming at this situation, a new method named muti-singular threshold value revising can solve the ill-condition model directly is set out, compared with the current revising singular value method, the new method divides all the singular values into different parts, according the new method ,each singular value can be revised with a suitable constant, Thus, the new method can not only avoid to calculating the normal maritx ,but also takes the distribution of the singular value of the design matrix into account ,in fact it is testified to be a valuable method ,and it can acquire the generalized inverse LS solution which is superior to others when the"partition effect"is notable.6.The theory on how"error transfer method"solves the ill-condition problems is analyzed in detail, and this method is used to solve the ill-conditioned functional model in surveying problems for the first time. after researching, the author conclude that the"error transfer"method can only solve the ill-conditioned of linear system of equations, but can not solve the ill-condition of adjustment model of surveying ,because the"error transfer"method must base on the strict equation but the surveying adjustment model content observe errors, the equation are not the strict equations, and these errors can't known beforehand. The research provides some reference for this method using in practice.7.It is well-known that how to find ways to obtain the integer ambiguity in GPS positioning is the key problem. In the paper, the biased estimator methods are used to GPS data processing and the author try to explore these methods to solve the ambiguity problem, expecting to attain the satisfactory float-point solution in the double difference adjustment model. The results of practical examples shows that the results by the biased estimation methods only have little deviation compared with the true values, which shows the biased estimator can be used to eliminate bad impact brought by the ill-conditioning, then we can obtain the relative high accuracy float result of integer ambiguity, which provide a good foundation for further obtain the integer ambigulity and the accurate baseline increment of coordinates.8. At last the paper summarizes the research productions and at the same time the next development direction is given.