Sammendrag
It is well known that the ordinary Global Positioning System (GPS) fails to provide location and time information under water. The reason is that the electromagnetic signals from the orbiting satellites are heavily damped in water and hence can not be detected by the receiver in most cases of interest. Acoustic waves are the canonical alternative, and there exist a variety of acoustically based systems. It is important to estimate the accuracy of position estimates and if possible correct for the errors. This is done systematically in the GPS case by the concept of Dilution Of Precision (DOP), and is also the natural approach underwater. The main issue presented below is a discussion and analysis of the concept of DOP in the context of Underwater Positioning Systems (UPS). This includes statistical models that differ from the one canonically used. Some of the sources of errors in the UPS case differs substantially from the GPS case. It is in particular demonstrated that the maximum likelihood estimator is biased. Alternative estimators, including an unbiased estimator and an optimal estimator, are discussed. The results are also briefly discussed in the context of an actual model experiment with ultrasound in air. A side effect of this is the demonstration of certain issues which have been ignored in the previous general discussion. The experiment also indicates that the results are relevant in other contexts. Other important classes of examples are given by Real Time Locating Systems as defined by the standard ISO/IEC 24730, and by Wireless Personal Area Networks as described in the IEEE 802.15 standard. These other contexts also provide most useful sources for research publications with results of relevance for UPS, and more generally for underwater communication systems.