徑向基函數(shù)神經(jīng)網(wǎng)絡(luò)在GPS衛(wèi)星鐘差預(yù)報(bào)中的應(yīng)用

王國(guó)成,柳林濤,徐愛功,蘇曉慶,梁星輝

1．中國(guó)科學(xué)院測(cè)量與地球物理研究所大地測(cè)量與地球動(dòng)力學(xué)國(guó)家重點(diǎn)實(shí)驗(yàn)室,湖北武漢 430077; 2．中國(guó)科學(xué)院大學(xué),北京 100049;3．遼寧工程技術(shù)大學(xué)測(cè)繪與地理科學(xué)學(xué)院,遼寧阜新 123000

徑向基函數(shù)神經(jīng)網(wǎng)絡(luò)在GPS衛(wèi)星鐘差預(yù)報(bào)中的應(yīng)用

王國(guó)成1,2,柳林濤1,徐愛功3,蘇曉慶1,2,梁星輝1

1．中國(guó)科學(xué)院測(cè)量與地球物理研究所大地測(cè)量與地球動(dòng)力學(xué)國(guó)家重點(diǎn)實(shí)驗(yàn)室,湖北武漢 430077; 2．中國(guó)科學(xué)院大學(xué),北京 100049;3．遼寧工程技術(shù)大學(xué)測(cè)繪與地理科學(xué)學(xué)院,遼寧阜新 123000

GPS衛(wèi)星鐘在空中很容易受到諸多因素的影響,導(dǎo)致其鐘差行為很難用線性模型、二次多項(xiàng)式模型、灰色模型等現(xiàn)有模型進(jìn)行描述和實(shí)現(xiàn)可靠的高精度預(yù)報(bào)。本文利用徑向基函數(shù)神經(jīng)網(wǎng)絡(luò)對(duì)4顆GPS衛(wèi)星鐘差連續(xù)進(jìn)行了5 min、1 h和1 d的預(yù)報(bào),分別取得了均方根誤差優(yōu)于0．8 ns、0．6 ns和1 ns的預(yù)報(bào)精度,證明了文中徑向基網(wǎng)絡(luò)結(jié)構(gòu)在鐘差預(yù)報(bào)方面的可靠性。

GPS衛(wèi)星鐘差;徑向基函數(shù)神經(jīng)網(wǎng)絡(luò);鐘差預(yù)報(bào)

1 引 言

GPS衛(wèi)星鐘差是GPS導(dǎo)航系統(tǒng)的主要誤差源之一[1],在大地測(cè)量學(xué)以及GPS應(yīng)用中有著非常重要的意義。目前,廣播星歷的軌道和鐘差性能已經(jīng)達(dá)到了均方根誤差為1 m和5 ns的精度[2]。自2007年以來,超快速星歷(IGU)的軌道產(chǎn)品精度得到了顯著提高,達(dá)到了5 cm和3 ns的精度[3],但是其鐘產(chǎn)品僅僅達(dá)到廣播星歷水平[4],依然沒有較好的預(yù)報(bào)性能,原因之一就是空間中的衛(wèi)星鐘很容易受到各種因素的影響[4],例如溫度、電離層、磁場(chǎng)等環(huán)境因素,這些影響導(dǎo)致衛(wèi)星鐘存在復(fù)雜的現(xiàn)象。

為了提高GPS衛(wèi)星鐘差的預(yù)報(bào)精度,許多學(xué)者作了一系列的研究,例如線性模型、二次多項(xiàng)式模型[5-7]、灰色模型[8-9]、卡爾曼濾波模型[10]、隨機(jī)模型[11]等,但這些模型均僅適合GPS衛(wèi)星鐘在平穩(wěn)的情況下,而GPS衛(wèi)星鐘的性能卻是非平穩(wěn)的[4]。另外,文獻(xiàn)[12]通過利用SLR和偽距資料對(duì)GPS衛(wèi)星鐘差進(jìn)行預(yù)報(bào),取得了優(yōu)于3 ns的精度,文獻(xiàn)[13]利用開窗分類因子抗差自適應(yīng)序貫平差方法控制了鐘差粗差的影響,其擬合精度和預(yù)報(bào)精度與沒有進(jìn)行抗差處理的自適應(yīng)序貫平差相比,分別提高了78．9%和60．4%。神經(jīng)網(wǎng)絡(luò)由于其在非線性系統(tǒng)建模與優(yōu)化求解方面的優(yōu)勢(shì),被廣泛應(yīng)用于預(yù)測(cè)控制中,形成了各種各樣的神經(jīng)網(wǎng)絡(luò)預(yù)測(cè)控制算法[14],其中,徑向基函數(shù)神經(jīng)網(wǎng)絡(luò)由于具有良好的逼近非線性模型的性能和有能力得到非常穩(wěn)定的結(jié)果而得到了廣泛應(yīng)用[15],這些優(yōu)點(diǎn)也同樣使得徑向基函數(shù)網(wǎng)絡(luò)對(duì)GPS衛(wèi)星鐘差預(yù)報(bào)方面具有較大的應(yīng)用價(jià)值,文獻(xiàn)[16]首次利用徑向基函數(shù)(radial basis function,RBF)神經(jīng)網(wǎng)絡(luò)對(duì)GPS衛(wèi)星鐘差建模預(yù)報(bào)進(jìn)行了初步嘗試,取得了比灰色模型更好的預(yù)報(bào)結(jié)果。本文利用重新設(shè)計(jì)的徑向基函數(shù)神經(jīng)網(wǎng)絡(luò)結(jié)構(gòu)對(duì)一系列的GPS衛(wèi)星鐘進(jìn)行了連續(xù)5 min、連續(xù)1 h和連續(xù)1 d的高精度的預(yù)報(bào),得出了一些有益的結(jié)論。

2 徑向基函數(shù)神經(jīng)網(wǎng)絡(luò)原理

徑向基函數(shù)神經(jīng)網(wǎng)絡(luò)(簡(jiǎn)稱徑向基網(wǎng)絡(luò))是一種性能良好的前饋型人工神經(jīng)網(wǎng)絡(luò),具有較高的運(yùn)算速度和較強(qiáng)的非線性映射能力,能以任意精度逼近一個(gè)非線性函數(shù),因此在許多領(lǐng)域得到了廣泛應(yīng)用[17-23]。徑向基網(wǎng)絡(luò)由輸入層、隱含層和輸出層3層構(gòu)成,如圖1所示。

圖1 徑向基函數(shù)神經(jīng)網(wǎng)絡(luò)結(jié)構(gòu)Fig．1 The RBF network structure

在RBF網(wǎng)絡(luò)中,隱含層神經(jīng)元采用徑向基函數(shù)作為激勵(lì)函數(shù),通常采用高斯函數(shù)作為徑向基函數(shù)。徑向基網(wǎng)絡(luò)傳遞函數(shù)的原型函數(shù)為

式中,n為自變量。該傳遞函數(shù)為輸入層和隱含層之間的距離的映射函數(shù),因此該變換是非線性的。

隱含層第j個(gè)神經(jīng)元輸出為

式中,X為輸入向量;uj為隱含層第j個(gè)神經(jīng)元的中心矢量;σ為徑向基函數(shù)的分布密度(SPREAD),一般根據(jù)經(jīng)驗(yàn)值確定。

由于隱含層與輸出層間的激勵(lì)函數(shù)為純線性函數(shù),因此輸出層第k個(gè)節(jié)點(diǎn)輸出為

式中,nk為輸出層第k節(jié)點(diǎn);yj為隱含層第j個(gè)神經(jīng)元的值;wkj為隱含層第j個(gè)神經(jīng)元與輸出層第k個(gè)神經(jīng)元相連的權(quán)值,該值一般按照學(xué)習(xí)方法獲得,如隨機(jī)選取中心法、自組織選取中心法、正交最小二乘法等。

3 徑向基函數(shù)神經(jīng)網(wǎng)絡(luò)對(duì)GPS衛(wèi)星鐘差的預(yù)報(bào)研究

選擇合適的徑向基網(wǎng)絡(luò)結(jié)構(gòu)對(duì)GPS衛(wèi)星鐘差的預(yù)報(bào)至關(guān)重要,針對(duì)不同的預(yù)測(cè)步長(zhǎng),需要選擇不同的樣本長(zhǎng)度、樣本量以及樣本之間的間隔,目前這些參數(shù)的確定缺少理論根據(jù),只能依賴經(jīng)驗(yàn)確定。由于GPS衛(wèi)星鐘差數(shù)據(jù)為一維時(shí)間序列,因此這里徑向基網(wǎng)絡(luò)采用前N個(gè)數(shù)據(jù)預(yù)測(cè)后M個(gè)數(shù)據(jù)的模式,即選取N個(gè)數(shù)據(jù)作為輸入層向量,M個(gè)數(shù)據(jù)作為輸出層向量,再選取H個(gè)樣本進(jìn)行訓(xùn)練,利用訓(xùn)練得到的網(wǎng)絡(luò)結(jié)構(gòu)進(jìn)行預(yù)測(cè)即可得到預(yù)測(cè)值。

本文從IGS發(fā)布的精密衛(wèi)星鐘差數(shù)據(jù)(ftp:∥igscb．jpl．nasa．gov/)中選取衛(wèi)星2、衛(wèi)星12、衛(wèi)星15以及衛(wèi)星16從歷元670到歷元714之間間隔為5 min的數(shù)據(jù),利用徑向基網(wǎng)絡(luò)分別建模,對(duì)歷元715以及歷元715以后的數(shù)據(jù)分別依次進(jìn)行288次連續(xù)5 min(1 d)預(yù)報(bào)、24次連續(xù)1 h(1 d)預(yù)報(bào)、7次連續(xù)1 d(1周)預(yù)報(bào),取得的結(jié)果分別見圖2—圖5,由于篇幅有限,表1—表6僅給出了部分結(jié)果。

(1)5 min連續(xù)預(yù)報(bào):為了檢測(cè)徑向基網(wǎng)絡(luò)對(duì)衛(wèi)星鐘差的超短期預(yù)報(bào)能力,這里首先對(duì)GPS衛(wèi)星鐘差連續(xù)進(jìn)行5 min預(yù)報(bào),為了利用離預(yù)測(cè)點(diǎn)較近的樣本,設(shè)置樣本間隔為36,樣本數(shù)為136,樣本長(zhǎng)度即N為16,預(yù)測(cè)長(zhǎng)度M為1,即為5 min,依次得到衛(wèi)星2、衛(wèi)星12、衛(wèi)星15以及衛(wèi)星16連續(xù)預(yù)測(cè)結(jié)果見表1。

從表1以及圖2對(duì)4顆衛(wèi)星連續(xù)288次5 min的預(yù)報(bào)結(jié)果可以看出,該徑向基網(wǎng)絡(luò)結(jié)構(gòu)對(duì)5 min的短期預(yù)報(bào)有著非常強(qiáng)的可靠性,可以實(shí)現(xiàn)這4顆衛(wèi)星上述時(shí)段的0．8 ns以內(nèi)的預(yù)報(bào)精度。

圖2 4顆衛(wèi)星1 d內(nèi)每5 min連續(xù)預(yù)報(bào)誤差Fig．2 The every five-minute prediction error within 1 day for four satellites respectively

圖3 4顆衛(wèi)星連續(xù)24次1 h預(yù)報(bào)誤差Fig．3 The every one-hour prediction error in a day for the four satellites respectively

圖4 4顆衛(wèi)星連續(xù)24次1 h預(yù)報(bào)均方根誤差Fig．4 The every one-hour prediction rmse in a day for the four satellites respectively

圖5 4顆衛(wèi)星連續(xù)7次1 d預(yù)報(bào)誤差Fig．5 The every one-day prediction error in a week for the four satellites respectively

表1 4顆衛(wèi)星1 h內(nèi)每5 min連續(xù)預(yù)報(bào)精度Tab．1 The every five-minute prediction accuracy within 1 hour for four satellites respectively

(2)1 h連續(xù)預(yù)報(bào):針對(duì)這4顆衛(wèi)星鐘差的1 h預(yù)報(bào),徑向基網(wǎng)絡(luò)結(jié)構(gòu)設(shè)計(jì)為,樣本的數(shù)據(jù)長(zhǎng)度即N為12,預(yù)報(bào)時(shí)間為1 h即M為12,樣本數(shù)為132,樣本間隔為64,按照該模型建立網(wǎng)絡(luò),同樣利用歷元715之前的數(shù)據(jù)對(duì)歷元715的數(shù)據(jù)進(jìn)行連續(xù)24次1 h預(yù)報(bào),依次得到各顆衛(wèi)星連續(xù)預(yù)測(cè)的結(jié)果見表2。

從表2、圖3和圖4對(duì)4顆衛(wèi)星鐘差連續(xù)24次1 h預(yù)報(bào)結(jié)果來看,雖然對(duì)應(yīng)的徑向基網(wǎng)絡(luò)結(jié)構(gòu)對(duì)各顆衛(wèi)星不一定為最優(yōu),但也實(shí)現(xiàn)了模型均方根誤差在0．6 ns之內(nèi)的預(yù)報(bào)精度,驗(yàn)證了該徑向基網(wǎng)絡(luò)結(jié)構(gòu)對(duì)1 h預(yù)報(bào)的可靠性。

(3)1 d連續(xù)預(yù)報(bào):針對(duì)衛(wèi)星2GPS衛(wèi)星鐘差的1 d預(yù)測(cè),即M為288,設(shè)計(jì)徑向基網(wǎng)絡(luò)為:樣本量為212,樣本間隔為24,樣本長(zhǎng)度即N為9,同樣利用歷元715之前的數(shù)據(jù)對(duì)歷元715及歷元715之后的數(shù)據(jù)進(jìn)行連續(xù)7次1 d預(yù)報(bào),依次得到衛(wèi)星2連續(xù)預(yù)測(cè)的結(jié)果見表3。

利用與衛(wèi)星2鐘差同樣時(shí)段的數(shù)據(jù)對(duì)衛(wèi)星12進(jìn)行1 d預(yù)報(bào)即M為288,采用樣本量為145,樣本長(zhǎng)度即N為9,樣本間隔為32,依次得到衛(wèi)星12連續(xù)預(yù)測(cè)的結(jié)果見表4。

利用與衛(wèi)星2鐘差同樣時(shí)段的數(shù)據(jù)對(duì)衛(wèi)星15進(jìn)行1 d預(yù)報(bào)即M為288,采取網(wǎng)絡(luò)結(jié)構(gòu)為:樣本量為142,樣本長(zhǎng)度即N為25,樣本間隔為24,依次得到衛(wèi)星15連續(xù)預(yù)測(cè)的結(jié)果見表5。

利用與衛(wèi)星2鐘差同樣時(shí)段的數(shù)據(jù)對(duì)衛(wèi)星16進(jìn)行1 d預(yù)報(bào)即M為288,采取網(wǎng)絡(luò)結(jié)構(gòu)為:樣本量為132,樣本長(zhǎng)度即N為12,樣本間隔為24,依次得到衛(wèi)星16連續(xù)預(yù)測(cè)的結(jié)果見表6。

表2 4顆衛(wèi)星1 h連續(xù)預(yù)報(bào)精度Tab．2 The every 1-hour prediction accuracy for four satellites respectively

表3 衛(wèi)星2連續(xù)7次1 d預(yù)報(bào)精度Tab．3 The every one-day prediction accuracy in a week for satellite 2

表4 衛(wèi)星12 1 d連續(xù)預(yù)報(bào)精度Tab．4 The every one-day prediction accuracy in a week for satellite 12

表5 衛(wèi)星15 1 d連續(xù)預(yù)報(bào)精度Tab．5 The every one-day prediction accuracy in a week for satellite 15

表6 衛(wèi)星16 1 d連續(xù)預(yù)報(bào)精度Tab．6 The every one-day prediction accuracy in a week for satellite 16

從表3—表6以及圖5對(duì)4顆衛(wèi)星連續(xù)7次1 d預(yù)報(bào)結(jié)果可以看出,針對(duì)不同的衛(wèi)星的1 d預(yù)報(bào)需要設(shè)計(jì)不同的網(wǎng)絡(luò),連續(xù)預(yù)測(cè)取得的均方根誤差均在1 ns內(nèi),可以說明各徑向基網(wǎng)絡(luò)結(jié)構(gòu)在時(shí)段上對(duì)各顆衛(wèi)星鐘差有較強(qiáng)的可靠性,也說明各顆衛(wèi)星1 d的相關(guān)性行為存在較大的差異。

4 結(jié) 論

通過設(shè)計(jì)不同的徑向基網(wǎng)絡(luò)結(jié)構(gòu)對(duì)不同衛(wèi)星的鐘差連續(xù)進(jìn)行5 min、1 h以及1 d的預(yù)報(bào)試驗(yàn)發(fā)現(xiàn),選擇合適的徑向基網(wǎng)絡(luò)結(jié)構(gòu)至關(guān)重要。針對(duì)5 min和1 h的超短期預(yù)報(bào),只要構(gòu)建合適的徑向基網(wǎng)絡(luò)結(jié)構(gòu),對(duì)不同的衛(wèi)星在不同的時(shí)間均可得到高精度的預(yù)報(bào)結(jié)果,同時(shí),這也反映出GPS鐘差序列盡管有其復(fù)雜性,但由于持續(xù)受空中環(huán)境的影響,使其前后有著較強(qiáng)的短期相關(guān)性;針對(duì)1 d的預(yù)報(bào),對(duì)不同的衛(wèi)星需要不同的網(wǎng)絡(luò)結(jié)構(gòu),這也說明不同衛(wèi)星1 d的行為特征具有較大的差異,但每顆衛(wèi)星連續(xù)預(yù)報(bào)的高精度結(jié)果說明其1 d的行為具備可預(yù)測(cè)性。上述網(wǎng)絡(luò)結(jié)構(gòu)均是不斷試驗(yàn)獲得,因此針對(duì)衛(wèi)星鐘差預(yù)報(bào),構(gòu)建合適的徑向基網(wǎng)絡(luò)結(jié)構(gòu)至關(guān)重要,直接決定著預(yù)測(cè)的精度,由此可見徑向基選擇網(wǎng)絡(luò)結(jié)構(gòu)的設(shè)計(jì)需要進(jìn)一步的理論研究。通過本文的試驗(yàn)結(jié)果,可以大膽假設(shè),若針對(duì)鐘差的先驗(yàn)信息找到徑向基函數(shù)神經(jīng)網(wǎng)絡(luò)結(jié)構(gòu)建立的理論根據(jù),那么徑向基網(wǎng)絡(luò)在GPS衛(wèi)星鐘差預(yù)測(cè)中將會(huì)產(chǎn)生巨大的應(yīng)用價(jià)值。

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(責(zé)任編輯:宋啟凡)

The Application of Radial Basis Function Neural Network in the GPS Satellite Clock Bias Prediction

WANG Guocheng1,2,LIU Lintao1,XU Aigong3,SU Xiaoqing1,2,LIANG Xinghui1
1．State Key Laboratory of Geodesy and Earth’s Dyanamics,Institute of Geodesy and Geophysics,Chinese Academy of Sciences,Wuhan 430077,China;2．University of Chinese Academy of Sciences,Beijing 100049,China;3．School of Geomatics,Liaoning Technical University,Fuxin 123000,China

Satellite atomic clocks can be easily influenced by various factors in space,so the clock behaviour is not sufficiently described and cannot achieved a reliable high-precision prediction by the existed model,such as a linear model,a quadratic polynomial model,grey model and so on．Radial basis function neural network was used in the continuous prediction of four GPS satellite clock bias with five minutes,one hour and one day in this paper,the root mean square error was better than 0．8 ns,0．6 ns and 1 ns,respectively,these prove the reliability of the radial basis network structure on the clock error forecasting．

GPS satellite clock bias;radial basis function;clock bias prediction

WANG Guocheng(1980—),male,PhD candidate,majors in geodesy and survey Engineering．

P228

A

1001-1595(2014)08-0803-05

國(guó)家自然科學(xué)基金(41021003)

2013-03-15

王國(guó)成(1980—),男,博士生,研究方向?yàn)榇蟮販y(cè)量學(xué)與測(cè)繪工程。

E-mail:guocgeng51796＠163．com

WANG Guocheng,LIU Lintao,XU Aigong,et al．The Application of Radial Basis Function Neural Network in the GPS Satellite Clock Bias Prediction[J]．Acta Geodaetica et Cartographica Sinica,2014,43(8):803-807．(王國(guó)成,柳林濤,徐愛功,等．徑向基函數(shù)神經(jīng)網(wǎng)絡(luò)在GPS衛(wèi)星鐘差預(yù)報(bào)中的應(yīng)用[J]．測(cè)繪學(xué)報(bào),2014,43(8):803-807．)

10．13485/j．cnki．11-2089．2014．0078

修回日期:2014-02-18