Reducing GPS Carrier Phase Errors with EMDWavelet for Precise Static Positioning Jian Wang, Jinling Wang and Craig Roberts School of Surveying and Spatial Information Systems The University of New South Wales Sydney NSW 2052, Australia Contact:
[email protected] Tel : 61-2-9385 4203 Fax : 61-2-9313 7493
ABSTRACT In this paper, a new EMD-Wavelet based model is used to reduce carrier phase errors for precise static GPS positioning. The EMD is a new signal processing method for analysing non-linear time series, which decomposes a time series into a finite and often small number of Intrinsic Mode Functions (IMFs). The decomposition procedure is adaptive and data-driven. The IMFs are stationary which are more suitable for wavelet analysis. The merits of both the EMD and Wavelets are combined to produce an improved EMD-Wavelet systematic error extraction model. Thereafter, a method of the GPS baseline calculation based on the EMD-Wavelet systematic error mitigation model is suggested. The experimental results show that the proposed scheme dramatically improves the reliability of ambiguity resolution with a larger F-ratio and Wratio indexes after systematic error elimination. INTRODUCTION Unmodeled systematic error is one of the most dominant factors inducing the failure of carrier phase based high precision baseline solutions and their stability. There are many complicated factors causing systematic errors which are impossible to eliminate completely including ionospheric and tropospheric errors, orbital errors, multipath and other unmodelled biases. Many models have been employed to study the mitigation of the systematic errors for GPS baseline processing. Ionospheric and tropospheric error modeling has been paid much attention. [1] investigated different tropospheric models on GPS baseline accuracy. Ionospheric error correction improvements of differential GPS for long baselines is presented in [2]. The orbit bias is a baseline length dependent bias which can be minimized by Kalman filter modelling and carrier phasedifference modeling [3,4]. Multipath is another important systematic error source. Finite impulse response (FIR) filters are tested with the limitation of dividing mixed multipath errors with the same frequency band [5]. Other unmodelled biases can also be eliminated to some extent with stochastic modelling. Wavelet noise reduction modelling is one of the most effective techniques with respect to complicated signal analysis. Recently, it has been introduced into the field of GPS data processing for signal de-noising, outlier detection, bias separation and data compression as well as other models introducing wavelets for multipath analysis and mitigation for baseline solutions have also been studied [6,7]. Although wavelet transforms are suitable for noise reduction, in other words, for signal trend extraction, its pre-divided frequency feature limits its ability to decompose the signal into different frequencies according to the inherent characteristics of the signal. Empirical Mode Decomposition (EMD) is a new signal analysis technique showing great promise for signal trend extraction [8]. EMD techniques have already been successfully applied to several other scientific problems. It provides an adaptive representation of non-linear signals, which ensure the non-linear signal can be converted into an Intrinsic Mode Function (IMF) more easily for wavelet analysis. Based on EMD and the wavelet shrinkage noise reduction model, a new trend extraction model called the EMD-Wavelet model is presented here. In this paper, systematic errors are isolated with the EMD-Wavelet model from DD observation residuals and integrated into the baseline solution for a higher reliability baseline solution. Experimental results show improvements in reliability and the effect of systematic errors is reduced.
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THEORETICAL BACKGROUND Empirical Mode Decomposition (EMD) An Intrinsic Mode Function (IMF) is defined as any function having the number of extrema and the number of zero-crossings equal (or differing at most by one), and also having symmetric envelopes defined by the local minima and maxima, respectively. By decomposing a given signal into different IMFs, the signal x(t ) can be expressed as [8,9]: n
x(t ) = ∑ imf i (t ) + rn (t )
(1)
i =1
Where imf i (t ) denotes the ith IMF components constrained to be zero-mean and rn (t ) stands for a residual “trend”. The Intrinsic Modes decision is based on automatic and adaptive (signal-dependent) time-variant filtering and their high vs. low frequency discrimination. It applies locally and corresponds in no way to a pre-determined sub-band filtering (as, for example, in a wavelet transform). The effective algorithm of the EMD can be summarised as follows [8]. ① Identify all extrema of x(t ) , ② Generate envelope e min (t ) (resp. emax (t ) ) by interpolating between minima (or maxima), ③ Calc. mean value with m(t ) = (emin (t ) + emax (t )) / 2 ,
= x(t ) − m(t ) , ⑤ Iterate steps ① to ④ on d (t ) until it is zero-mean according to some stopping criterion. The obtained d (t ) is referred to ④ Extract detail with d (t )
as an IMF, ⑥ Calculate m(t ) = x(t ) − imf i (t ) , ⑦ Iterate step ① to ⑥ until no more IMFs are available. Further investigation of the algorithm is demonstrated below with regard to the multi-resolution standpoint. The IMF calculation operator and residual calculation operator are defined as Fimf (•) and Fresidual (•) , where the two operators are similar to high frequency and low frequency filters. Operator Fimf (•) includes EMD decomposition steps from ① to ⑤ with the view
of obtaining the high frequency component of the specific scale. Fresidual (•) represents step ⑥, which calculates the residual, low frequency component of the corresponding scale. Then, the multi-solution structure is realised by decomposing the low frequency step by step. The decomposition formula from the ith to (i+1)th is given as:
imf i +1 (t ) = Fimf (mi (t ) )
mi +1 (t ) = Fresidual (mi (t ) )
(2) (3)
and the reconstruction is expressed as: −1 −1 mi (t ) = Fimf (imfi +1 (t )) + Fresidual (mi +1 (t ))
(4)
Where, F −1imf (•) and F −1residual (•) denote the inverse operator of Fimf (•) and Fresidual (•) respectively. Wavelet Shrinkage Noise Reduction Model “Waveshrink” uses orthogonal wavelets to decompose a given signal and the obtained wavelet decomposition coefficients w j ,k (i,j present the wavelet decomposition level and scale) are shrunk by considering threshold rules based on the idea that only very few coefficients contribute to the signal, which is given by [10]. The updated wavelet coefficients
wˆ j ,k given with
the hard-threshold and the soft-threshold noise reduction models are presented as: hard-threshold:
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⎧w j , k ⎪ wˆ j ,k = ⎨ ⎪⎩0
w j ,k ≥ λ (5)
w j ,k < λ
Soft-threshold:
⎧sign( w j ,k ) ⋅ ( w j ,k − λ ) w j ,k ≥ λ ⎪ (6) wˆ j ,k = ⎨ w j ,k < λ ⎪⎩0 where λ = σ 2 log(n) / n is a predefined threshold value，n is the length of the discrete signal. σ can be estimated by σˆ = MAD / 0.6754 ，MAD is the median of the absolute deviation value. Thereafter, wavelet reconstruction algorithm is applied to obtain the de-noised signal. We focus on soft-thresholding noise reduction in the later proposed model. EMD-WAVELET Noise Reduction Model The noise included signal
x(t ) is firstly decomposed into imfi (i = 1,2L, n) standing for the IMFs of EMD results. High
frequency IMFs selected according to later mentioned criteria are chosen for wavelet noise reduction and the trend of the signal xˆ(t ) (or the nonlinear optimization estimation of the signal) is obtained with inverse EMD performance. The criteria of the scale selection for the systematic errors extraction is given by defining the means of the accumulated standardized modes(MSAM) by the formula:[8,11]：
hˆm = mean{
m
∑[ imf (t) − mean (imf (t)) / std(imf (t)) ] } i=1
i
i
i
m≤n
(7)
where imf i (t ) is called the ith scale mode, if hˆm is obviously biased from zero, the scale m is fixed as the mark to distinguish the high frequency mode from the low one. It will be shown that the estimated systematic errors can be expressed as: m −1
n
i =1
i=m
xˆ (t ) = ∑ de(imf i (t ))∑ imf i (t ) + rn (t )
(8)
Where de(•) denotes wavelet shrinkage noise reduction. The scheme of EMD-Wavelet based GPS baseline solution can be illustrated as Figure 1.
imf1 x(t )
E imf2 M D …
(HF) imf
imfn
(LF) imf
Wavelet denoising
xˆ (t )
Figure 1: The flowchart of EMD-Wavelet trend extraction model EMD-WAVELET based baseline solution scheme The DD baseline solution equation is given by:
∇ ΔLi = ∇ Δρ − λ i ∇ ΔN i + ∇ Δtrop + ∇ ΔV sys + ∇ ΔVσ
(9)
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Where ∇Δ is the double-difference (DD) operator. i indicates a certain frequency of signal (i=1 or 2 for GPS receiver), ∇ΔLi , ∇Δρ stands for DD carrier phase and pseudo-range observations respectively. λi is wavelength of the carrier phase and ∇ΔN i denotes DD integer ambiguity. ∇Δtrop is pre-calculated with an appropriate model, precise ephemeris is used to mitigate errors from the satellites, and this is a short baseline, ionosphere errors can be omitted in the calculation. The residual series of DD carrier phase observations reflect the systematic errors ∇ΔVsys separate from random noise ∇ΔVσ . The EMD-WAVELET model is applied to extract the systematic errors from the residual series.Thereafter, the DD carrier phase observations are modified by the extracted systematic errors epoch by epoch. The DD observation equation is recomposed and the float solution is calculated after that. Then the Lambda algorithm is used to fix the integer ambiguity and the final baseline solution is given at the same time. The recalculated residual series and the validation of integer ambiguity resolution are used to evaluate the effectiveness of the suggested scheme (Figure 2). RINEX file
RINEX file
Data input
Constitute DD observation equation Modifying DD observations
Residuals of DD observations
Float resolution
EMD-Wavelet based systematic error extraction
LAMBDA
Fix Integer Ambiguities Validation test Ambiguity fixed baseline solution
Figure 2: Functional block diagram of the EMD-Wavelet based GPS baseline solution algorithm EXPERIMENTAL RESULTS AND ANALYSIS The data sets were measured using a NovAtel GPS receiver from 3:12:00 to 3:32:00 on Dec. 5th, 2001 with a 1s sampling span using 1024 epochs. PRN 21, PRN17, PRN3, PRN18 and PRN14 are observed simultaneously all along the selected span with the mean elevation angle of 61.0, 44.3, 43.5, 56.7 and 36.0 degrees respectively. The L1 band is used and PRN 21 with the highest mean elevation angle is selected as a reference satellite for double-difference time series calculation in order to avoid systematic effects. Figure 3 demonstrates the DD system error series obtained from the PRN21-17, PRN21-3, PRN21-18 and PRN21-14 satellite pair data.
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Figure 3. Baseline systematic error series Systematic Error Extraction Each residual series is used to extract systematic error using the EMD-Wavelet model. Due to the space limitation, only the PRN21-17 residual is presented. This paper presents the details of the extraction procedure. The eight-scale EMD decomposition of the PRN21-17 residual series is shown in Figure 4. According to the criteria of the scale selection, the relationship between the means of the accumulated standardized modes (MSAM) and the scale is given in Figure 5. The appropriate scale for the high frequency IMFs and low frequency IMFs discrimination is 4, that is, m equal to 4 in equation (7). Each high frequency IMF (from IMF1 to IMF3) is de-noised with a ‘db3’ wavelet shrinkage noise reduction model and then the EMD reconstruction gives the extracted systematic error. Another three DD systematic trends are extracted with the same procedure (Figure 6). The extracted trend is input into the DD observation equation for baseline calculation. The histogram of the residual series shows an apparent random characteristic and most of the trend is isolated (Figure 7). 0.02
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Figure 7: Residuals of PRN21-17, PRN21-3 , PRN21-18 and PRN21-14 Analysis on Improvement of Baseline and Ambiguity Table 1 summarises the basic results calculated before and after the application of the EMD-Wavelet model for systematic error mitigation. The F-ratio and W-ratio tests of the ambiguity verify the stability of the baseline is improved significantly. Figure 8 shows the recalculated DD residual series of the baseline fixed solution showing obvious random distribution characteristics, which demonstrates the systematic error has been eliminated. The best and second best ambiguity combination is significant for ambiguity discrimination. F-ratio is a standard test for this comparison and W-ratio [12] is also compared. Although the fixed solution shows little change before and after systematic error elimination, the float solutions are affected by systematic errors to a great extent. After the systematic error mitigation, the float solutions are close to the fixed solution, and the stability of the ambiguity resolution is improved significantly. Therefore, a conclusion based on the results shown in Table 1 and Figure 8 is that most systematic errors in the residual series have been detected and eliminated by the proposed baseline calculation scheme and the stability of the fixed solution of the baseline has been improved. 0.1
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Figure 8. Recalculation of the DD equation fixed solution residuals series after systematic error elimination CONCLUDING REMARKS
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In this paper, Empirical Mode Decomposition (EMD) and wavelet shrinkage noise reduction models have been briefly reviewed. The EMD-Wavelet trend extraction model is presented and analysed in detail with both simulated and real signal data. The suggested procedure using the EMD-Wavelet model can eliminate trend effects in baseline solutions and the example shows satisfactory results. However, the model is still faced with the difficulty of parameter confirmation in some cases, which should be conducted. Table 1: Baseline solution before and after applying the DD system error extraction algorithm Vector x y z
Float solution(m) before after 0.0905 0.1443 -9.1982 -9.1775 -6.6749 -6.6801
Fixed solution(m) before after 0.1499 0.1498 -9.1757 -9.1756 -6.6804 -6.6805
F-ratio before After
W-ratio before after
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