OMEGA PROPAGATION PREDICTION IN A SIMPLE INTEGRATED NAVIGATION SYSTEM FOR OCEANOGRAPHIC USE

Trabajo recibido el 3 de julio de 1980 y aceptado para su publicación el 18 de agosto de 1980.

RESUMEN

INTRODUCCIÓN

Next decade will see the implementation of the Global Positioning System (GPS). This satellite based system will provide continuous and precise timing and position information at all points of the globe. Its implementation and the availability of relatively inexpensive user equipment will have made out of Omega navigation a short lived phenomenon with a useful. life of the order of ten years.

However, in the meantime, Omega navigation is the only radio navigation systeni that is available on a continuous and global basis. Its continuity in time, as compared to present Transit system, makes it an important candidate for several oceanographic uses where knowledge of the relative motion of a slow moving observation platform is a necessary portion of the data collection. Examples of this need are current velocity profiling from a drifting platform and continuous zig-zag profiling of hydrographic parameters from a relatively slow moving ship. Both techniques, that have become of fairly comnion use in oceanographic research, require a relative position accuracy for which the present Omega system accuracy is only marginally useful.

Besides the problem of lane ambiguity which will not concern us here, the problem of propagation error variations is the main source of accuracv degradation in Omega positioning. Presently available propagation error correction (PEC) tables are derived from a combination of theoretical models and an extensive data base of propagation and geophysical parameters (Swanson and Brown, 1972). Although these tables are, at present, the most Precise source of information on propagation errors on a global basis, there are two basic problenis involved with their use: 1) The enormous volume of data requires rather large computing and memory resources to access it in real time applications, 2) The inherent inaccuracy, due to the long term and global character of the prediction: it can not incorporate predictions of local permanent errors, at least not before the calibration/validation program is completed (Vass, 1978), nor variations with characteristic times shorter than a year due to geophysical perturbations that may, or may not, be local. As an example of this inaccuracy, figure 1 shows a comparison between PEC published values and observed values during five arbitrarily chosen days at the corresponding location and date.

Several solutions have been proposed to overcome these problems. In relation to the first problem, simplified algorithms for the generation of PEC values have been proposed (Beavers et al, 1975) for use in automatic Omega navigation. These algorithms produce PEC values almost as precise as those in published tables but with order of magnitude simpler memory and computational requeriments. The degradation of the values so obtained is of the order of the inherent inaccuracies of the published tables.

Fig. 1. Observed vs. theoretical propagation correction values for five consecutive days chosen at random. ----Theoretical values. ___Observed values for the same position and dates.

The second problem is a more fundamental one. lt defines the real accuracy limitation of the system. The most promising solution is differential Omega. This system uses the fact that propagation errors have a space correlation distance of the order of 1500 km for areas not crossed by the day-night line. In differential Omega, a fixed receiver station monitors the propagation variation and this information is used as PEC by mobile stations while they are in the area where the propagation error is assumed to be strongly correlated with that at the fixed station. An accuracy improvement of the order of 50% can be achieved by differential Omega as compared to direct Omega (see, for example Baxa and Piserchia, 1975). The limitations of differential Omega are operational: the need for a fixed nearby receiver station and the need, for real time applications, of transmitting and receiving equipment.

Other costlier solutions to the problems of continuous and precise navigation by means of sophisticated integrated navigation systems combine several positioning instruments (Omega, Transit, sonar, radar and even inertial navigation) with large, dedicated memory and computing resources.

In this paper, an algorithm for an Omega based integrated navigation system is presented. It is a computationally simple prediction scheme that does not require a fixed data base and produces accuracies comparable to that of differential Omega while being independent of a fixed station. It is of the type of integrated navigation systems in the sense that, to produce predictions of propagation error, it needs an external sporadic source of precise positioning. This shortcoming, common to all integrated navigation systems, is considered minor due to the extensive and growing availability of Transit equipment (more than 2000 users in 1977).

In the proposed algorithm a prediction technique suited for short time series is used to obtain Omega propagation errors ahead in time after each external "fix". In the experiment presented, PEC published values are compared to those predicted from fixes given at a frequency of occurrence of the order of Transit fixes (around 90 minutes). Accuracy improvements of the same order as those achieved by differential Omega are obtained. The implementation of the algorithm for real time applications requires very modest computing capabilities as compared to other integrated navigation systems or to methods of in situ estimation of PEC tables with simplified algorithms, as the one proposed by Beavers et al , 1975.

The object of the experiment is to determine a means of efficiently use sporadic precise fixes, of a moving vehicle, to improve the accuracy of continuos Omega positioning and to determine if the frequency of occurrence of fixes produced by present satellite navigation is sufficient to obtain a significant increase in accuracy.

The technique used is self-prediction of PEC values. The external fixes are used as a source of discrete PEC samples. From these samples, a prediction filter that incorporates the statistical information contained in them is computed. The filter is then used to predict PEC values ahead in time.

The data used in the experiment corresponds to two time series of PEC values monitored continuously by fixed recciver stations at two different locations (about 1000 km distant) and taken with a time separation of two months. The time series are ten days long and correspond to the 10.2 khz. signals of the Hawaii North Dakota (C-D) pair.

Both the predicted signal and published PEC values are compared (means removed) to observed values. This comparison has to be seen in the light of the basic difference between a self-Prediction and a parametric prediction, such as that used in the computation of PEC tables or the one proposed by Beavers et al . (1975). While parametric prediction of a signal requires some knowledge of each of the factors that contribute to it, in a self-Prediction scheme, those factors only appear through their net effect on the signal and are predicted by means of direct or indirect estimation of the time autocorrelation of the signal. Thus, in a self-prediction scheme, global and local events are undistinguishable from otherwise unpredictable events such as instrument miscalibration or bias, immediate vecinity antenna disturbances, etc. As long as the contribution of a phenomenon is not white noise, it has an effect on the filter. Thus, except for constant perturbations, that are easily eliminated in both cases, selfprediction has an inherent advantage over pararametric prediction (at the expense of the need of an external source of sporadic fixes) by including a continuous system calibration effect.

A further consideration is necessary for the interpretation of results of the comparison experiment. Expected navigation accuracies are given in phase units of centicycles (cec) and not in their equivalent and more useful distance units. This is done to keep proper generality of the conclusions. Indeed, for a given error in phase measurement, the corresponding radial distance error depends strongly upon the geometry of the transmitters-receiver system and the number of transmitter stations used. Using error estimation formulae for hyperbolic positioning systems derived by Lee (1975), Thompson (1977) shows that for a given geographic location and for a given phase error, actual distance inaccuracies may differ by a factor of two depending upon the factors mentioned above.

The time series to be predicted is non-stationary and has a large spectral peak around 24 hour periods. Thus, a series 24 hours long or more is necessary to do any kind of prediction. A much longer series will improve the prediction of the low frequency energy content but would average out the non-stationary events that are the main source of positioning error. The limit of high frequency predictability is given by the sampling rate of the past data used in the prediction and this is a system constraint given, for example, by Transit fixes frequency.

Within these limitations, a prediction algorithm will be better than others if it estimates better the time autocorrelation of the signal for time lags longer than the length of the vigen time series.

It has been shown (Burg, 1972) that for short time series, the error prediction filter obtained by Levinson's recursion algorithm produces consistently better spectral estimates over truncation methods in which it is assumed that the autocorrelation is zero for time lags longer than the length of the given time series.

Levinson recursion produces a filter whose convolution with the signal has a white spectrum; this means that the residue of the prediction has a zero information content. Although Levinson recursion filters are obtained simply by minimizing the mean squared prediction error of forward and backward prediction, the algorithm guaranties that the filter is minimum phase and, thus, that the corresponding spectrum is positive (Clearbout, 1975).

Following Levinson (1947), the recursion algorithm can be succinctly described as follows: a filter of length one is obtained by minimizing the sum of its foward and backward prediction errors:

where xt is the signal at time t and (a, 1) and (1, a) are the forward and backward prediction filters of length. one. The next step in the Tecursion is to assume that the filter of length two is a linear combination of the previous forward and backward filters:

and the corresponding mean squared error is minimized with respect to c. Longer filters are obtained by recursion of the same scheme. Using this algorithm requires n² multiply-adds to compute a filter of length n.

Once the prediction algorithm has been established, one has to decide the length of the filter. The optimum length depends on properties of the physical processes that produce the signal. Several lengths were tested that are consistent with the roposed use of the algorithm, and with the non-stationary character of the signal. Results are shown in later sections.

Events with a characteristic time of the order of the sampling period can not possibly be predicted, but non-stationary features of longer life can be "followed" by the prediction scheme if the filter is recomputed every time a new data enters. Following this idea, Levinson recursion is used in a data-adaptive filter scheme incorporating into the filter the information contained in every new data point.

In order to simulate realistic applications, all predictions were made with published PEC values as initial conditions. The prediction filter is first computed from those values and afterwards is recomputed every time a new data point enters. Thus, after a time period equal to the length of the filter the system has completely forgotten the initial theoretical information.

To test non-trivial character of the adopted prediction scheme, four different predictions were compared to the observed values: published PEC values, Levinson recursion filter prediction, low order polynominal extrapolation and permanence of values from one day to the next. The results of the last two proved to be consistently worse than PEC values according to all statistical criterions used; results are only given for comparisons between observed values and the first two predictions.

All runs were made with fully adaptive filters in an attempt to reproduce the non-stationary character of the signal. Several filter lengths were tested: from 24 hours to 5 days. The results obtained for those filter lengths reflect the characteristic times of the dominant perturbations in the signal. For 24 hours, the prediction tends to overestimate the permanence of short lived perturbations. In this sense, its prediction is very similar to the one obtained by simple permanence: it reproduces almost faithfully the large events occurring a day earlier giving an overabundance of large errors as compared to PEC tables.

As the filter length increases, the estimation of time autocorrelation is improved and the filter tends to give the proper weight and permanence to short lived events until, for very lono, filters (5 days or more), events begin to be averaged out and the filter's prediction, although different in its details to PEC values, has a prediction error statistically equivalent to them.

Fig. 2. Probability distribution and probability density of prediction errors.---Theoretical values.___Prediction produced by 36 hours long filter. Confidence intervals at midpoint (50%) of the probability distribution curves is 3%.

Between the extremes of long and short filters, there is an optimum length for which prediction errors are minimized. This prediction is significantly better than PEC tables. However, for a whole range of filter lengths, from 36 to 100 hours, there is an improvement over PEC published values. In figure 2 a comparison is made between the statistics of prediction errors of PEC published values and those obtained with a filter 36 hours long. For both, data sets the probability density and the probability distribution are given in percent of occurrence. It can be seen from figure 2 that even for this short a filter there is a significant increase in the occurrence of small errors while for large errors, the theoretical prediction is better. This is due to overestimation of the permanence of short lived large events. In figure 3, PEC published values statistics are compared to the statistics of the best prediction filter, the one with a length of three days. In this case Levinson recursion prediction is better for all sizes of the prediction error and the improvement is most significant at small errors. While for published PEC values, 50% of the time the errors are less than 6 cec, for a 72 hours long filter they are less than 3 cec. The expected error mean, rms value and some percentile values corresponding to figures 1 and 2 are given in table 1

Fig. 3. probability distribution and probability density of prediction errors.---- Theoretical values.___Prediction produced by 72 hours long filter. Confidence interval as in figure 2.

TABLE 1 ENCAPSULATED STATISTICS FOR PREDICTION ERRORS OF PUBLISHED VALUES AND PREDICTIONS MADE WITH FILTERS 36 AND 72 HOURS LONG.

Oceanographic observations, such as current velocity profiling from shipboard, or continuous monitoring of hydrographic parameters from a slow moving ship, require continuous position information with greater accuracy than that provided by Omega navigation system. This accuracy can be obtained using differential Omega navigation. However the implementation of this is not always practical. The alternative proposed and tested is based on the use of external sporadic fixes for the computation of a prediction filter. The predictions so produced not only obviate the storage and use of published PEC, but improve over them producing accuracies of the order of those of differential Omega.

For a frequency of external fixes of the order of that provided by Transit system (18 per day) it is found that filters with lengths over 24 hours produce acceptable predictions. Prediction error is sensitive to filter length: longer filters are more stable and tend to average out large disturbances. A proper balance between sensitivity and length is obtained for a 72 hours long filter that produces an accuracy improvement of the order of 40% over PEC published values. Longer filters produce predictions whose error statistics tend to those of PEC published values.

Although the experiment was made with fixed receiving stations, results are equally valid for a slow moving vehicle: from the point of view of the adaptive filter, the changes in PEC values due to station motion are undistinguishable from time variations in a fixed receptor. As long as the vehicle moves less than the PEC space correlation distance in a time of the order of the filter length, the motion induced variation will be properly followed by the filter. For faster vehicles, a prediction comparable to PEC published values can be obtained with shorter filters. A useful criterion for the relation between vehicle velocity and proper filter length is given by the fact that while perturbations have a correlation distance of the order of 1500 km., the mean propagation variation has a negligible change in distances of the order of 500 km. as can be seen from PEC tables.

The author expresses appreciation to Eduardo Sainz of the Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM) for his help in data processing work and to Agustin Fernández of the Instituto de Ciencias del Mar, UNAM, for his collaboration along the realization of this study.

Aviso de Privacidad

Cerrar esta ventana