A New Technique to TEC Regional Modeling using a Neural Network.

Transcription

1 A New Technique to TEC Regional Modeling uing a Neural Network. Rodrigo F. Leandro Geodetic Reearch Laboratory, Department of Geodey and Geomatic Engineering, Univerity of New Brunwick, Fredericton, Canada BIOGRAPHY Rodrigo Leandro i Ph.D. candidate at the Department of Geodey and Geomatic Engineering, Univerity of New Brunwick, Canada under uperviion of Dr. Marcelo C. Santo. He hold a M. Sc. Eng. Degree in Civil Engineering from the Univerity of São Paulo, in São Paulo, Brazil. He ha been involved in reearch in the field of Geodey and atellite navigation. ABSTRACT In thi paper we preent a new technique of regional modeling of TEC (Total Electron Content), uing a Neural Network model. Thi new model ha the capability to predict TEC value derived from a GPS tracking network. Preliminary tet and repective reult are hown. One of the main ource of error of GPS meaurement i the ionophere refraction. A a diperive medium, the ionophere allow it influence to be computed by uing dual frequency receiver. The ue of two frequencie allow etimating the influence of ionophere on GPS ignal by the computation of TEC value, which have a direct relationhip with the magnitude of the delay caued by the ionophere. In the cae of ingle frequency receiver it i neceary to ue model that tell u how large the ionopheric refraction i. Such i the cae of which the GPS broadcat meage carrie parameter of the Klobuchar model. One other alternative to ingle frequency uer i to create a regional model baed on a network of dual frequency receiver. In thi cae, the regional behaviour of ionophere i modeled in a way that it i poible to etimate the TEC value inide or near thi region. Thi regional model can be baed on polynomial, for example. We have invetigated a Neural Network-baed model to the computation of regional TEC. The advantage from the ue of thi Neural Network model i that with the ame model we can predict value for a tation either within or outide the network, due to the adaptation capability of neural network training proce, that i an iterative adjut of the ynaptic weight in function of reidual, uing the training parameter. We have ued data from the permanent GPS tracking network in Brazil (RBMC). We have teted the accuracy of the new model at all tation. To perform the tet TEC value were computed for each tation of the network, except for a tet tation. After that the training parameter data et for the tet tation wa formed, baed on the TEC value of all other tation of the GPS network. The Neural Network wa trained with thee parameter, and teted by computing the TEC for the tet tation. Thi aement wa carried out everal time, one for each tation of the network. Preliminary aement of reult uing our new technique how a capability of retrieving around 85 % of TEC value for all tation. Thi mean that we can correct the ionopheric delay at the ame amount, due the direct relationhip between both TEC and ionopheric delay. INTRODUCTION Ionopheric refraction i one of the mot damaging effect on GPS ignal. Thi effect i proportional to the total electron content (TEC), which i the number of free electron contained in the ionopheric layer. Electron of atmophere are generated due to everal factor, including olar activity. Figure how how olar radiation can create electron in the atmophere, forming the ionopheric layer. Figure. Creation of an oxygen ion and a free electron. Once the TEC i known, it i poible to determine the delay caued by the ionophere on GPS ignal. Due to the diperive characteritic of the ionophere, the delay i a function of the frequency. It i poible to know the value of TEC uing a dual frequency GPS receiver. Uing the obervation at both frequencie it i poible to compute the TEC value for the local where the tation i. Thi

2 computation will be explained in more detail in the next ection. One alternative for ingle frequency receiver uer i to ue a regional model of TEC, generated by uing data from a tracking network of dual frequency receiver. There are everal way to create uch model. A network of receiver can generate a patially ditributed grid of TEC value. Uing thi grid it can be created a model from which i poible to etimate a TEC value to any poition inide or near the region covered by the tracking network. Once the local TEC value i etimated, it i poible to correct the ingle frequency receiver obervation. In thi paper we preent a new technique to regional TEC modeling, uing a Neural Network approach. Thi new technique ha the capability to predict TEC value derived from a GPS tracking network. Preliminary tet uing the new technique indicate an average accuracy in the TEC value etimation of 97.5 %. In other word we can correct the ionopheric delay by the ame amount, due to it direct relationhip with TEC. Thee preliminary tet and repective reult will be hown later in the paper. TEC COMPUTATION USING A DUAL FREQUENCY RECEIVER Thi ection deal with the firt tep of our technique, that i the computation of the Vertical TEC (VTEC), uing dual frequency obervation. Thi computation allow the determination of VTEC value for each tation of the tracking network. The model for VTEC computation preented here i a imple model, becaue our final objective i not to get a great preciion in the VTEC determination for the tracking tation itelf, but a good etimation uing our regional model for void area, which i the main ubject of thi work. Thee ame value can be computed uing different technique, probably providing a better quality input data to the regional model. However it will be hown that the final reult obtained uing our approach are atifactory. Let the equation of the carrier phae meaurement on two frequencie (L and L) be: λ λ φ r (t) = ρ r (t) I r (t) + Tr (t) + δ r (t) c + λ N r φ r (t) = ρ r (t) I r (t) + Tr (t) + δ r (t) c + λ N r, (), () where λ and λ are the carrier phae wavelength, in meter, φ φ r (t) and r (t) are the carrier phae meaurement for a receiver r and a atellite, in cycle, ρ r (t) i the geometric ditance between the receiver and atellite antenna, in meter, r (t) and Ir (t) are the effect caued by the ionopheric refraction, in meter, Tr (t) and Tr (t) are the effect caued by the tropopheric refraction, in meter, δr (t) and δr (t) are the combination of the atellite and receiver clock error, in econd, c i the peed of light, in meter per econd, and N r and N r are the carrier phae ambiguitie, in cycle. Taking the difference between () and () we get: λ φ r (t) λ φ r (t) = I r (t) I r (t) + λ N r λ N r, where the geometric ditance, tropopheric delay and clock error term were cancelled out due their ame behaviour in both frequencie. If there i not a cycle lip the ambiguity term are contant. Let u combine the ambiguity term of both frequencie into a contant, a follow: C r r r I (3) = λ N λ N, (4) where C r i the combination of the ambiguity term of the two frequencie. Subtituting (4) into (3) we get: λ φ r (t) λ φ r (t) = I r (t) I r (t) + C r (t). (5) The influence of ionophere on GPS ignal ( I r and I r ) can be computed according to (Hoffmann-Wellenhof, 00): 40.3 TEC Ir =, (6) (f) and 40.3 TEC Ir =, (7) (f ) where f and f are the frequencie of the L and L carrier ignal, in unit of Hz, and TEC i the total electron content in 0 6 electron m. Subtituting (6) and (7) into (5) we will obtain the following expreion: λ φ r (t) λ φ r (t) = TEC + C r. (8) (f) (f)

3 For implification we aumed the TEC a a contant value during the period ued for the computation. The choice of the ize of uch period i arbitrary, but it need to be large enough to provide a good number of degree of freedom in the adjutment and mall enough to atify the aumption that TEC i contant over that period. In thi work, we ued period of one hour for each determination of TEC. We can evaluate (8) a follow: φr(t) λ φr(t) = TEC Cr λ +, (9) ( λ φ (t) λ φ (t)) TEC Cr = 9.5 r r. (0) TEC i defined a being the number of free electron contained in a column with one meter quared of tranveral ection, along the path of the ignal through the ionopheric layer. It i a number aociated to an inclined trajectory with repect to the local zenith, a a function of the elevation angle of the atellite. In addition to that, the ignal goe through the ionophere at coordinate different from thoe of the tation, at the ionopheric piercing point. To correct for the inclination and the poition of the piercing point we can ue mapping function, a follow: TEC = M VTEC, () where M =. () in(el) where el i the elevation angle of the atellite at the oberving tation. Equation () and () allow u to determine VTEC intead of TEC. The mapping function ued in thi work i a imple bilinear model, a follow: ( a + a φ + a λ) TEC = M 0, (3) where φ i the latitude difference between the obervation point and the ionopheric piercing point, λ i the longitude difference between the obervation point and the ionopheric piercing point, and a 0, a and a are the coefficient of the bilinear model to be adjuted. Subtituting (3) into (0) we will get the final expreion for TEC computation ued in thi work: M ( a + a φ + a λ) 0 = C r = ( λ φ (t) λ φ (t)) r r. (4) With thi expreion i poible to compute the parameter a 0, a and a and to determine VTEC for the tracking tation uing meaurement of everal atellite during a certain period of time. The parameter of the model are the ame to any atellite, but for each atellite included to the adjutment we will have an additional term C r. Therefore, in thi adjutment we will have (3 + n) unknown, where n i the number of atellite ued in the computation. The number of obervation can be determined according to: n no = no(), (5) total = where no total i the total number of obervation ued in the adjutment, and no() i number of tracked epoch of the atellite during the period of time ued to the computation. If a cycle lip occur during thi period, it i neceary to add another term to determine the combined ambiguitie, looing one degree of freedom. Depending on the ize of the period it may be better to ignore uch atellite to avoid the increaing of the ambiguity term quantity. The linear ytem formed by the everal obervation according to the equation (4) can be olved uing the Parametric Leat Square Method. Performing thi computation for each tation of the GPS tracking network we will have a VTEC value aociated to a coordinate wherever we have a tation of the network. Thee value will be the input parameter of our Neural Network Model, which will perform the etimation of VTEC for any other point in or near the region covered by the network. The Neural Network Model will be dicued in the following ection. THE NEURAL NETWORK MODEL A Neural Network i an information proceing ytem formed by a big number of imple proceing element, called artificial neuron, or imply neuron. Figure. Nonlinear artificial neuron model (Adapted from Haykin, 999).

4 A neuron compute it input a a linear combination of it input ignal by uing the ynaptic weight. The ynaptic weight play the role of parameter, which are adjuted at the training proce (thi procedure will be dicued later in thi ection). After that an activation function i applied to the neuron input to generate the neuron output (in the cae of a ingle neuron it i already the output ignal). One neuron may have one or more output, with the ame value. In the cae of a linear activation function, the neuron play the role of a regreion linear model. The proceing of a neuron k can be repreented by: m y k = ϕ (xi w ki ) + bk, (6) i= where y k i the neuron output, ϕ i the activation function, m i the number of input parameter, x i i the i- th input parameter, w ki i the i-th ynaptic weight and b k i the bia. Typically the order of normalized amplitude of a neuron output i within the range [0,], or alternatively [-,]. Thi range depend on the type of activation function ued. The neural model alo include a term that i applied externally, called bia and repreented by b k. The bia ha the function of increae or decreae the neuron input. It i poible to introduce a functional link into the network a an additional layer of neuron, called a hidden layer. Thi layer can be compoed of one or more neuron The input ignal of the hidden layer neuron i generated by the output ignal of the input layer. The output ignal of the hidden layer i ued to generate the input ignal to the output layer. It i alo poible to introduce not jut one, but everal hidden layer into the model. Figure 3. Neural Network Multilayer Perceptron Figure 3 how a cheme of a neural network with one hidden layer. In thi example x(), x() and x(3) are the input parameter and y(t) i the output parameter. Each element, excepting the biae, i a neuron. Each of thee neuron i a proceing element that work according to equation (6). The ynaptic link (the line in the draw) connect the different layer, carrying the output ignal of a previou one to generate the input ignal of the next one. Each ynaptic link of the network ha a correponding ynaptic weight that i applied to the flowing ignal that i going through it. Another iue of a neural network model i the number of neuron of each layer. Thi number i fixed to the input and output layer, in function of the input and output parameter. For the hidden layer thi number i arbitrary. The model reulting from adding hidden layer between the input and output layer i called Multilayer Perceptron (MLP). The MLP i not the only type of neural network model, but i one of the mot popular one. In thi work we have ued a MLP. It i neceary not jut to know which model will be ued (in our cae the MLP), but alo all it characteritic, uch a the number of hidden layer, the number of neuron in each hidden layer, the activation function of each layer, etc. There are other more pecific characteritic that will not be dicued here. The characteritic of the new model will be preented later in thi ection. Once we have a model defined, it i neceary to train the neural network with data. Such data i compoed by a et of know input and output parameter. The training proce i not more than an adjutment of the ynaptic weight to the data et. Thi adjutment attempt to decreae the reidual of the output of the network. The reidual are the difference between the computed output and the known output. Baed on thee reidual i performed an actualiation of the ynaptic weight. Due to the complexity of neural network the adjutment cannot be done with a direct computation. Therefore the o called training algorithm, which are a type of iterative adjutment of the ynaptic weight, are ued. One of thee algorithm i the Backpropagation Training Algorithm, which i compoed by two tep. The firt one i the feedforward, when the ignal i propagated through the network, from the input layer to the output layer. After that the output value i compared with the known output and the reidual are computed. The econd tep i the feed-backward. In thi tep the error are propagated through the network from the output layer to the input layer. During the feed-backward tep the ynaptic weight are adjuted. It i made everal time to each parameter up to the reidual converge to a deired threhold value. After the training proce we have a Neural Network

5 Model with adjuted ynaptic weight according to the training parameter. The preented model wa created to etimate the VTEC for a certain poition. The input parameter of the neural network model are Latitude and Longitude, while the output parameter i the VTEC. In thi way, once the network i trained, it i poible to get a VTEC value for any location. The training parameter are the known coordinate and VTEC value of each tation of the GPS network at a given time. Once the model i adjuted we can etimate a VTEC to any poition inide or near the region covered by the GPS network to the given time. Two hidden layer were ued, each one with five neuron. The activation function of all layer (except the input one) i the hyperbolic tangent igmoid function, repreented in equation (7). ϕ(x) = + e x, (7) where x i the input ignal of the neuron. Figure 4 how a cheme of the neural network model ued. The technique to apply the model with the GPS tracking network data are dicued in the following ection. Figure 4. The Neural Network Model. Figure 5. Station of the RBMC The advantage of uing that network i due to the continental dimenion of Brazil, what can be conidered one additional factor to tet the capability of the model to etimate the TEC to long ditance. Due to operational retriction (not all tation are alway operational) we did not ue the whole network, but the tation BOMJ, BRAZ, CRAT, CUIB, IMPZ, PARA, POAL, RIOD, SALV, SMAR and UEPP, in a total of tation. All tation were ued either to calibrate the model or a a tet tation. For each determination the tet tation data wa not ued during the training proce of the neural network. After the training proce the model wa ued to etimate the VTEC value for the tet tation poition. Thi value i then compared with the known VTEC value obtained with the technique expalined in previou ection. The difference between them how the error of the prediction of the Neural Network Model. Performing thi procedure to each of the tation we could acce the eficiency of the model everywhere. Uing thi technique we could analie the performance of the model for prediction inide and at the edge of the area covered by the network. Figure 6 how a flowchart of the data proceing for each given time. ANALYSIS STRATEGY The data ued in thi work wa obtained from the RBMC (Brazilian Continuou Monitoring Network), which i a GPS tracking network in Brazil. Figure 5 how the configuration of uch network.

6 where VTEC i the computed value of VTEC, in TECU, and VTEC e i the etimated value of VTEC, in TECU. The relative error can be computed according to: Re lative Error Abolute Error = 00. (9) VTEC The le the abolute and relative error are (a given by equation (8) and (9)), the cloer are the predicted VTEC e (given by our Neural Network Model) and the computed VTEC (determined from dual frequency receiver) ued a reference. Figure 6. Flowchart of the data proceing. We performed thee tet for two different period of five day: One with low olar activity and the other with high olar activity. The low olar activity period covered day form February t 004 to February 5 th 004. The high olar activity period covered day from October 6 th 003 to October 30 th 003. The index ued to the analyi of olar activity wa the olar radio flux. Figure 7 how the behaviour of thi index, within the choen period. Due to the direct relationhip between TEC and ionopheric delay, according to equation (6) and (7), we can correct the ionopheric delay with a imilar accuracy of the etimation of TEC. The reult of VTEC etimation can be regarded a an etimated accuracy for correcting the ionopheric delay to ingle frequency receiver. In thi invetigation 38 etimation were made with our new model, involving different tation, day and time of the day. The average abolute error of all etimation i equal to 3.7 TECU with tandard deviation of.7 TECU ( igma). The average relative error wa 4.9 %, with tandard deviation of 0.9 % ( igma). Figure 8 and 9 how the minimal abolute and relative error for all tation, repectively. Figure 7. Solar radio flux. For each day tet were performed to compute VTEC at, 4 and 6 hour (local time), correponding to three prediction per day per tation, reulting in a total of 38 prediction. The period ued for the TEC computation wa hour. Prediction reult are hown in the following ection. ANALISYS OF RESULTS Figure 8. Minimal abolute error for all tation. Reult were analyed by auming both abolute and elative error. The abolute error can be computed according to: Abolute Error = VTEC VTEC, (8) e

7 Figure 9. Minimal relative error for all tation. The wore average reult were obtained for tation IMPZ. It could be expected, ince thi tation i the farthet from the other. The cae of tation IMPZ i an extrapolation cae. The average reult obtained for thi tation are 5.5 TECU and 8 % for abolute and relative error, repectively Figure. Relative minimal error during the low olar activity period. Figure and 3 how the abolute and relative minimal error, repectively, for all tation during the period of high activity. Figure 0 and how the abolute and relative minimal error, repectively, for all tation during the period of low activity. Figure. Abolute minimal error during the high olar activity period. Figure 0. Minimal average error during the low olar activity period.

8 tation it would be expected an even greater tability and confiability of the etimation of the model. The wore average reult were obtained for tation IMPZ. It could be expected, ince thi tation i the farthet from the other. The cae of tation IMPZ i an extrapolation cae. Since the reult are not bad (average of 5.5 TECU and 8 % for abolute and relative error, repectively) the new model i a good model to etimate TEC value not jut inide the region covered by the tracking network, but alo outide it. An important conideration i the ditance between IMPZ and the nearet tation, of the order of eight hundred kilometre. It how the capability of our model to extrapolate. Figure 3. Relative minimal error during the high olar activity period. The abolute error in both cae (low and high olar activity) were obtained in the ame order of magnitude. Becaue the larger the TEC value during high activity period, the maller the relative error in thi ituation than thoe obtained during the low activity period are. CONCLUSIONS AND FUTURE RESEARCH The model performed etimation with an average error of 3.7 TECU with tandard deviation of.7 TECU ( igma). The average relative error wa 4.9 %, with tandard deviation of 0.9 % ( igma). Thi mean that according to thee preliminary reult the new model allow to correct approximately 85 % of the ionopheric refraction to a ingle frequency receiver inide or outide the region. It can be concluded that the new model i adequate to predict VTEC value. The value of the tandard deviation allow u to conclude that there wa not great difference when comparing different tation, day, time or even olar activity level. The wore abolute and relative reult were obtained in period of high and low olar activity, repectively. A aid before, thi i due to a lower denominator value in equation (9), during low olar activity period. It can be alo concluded that there i a tability of the etimation in term of abolute value, during different ituation with repect to olar activity. Therefore the average relative reult for the high olar activity are better than thoe for low activity. However, the abolute error during high olar activity period are not o good a thoe for calm period. Eventhought the pacing of tracking tation of the network ued in thi reearch i pare, the model produced good etimation. With a larger number of Future reearch i required to a complete validation of the model, aeing the efficiency of the new technique to different condition of geomagnectic and olar activity. Comparion of the etimation of thi new model with current model i another way to validate of technique. Since it i concluded that the technique i a good way to modeling regional TEC, it can be poible go ahead and invetigate imilar technique to global TEC modelling. Probably it would be neceary change in the neural network configuration, baed on the complexity of the problem. ACKOWLEDGEMENTS I would like to acknowledge the Brazilian Intitute of Geography and Statitic (IBGE) for the ue of the GPS data. Particular thank to Mr. Katia Pereira. Thank to my upervior Dr. Marcelo Santo. Fund for thi reearch provided by NSERC. REFERENCES Haykin, S. (999). Neural Network A Comprehenive Foundation. Prentice Hall Upper Saddle River, New Jerey. Hoffmann-Wellenhof, B. et. al. (00). Global Poitioning Sytem: Theory and Practice. Springer-Verlag Wien New York. Klobuchar, J.A. (987). Ionopheric Time-Delay Correction for Advanced Satellite Ranging Sytem. AGARD Conference Proceeding Propagation Limitation of Navigation and Poitioning Sytem. Komjathy, A. and Langley, R. B., (996). An Aement of Predicted and Meaured Ionopheric Total Electron Content Uing a Regional GPS Network. Proceeding of

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