GB2289587A - Model generation - Google Patents

Abstract

A model of the ionosphere for use in the operation of a skywave over the horizon radar is generated by taking a set of maps of heights of various ionospheric layers against location for corresponding sunspot numbers, comparing these to the measured value of the layer height at at least one measurement point, and calculating the value of the sunspot number which would give the closest fit of the ionospheric layer heights at those points with the measured value of height, and using this calculated value of sunspot number to produce a second map of ionospheric layer heights against location. A respective local correction term is then applied to the height values in the vicinity of those locations on the map at which measurements have been made, to bring map and measured heights into agreement, the local correction term being derived from the covariance between the heights, the locations of the measured points and the difference between the measured values and the values predicted by the second map at the measurement points. <IMAGE>

Description

MODEL GENERATION This invention relates to a method of model generation and particularly to a method of generating a model of the ionosphere.

The phenomenon of the bouncing of radio waves from the ionosphere is generally referred to as reflection, and the ionospheric layers from which radio waves are bounced back can be thought of as reflecting surfaces, but the actual phenomenon which causes the bouncing back is refraction within layers of the ionosphere having different refractive indexes.

In order to operate, skywave over the horizon radars depend on the bending of radio waves due to refraction by different layers of the ionosphere. In order for such radars to function accurately and produce useful information it is essential that the composition and vertical height of these layers be known so that the position of a target can be calculated from the received radar signals.

Unfortunately the ionosphere is not a fixed structure but undergoes changes due to solar activity, generally sun spots, and seasonal and diurnal influences on the earth which cause ionospheric conditions to vary relatively slowly. There are also transient phenomena which cause the ionosphere to alter over a much a shorter time scale. For example ionospheric ionisation caused by solar flares, generally referred to as short wave fadeout, and due to weather conditions in the lower atmosphere influencing ionospheric conditions, generally referred to as gravity waves. In order to allow a skywave over the horizon radar to operate effectively it is essential that an accurate model of the ionosphere over a large area can be generated in real time.

Ionospheric models have been produced by observation and measurement which plot global changes in ionospheric conditions as a function of sun spot number, month and time of day, but these are inevitably averages generated by measurement at different locations at different times and cannot be made sufficiently precise to allow accurate radar operation and in any case certainly cannot allow for transient effects on the ionosphere which are by their very nature unpredictable.

This invention arose from an attempt to produce a method of generating an ionospheric model in real time with sufficient accuracy to allow effective skywave over the horizon radar operation.

This invention provides a method of generating a model, the model comprising a map or values of at least one variable parameter, the model varying in a manner which is a priori statistically correlated with the value of a first variable, a method comprising the steps of; a) measuring the value of a parameter at at least one noint; b) using the value or values determined in step a) to estimate the value of the first variable; c) determining a prototype model corresponding to the value of the first variable determined in step b) utilising the a priori known correlation between the model and the first variable.

This allows the appropriate version of the model to be determined by determining the value of the first variable from one actual measurement of a variable parameter rather than using a model determined by a prior estimation of the appropriate version of the model without making measurements to test the accuracy of the estimation. As the relationship between the value of this variable at a point of measurement and the version of the model is known, measuring the value of the variable at a measurement point allows the version of the model corresponding to the situation prevailing at the time the measurement was made to be estimated with an improved degree of confidence.

From this model the values of the first variable at points remote from the point of measurement may be determined. Values of other non-measured variables at all points may also be determined from the model.

The method may also comprise the steps of: d) for each measurement point, determining a set of local correction terms; and e) generating a corrected model by using the correction terms to locally modify the prototype model in the vicinity of each measurement point such that, at each measurement point, the model value equals the measured value and the amount of correction progressively decreases with increasing distance from the measurement point.

In the case where a number of measurements are made either at a single point or a number of spaced-apart points, it may be that it is not possible to fit the measurements exactly to any version of the model. In such cases that version of the model which provides the best fit to the measurements can be used as the prototype model. However, as the measurements do represent the actual conditions pertaining at the measurement points, the provision of local correction terms allows a more accurate version of the model to be generated. Progressively reducing the amount of correction with increasing distance from the measurement point reflects the fact that the prototype model represents the best estimate of the actual conditions at points remote from a measurement point.

The local correction term may be a function of the covariance between the parameters, the location of the measurement points and the difference between the measured values and the values of the prototype model.

This allows the local correction value to provide a more accurate representation of the actual conditions pertaining. The model may be a model of the ionosphere, the map of values comprising parameters of ionised layers capable of refracting electromagnetic radiation. The parameters may include the height of a layer and/or the maximum frequency which the layer can reflect. The first variable may be the sunspot number.

A model generation method employing the invention will now be described by way of non-limiting example only with reference to the drawings in which; Figure 1 illustrates graphically some steps in the method in accordance with the invention; and Figure 2 illustrates the application of corrections to a model in accordance with the invention.

The need can arise to generate a real time model where it is not possible to measure the system being modelled quickly or extensively enough to allow a real time picture of the system to be produced.

For example, in order to operate a skywave over the horizon radar the characteristics of the ionosphere over a very large area must be either measured or predicted accurately. Due to the very large area involved it is simply impossible to build up a picture of the ionosphere by direct measurement in real time.

What has been done is that over a period of time enough measurements have been made to allow maps of the average ionosphere parameters over the globe at a number of different times of day in each month of the year. These maps are generally produced from the historical data showing the average ionospheric parameters for each point and time at two separate extreme levels of sun spot activity, generally referred to as sun spot number 0 and sun spot number 100.

By observing the actual number of sun spots the average expected ionospheric parameter values at any points and times given on the maps can be calculated by interpolation between these two extremes.

Although such global maps are too crude to allow useful over the horizon radar operation due to their scale this technique can be used to generate more detailed maps of the area of operation of the over the horizon radar. However in order to find out how the actual parameters vary from these predicted averages, to produce ionospheric parameter values for times and locations between those at which measurements have been made and in order to spot and correct for transient phenomena it is necessary to combine real time measurements of the ionospheric parameters with the historic map.

Real time ionospheric parameter measurements are made by a network of vertical incidence sounders (VIS) and oblique incidence sounders (OIS) which operate by transmitting a swept frequency signal upwards towards the ionosphere and then comparing the return signal bounced back by the ionosphere with the transmitted signal. Vertical incidence sounders transmit the signal vertically upwards and receive it at a receiver co-located with the transmitter while oblique incident sounders transmit a signal from one location to a receiver at a remote location after being obliquely bounced off the ionosphere.

Once the sounder data for ionospheric parameters has been received this is compared to the existing model of the ionosphere for the two extreme sun spot numbers and the time of day and time of year closest to the current time and date. Maps for sun spot numbers between 0 and 100 are generated by interpolation between the 0 and 100 sun spot number maps and the intermediate sun spot number m producing ionospheric parameter values which produce the best linear least squares fit to the measured parameters is selected as the basis for the real time model. Other forms of closest fit calculation could be used, but linear least squares is preferred.

This process is shown in Figure 1 where a first map 1 for sun spot number 100 and a second map 2 for sun spot number 0 of ionospheric parameters plotted against latitude and longitude have been produced by historical measurement, averaging and interpolation. Both maps correspond to the parameter values which are expected to be found at the same time of day at the same time of the year. Ionospheric parameters are measured directly at points 3 and 4 within the region covered by the maps 1 and 2 and interpolation is used to produce a map 5 of ionospheric parameters against latitude and longitude which provides a linear least squares fit to the actual parameter measurements at locations 3 and 4. This map corresponds to a sun spot number S.

It is found that deriving a map on which to base the real time model in this way is more accurate than attempts to directly measure sun spot number and then use the sun spot number to interpolate and generate a map as has been done in the past.

Once this scaled map has been derived to form the basis of a real time model the values on the map must be altered to agree with the measured values at the points measured by the sounders.

In order to do this a local correction term is calculated and applied to each point on the map, the local correction terms being derived from the ionospheric parameters at points where they have actually been measured by a sounder. The local correction term at a point where it is desired to know the ionospheric parameter values, referred to as a control point, is based on the covariance (or correlation) of the ionospheric parameters as a function of geographic location, solar activity (sunspot number), month and time of day.

The covariance of the ionospheric parameters between two locations can be characterised by a function of the form y p 1 Ax2+By2 where p = covariance V = variance coefficient for or between ionospheric parameters A = coefficient for latitude fit B = coefficient for longitude fit x = latitude separation y = longitude separation The coefficients V A and B can all be evaluated by analysis of historic ionospheric parameter records or from the real time scaled measurements obtained from an extended network of a plurality of sounders.

The local correction term dz applied at any point is computed as follows dz = g [P + 1 1d .......(1) where P is a matrix of co-variances between the ionospheric parameters at all of the sounder locations, i.e. element Pmjnk is the covariance between a parameter M measured at location J and a parameter N at location K. These covariance values are obtained using equation 1 from the measured or historic data.

# is a matrix of co-variances between the sounder measurement errors due to auto-scaling of the ionospheric parameters, noise etc which were not present when P was calculated. Its elements correspond to those of matrix P.

q is a vector of covariances between the ionospheric parameters at all of the sounder locations and the control point, i.e. element ennk is the covariance between parameter M measured at the control point and parameter N at location K. These values were obtained from equation 1.

d is the residual error vector. Element d, is the difference between the scaled predicted values from the scaled map and the measured values of parameter N at location K.

The complicated notation is required to accomodate the large number of parameters which may be observed (m,n) and the potentially large number of different locations (j,k).

The local correction dz obtained is then added to a scaled predicted value from the map of the ionospheric parameters and the point where it is desired to know the value of an ionospheric parameter.

At such a point P; = Pz + dz where P; is the real time ionospheric model predicted value.

This is illustrated in simplified form in Figure 2. Pzo, Pzl, Pz2s Pz3 represent actual measured values at locations 0, 1, 2, 3. Solid line 20 is a curve corresponding to the sunspot number S determined from the measured data and historical as described above. Pzo, P , Pz2, Pz3 are points lying on the curve 20 represent the expected values at the respective locations.

The dashed curve segments between points 21, 22; 23, 24; 25, 26; and 27, 28 represent positions of the curve to which corrections have been applied to make the curve pass through the actual measured values. It can be seen that the correction progressively reduces with increasing distance from the measurement point such that no correction is applied to the those parts of the curve lying between points 22 and 23, 24 and 2526 and 27, to the left of 21 or to the right of 28.

The shape is thus in general somewhat different from that of a curve which would be obtained by simply fitting a polynomial to the measured points. This reflects the assumption that, at points close to a measurement point, the value is more likely to be close to the measured value than the theoretical value, whereas at points remote from a measurement point, the theoretical value based on historical data is likely to be more accurate than a value obtained by either interpolating between adjacent measured values or fitting a curve to all measured values.

An advantage of this model generation method is that the model can be continually updated whenever new data is available by adjusting the local correction terms as well as by adjusting the sun spot number S of the scaled map to which the correction terms are applied.

Having now described the principles of the invention in general terms, one way of performing the invention will now be described in more detail in the context of the prediction of ionospheric parameters.

The method for prediction of the ionospheric parameters at an unknown point consists of the following steps. The first two steps are performed once only on historical data and can be considered as calibration steps; the remainder are performed in real time and represent operation steps.

1. Calculate the covariance matrix Using a large set of scaled sounder parameters derived from simultaneous measurements at several sounder locations, first partition them into several categories according to season and time of day. For each category, calculate a covariance matrix as follows.

On each simultaneous set of soundings first perform a least-squares fit to determine the effective sunspot number, then using this value calculate the difference between measured and predicted parameters at each sounder. Sum the differences and their productd over all the soundings in the category to obtain the means and the covariance matrix P and invert it.

2. Model the spatial variation of correlation Using the data from the previous calculation, devise a model which will estimate the variance at a single point and the covariance between two points within the region covered by the sounder. It may be found that these are relatively simple functions of separation (for example, ellipses) and possibly of position. This model will be necessary for constructing the vector q of covariances between the sounders and the unknown point and the variance r at the unknown point.

3. Determine the effective sunspot number Using a set of simultaneous sounder measurements x, perform a least-squares fit of the measurements and the numerical map predictions to determine the effective sunspot number.

4. Calculate the predicted values of the parameter Using this value of the sunspot number in the numerical maps, calculate the predicted values p and py, the variance a2 and the residual prediction errors d.

5. Determine the covariance for the unknown point Using the model created in 2 above calculate the variance r and covariance vector q for the unknown point, and the variance

6. Calculate the correction at the unknown point Using the results of 4 and 5 above, calculate the correction d y =qP~' d at the unknown point and add it to the predicted value py to give the final value y. The technique of singular value decomposition may for example be used to solve the resulting matrix equation. Sum the variances

an;

to give the total variance of y.

Formulation Assume that the scaled sounder parameters are the same quantities needed for the model, without further calculation. Thus if accurate sounder data were available for the location of interest (referred to in the following as the unknown point" - its location is known, but the ionospheric parameters there are not) the problem would be solved. The following analysis may then be applied independently to each sounder parameter. Let the unknown value of the wanted parameter at the position of interest be y. Let the number of sounded points be N.Assume that the sounder scaling algorithms return values Xj which fluctuate as a result of scaling errors, noise etc. in the short term but have independent Gaussian distributions with means equal to the true values m; of the parameters, and variances axi ,j 2, ...N. Then the vector x represents the measured values of the parameter at the sounders.

Linear interpolation In practice the sounder data are only known at a few locations which are not necessarily close to the region of interest. The simplest approach would be to interpolate linearly, by taking a weighted sum of the sounder data:

The weights will be derived from some measure of the known spatial correlation of the parameter as a function of distance and direction, such that at any of the sounded points one weight is unity and the others are zero. More complicated interpolation methods which fit some kind of 2-dimensional polynomial through the known points could also be used, but higher order does not necessarily lead to higher accuracy. Alternatively a lower-order polynomial adjusted to minimise some measure of the fitting error at the known points (e.g. a least-squares fit) could be used.Because the measured data are not independent it is difficult to estimate the variance of the predicted value using interpolation methods.

Clearly this kind of prediction model will have desirable global properites, in that if all the sounder measurements are high then the prediction will be high and vice versa. However, it will not take account of any systematic local variations e.g. the fact that the data at a certain point are consistently higher or lower than the average of surrounding points, and even if it can be calculated, the variance of the predicted value will not reflect any such systematic errors.

Furthermore, not only may the sounder points be distant from the region of interest, but they may well not be in a geometrically favourable configuration relative to the unknown point Under these circumstances, "interpolation" tends to extrapolation, and the errors become large and unquantifiable, and the method loses all validity.

Correlation techniques An alternative to the interpolation method is to apply the covariance matrix. It is assumed that the long-term means (x) and covariance matrix P of the sounder measurements x for the appropriate season, time of day, etc., can be determined by statistical analysis of previous measurements. Also, it is assumed that the covariance vector q between the x and y and the variance r of y about its means can be estimated, since they can probably be represented by simple functions of distance and direction between the unknown and the sounded points and of the location of the unknown point. then the deviation of y from its means, and the variance of the deviation (i.e. the error in the predicted value), can be determined. The deviation from the mean is dy = qP~id, where d=x-(x), and its variance is Ob =r - qp~lq The principal drawback is that the mean value of y must be detetmined by some other means.

Incorporation of predicted data Existing numerical maps of the ionosphere represent a higher-order interpolation method in which spherical harmonics are fitted through measured data points to provide a global model of median conditions. The predicted value p y from the maps can then be used as an approximation to the mean of y needed in the previous section. To allow for global systematic errors in the maps, the sounder data values x used as input should be corrected by subtracting the difference between the values p j predicted by the numerical maps at the sounder locations and the true means (xj) of the sounder data.Since the quantities p and (x) are fixed, this has no effect on the covariance matrix, but the input and output parameters of the method are then not the difference x-(x) between the measurement and its mean but the difference d=x-p between measured and predicted values. Even with this improvement, a potential problem with this method is that it does not model global effects when the individual correlations are low. For example, assume that the unknown point is surounded by sounder, but at such a distance that they are not strongly correlated, i.e. all elements of q are small. If this is the case, then naturally d, will always be small, so the predicted value of y will always be nearly equal to its mean.

However, if all the xj are higher than average, on the basis of interpolation one would expect y also to be higher than average even though the individual correlations are small, whereas this method indicates that y will still have its mean value.

Improvement of map predictions using a least-squares fit In accordance with the invention, the lack of global predictive power mentioned above can be overcome by noting that the numerical maps in effect depend on a free parameter, the smoothed sunspot number R. If this parameter is adjusted to achieve the best fit between measured and predicted values at the sounded points, it should produce changes in the predicted value at the unknown point which will reflect overall high or low conditions. This can be done in a straightforward way by a least-square fit, since the predictions of the numerical maps vary linearly with sunspot number.Therefore all that is necessary is to predict the sounded parameter at each sounder location for two values of R, say 0 and 100, and minimise the sum

whence

Since several different ionospheric parameters with different dimensions are being predicted at each point, in fact each parameter must be multiplied by a different weighting factor in the above sums. The weights should be chosen so that the quantities being summed have approximately equal magnitude for each parameter.

The predicted value at any point k will then be R Pk(R) = pt(O)+ 200b*(100)-p(0)] 100 If the value of R givven above is substituted back in to this, it becomes

where

and

Effect of measurement errors on the adjusted map predictions We need to estimate the effect of errors in the input parameters xj on the output parameters p,. The variance of p k (R) is given by

Some care is needed in specifying exactly what is being averaged in this case.Here we are considering fluctuations in the output p y of the deterministic function obtained from the leastsquares fit, as a result of errors in the input xj. Therefore the expectation value implied by the brackets () is the short-term average over noise, measurement errors or other fluctuations which are assumed to be independent for each measurement. Therefore the cross terms vanish and

and so

including the special case of the unknown point, for which k = y.

The variances of the Q can be found in the same way to be

but in fact they will not be needed.

Use of the correlation method to increase the improvement In general, the parameters pk(R) predicted by the improved numerical maps will not be equal to the measured values Xk, but there will be a residual error Q = x - Pk- The correlation technique described above can be applied to the known residual errors at the sounded points to predict a correction at the unknown point, which should be added to the predicted median py(R) from the adjusted numerical maps.The final estimate of y is then obtained by adding the predicted median py(R) at the unknown point to the difference dy = x - p(R) calculated by the method described above but using the differences d in the statistical analysis used to calculate the covariance matrix. Because the covariance matrix is calculated from the measured errors, which include the short-term variances a2, the variances of the dk should not be added to the covariance matrix P. With this definition of the covariance matrix, dy = #P-1d with variance

Fianlly we need to find the variance of y itself.This is obtained from the deterministic calculation of py, which includes short-term fuctuations and measurement errors, and the probabilistic calculation of d yX which accounts for long-term correlations, but does not include short-term fluctuations, except in so far as they dilute the calculated covariance matrix. It seems reasonable therefore to assume that the errors in these two quantities are independent, and to add their variances to give the variance of y:

Effect of location errors.

It has been assumed that the locations of the sounded points and the unknown point are known. In practice they are not known with certainty and this could introduce additional errors.

The positions enter into the above calculations in two ways: 1. They are used directly in the numerical map predictions p k The predicted median parameters vary smoothly and fairly slowly with position, so the effects of even quite large positional errors are likely to be negligible compared with the variations in the measured data x k and hence in the residual dk calculated from or added to the p k. This is fortunate sir.ce the numerical maps are nonlinear functions of position, which makes analytical estimation of the errors difficult.

2. They are used indirectly to estimate the variances q and r from what is likely to be a somewhat fuzzy empirical model. In view of the uncertainties in the construction of the model, the variances are also likely to be slowly-varying functions of position and so the resulting errors can again be ignored.

Although normally, even if only one set of sounder data were available it would relate to a plurality of parameters, if only one set of sounder data for only one ionospheric parameter is available the scaled map will be set to a sun spot number S which causes it to agree precisely with the parameter value measured at the sounder location. However the model will still be adjusted in other locations by the covariance coefficients related to that measurement location, allowing more accurate predictions of ionospheric parameters than are otherwise available. In this case the covariance data must be historic, reducing the accuracy of the system, so it is preferred that sounder data from a number of locations is used so as to allow the covariances to be calculated from the sounder data directly in real time as well as allowing a more accurate scaled map to be generated.

This model generation technique has been described in terms of forming a model of ionospheric parameters but it could apply to any process where real time observations are combined with maps of historic parameter variation in attempts to produce a real time model such as ocean currents or tropospheric weather patterns.

Claims (10)

1. A method of generating a model the model comprising a map of values of at least one variable parameter, the model varying in a manner which is a priori statistically correlated with the value of a first variable, a method comprising the steps of; a) measuring the value of at least one of the variable parameter at at least one point; b) using the value or values determined in step a) to estimate the value of the first variable; c) determining a prototype model corresponding to the value of the first variable determined in step b) utilising the priori known correlation between the model and the first variable.

2. A method as claimed in claim 1 futher comprising the steps of; d) for each measurement point, determining a set of local correction terms; and e) generating a corrected model by using the correction terms to locally modify the prototype model in the vicinity of each measurement point such that, at each measurement point, the model value equals the measured value and the amount of correction progressively decreases with increasing distance from the measurement point.

3. A method as claimed in claim 2 in which the local correction term is a function of the covariance between the parameters, the location of the measurement points and the difference between the measured values and the values of the prototype model.

4. A method as claimed in any one of claims 1-3 in which the model is a model of at least part of the ionosphere and the map of values comprises parameters of ionised layers capable of refracting electromagnetic radiation.

5. A method as claimed in claim 4 in which the parameters include the height of a layer.

6. A method as claimed in claim 4 or 5 in which the parameters include the limiting frequencies which a layer is capable of refracting.

7. A method as claimed in any one of claims 4 to 6 in which the first variable is the sun spot number.

8. A method as claimed in any preceding claim in which step a) comprises measuring the value of a plurality of parameters at a single point.

9. A method of generating a model substantially as described.

10. A method of generating a model of the ionosphere substantially as described.