Application of Kalman filter to noise reduction in multichannel data

Transcription

1 S C H E D A E I N F O R M A T I C A E NNNN ISSUE NN YEAR Application of Kalman filter to noise reduction in multichannel data ANDRZEJ LEŚNIAK, TOMASZ DANEK, MAREK WOJDYŁA Department of Geoinformatics and Applied Computer Science, AGH Uniersit of Science and Technolog, Miciewicza a,, Kraów, -59 Poland, PBG Geophsical Eploration Compan Ltd., Jagiellońsa 76, Warszawa, - Poland lesnia@agh.edu.pl Abstract. The results of application of multichannel Kalman filtering to reduction of uncorrelated noise in magnetotelluric recordings are discussed in this article. Magnetotelluric method of Earth structure recognition is shortl presented together with the its most popular measurement method called the remote reference method. The basic theor of nonstationar, discrete Kalman filter and its implementation to multichannel magnetotelluric data recorded in multi-site eperiment are also discussed with details. The practical eamples of Kalman filter application to the real D and D data illustrate the merits of presented technique. Kewords: Kalman filter, noise rejection, magnetotellurics,. Introduction Application of multichannel Kalman filtering to reduction of uncorrelated noise in magnetotelluric data is discussed in the article. Magnetotelluric recordings proide important information about structure of the Earth. The were successfull used to build or erif models that describe inner structure of the Earth. After specialized processing we can find so called transfer function that describes the conductiit distribution below the Earth surface. Usuall the useful signal is buried in arious inds noises therefore the first and most important step in data processing is noise cancelling.

2 Numerous methods of noise reduction in magnetotelluric data eist. The aries from simple deterministic band pass filtering to sophisticated methods that incorporate artificial intelligence methods. The first can be use to narrow frequenc noise [] while the other hae larger scope of application []. The methods used so far to noise reduction of such data usuall do not tae adantage of the fact that man of magnetotelluric data (especiall recorded in EMAP technique or multi-site sure) are multichannel. For the reason of that and because of the alternating spectrum and irregular appearance of the noise in magnetotelluric data we hae to use adaptie filtering to sole the problem. We propose to use the Kalman filtration of reference recordings to effectie noise reduction or een rejection.. Theor The Kalman filter is a nonstationar, recursie filter that allows estimation of the useful signal in nois time series in each moment of time [4]. In stead state Kalman approach to digital filtering signal { } is described b two linear difference stochastic equations: F Gw () H The first equation describes process of signal generation while the second process of represent the process and signal measurement. The random ariables { } w and { } measurement noises (respectiel). The are assumed to be independent (of each other), white, and with normal probabilit distributions. The coariance matri of process noise and coariance matri of measurement noise E E [ w w ] Q p( w ) ~ N(,Q) l l () [ ] R p( ) ~ N(,R) l l () might change with each time step of time series, howeer here we assume the are constant. The matri F in the first difference equation relates the state at the time step - to the state at the step, in the absence of either a driing function or process noise. The matri F might change with each time step, but we assume it is constant. The matri H in the measurement equation relates the state to the measurement. We also assume here that it is constant.

3 The Kalman filter estimates the process state at step and then obtains feedbac in the form of (nois) measurements. Therefore, the equations for the Kalman filter fall into two groups. The first is time update equations: ˆ F P ˆ FP F T Q (4) and the second measurement update equations: K ˆ P P ˆ H T K T [ H P H R] ( H ˆ ) [ I K H ] P (5) The time update equations are responsible for projecting forward (in time) the current state and error coariance estimates to obtain the a priori estimates for the net time step. Matri P is a priori estimate of error coariance: while matri [( ) ] T ˆ P (6) E ˆ P is a posteriori estimate of error coariance [( ˆ )( ) ] T P (7) E ˆ As we can see from the second filter measurement update equation a posteriori state estimate ˆ is a linear combination of an a priori estimate ˆ and a weighted difference between an actual measurement and a measurement prediction ˆ H (8) ˆ ˆ Starting from the initial estimates ˆ and P one can obtain estimates of ector for an time.. Magnetotelluric obserations and data

4 4 The adantages of multichannel Kalman filters can be shown in noise rejection in data recorded during measurements of natural electromagnetic field induced in the Earth. Magnetotelluric measurements are passie measurements of fluctuations of natural electric and magnetic fields at the Earth s surface [6]. Such fluctuations of electromagnetic field generate so called magnetotelluric waes. Periods of magnetotelluric waes usuall ranges from - s to 5 s what depends on the wae source. The shortest waes are generated b lightning stries while longer one b ionosphere resonance and solar wind ariations. Penetration of magnetotelluric waes ranges from 5m to more than 5 m. The essential aspect of interpretation procedure of magnetotelluric data is estimation of transfer function. A noise reduction is a critical moment of that procedure. An awward noise in magnetotelluric time series can be generated b measurement equipment (especiall sensors), meteorological phenomena and anthropogenic factors (industr installations and deices, railwas). One of the fundamental techniques that allows the noise elimination is a remote reference method []. In this method components of the magnetotelluric field are snchronicall registered in measurement and reference points. Application of appropriate processing procedures to this data allows uncorrelated noise to be reduced. The magnetic remote reference point has to be localized in a place free of disturbances. In practice onl uncorrelated noise eists in data recorded in places located in distance larger than seeral dozens of ilometers. Such reference is called the distance remote reference. If multi-site measurement instrumentation is used onl in relatiel small area it is possible to use mutual electrical components of magnetotelluric field as a reference components. The are called local remote reference. In practice the simplest method of the impulse-tpe disturbances reduction is applied. These spies are simpl rejected and the oids appears in recordings. It maes ealuation of the impedance function for low frequencies difficult and erroneous. The more appropriate method of noise reduction is a proposed Kalman filtering. In contrar to frequenc filtering the Kalman filter can effectiel reject uncorrelated disturbances because it presere the shape of noise-free registrations. Therefore the errors of impedance function estimation are significantl reduced. The filter enables also reduction of the noise uncorrelated between particular channels. As an output of the filter we obtain corrected electric or magnetic recordings consisted onl from components which eist in reference point. Consequentl, the filter can reduce the atmospheric and industrial noises if their sources ware located close to measurement points. The data used in our stud was recorded in area where noise was relatiel large comparing to other measurements in that region. Analzed measurement point was localized in area where noise intensit was particularl high. The disturbances were caused b electricit power lines and transformer stations, railwas, traffic and buildings. High-oltage lines were 5 meters awa, electric railwa lines were 5 meters awa and some inhabited buildings were in close icinit of the point. Furthermore, some of the disturbances were qualified as a meteorological ones, generated b wind or lightings. The data gathered in measurement point consists of electric field recorded b three dipoles aligned in north-south direction (E, E, E components), three dipoles aligned in west-east direction (E, E, E components) and magnetic field recorded b two magnetometers aligned in north-south (H component) and east-west direction (H component). Magnetic field was also recorded in reference point b two magnetometers aligned in north-south (HR component) and east-west direction (HR component).

5 Recorded time series are er long. The tpicall consist of seeral millions of data points, because measurements in one point can tae up to one wee of continuous recordings. We present the results onl from short window of 8 s length. The corresponding time series are shown in Fig.. 5 Fig.. Windowed magnetotelluric data of 8s length. 4. Kalman filtering of multichannel magnetotelluric data In remote reference method components of the magnetotelluric field are snchronicall registered in measurement and reference points. We compare two or three components of the same tpe (electric or magnetic). We assume that each component consists of useful signal and noise, appropriate for each component, as shown below:

6 6 (9) where,, -number of sample. We built the Kalman filter in the following wa [5]. As a reference signal for a first channel we used a signal recorded in second channel. In case of three component interpretation the useful signal for the first channel is an aerage of the signals from second and third channel. The Kalman filters for two component filter is gien as: w w,, () and for three component case as: w w w,, () In this approach useful signal is estimated from measurement signal if noise in reference channels is uncorrelated. The starting coariance matri is equal P * I. where I is unit matri. Different alues for process noise coariance and measurement noise coariance were applied to the calculation. Their alues were estimated in fie point moing window centered in the point of ealuation. Below we present the results of two components filtering for magnetic recordings (in measurement and reference points) ealuated in Matlab enironment. For electric field recordings we used three component filter. In Fig. A the time series presents local and remote components of magnetic field for time window of length 4s. The signals were recorded in two distant points and differ in shape. Remote signal recorded in noise free enironment (green cure) does not contain spies which are frequent in signal recorded in local point (blue cure). After filtering local and remote signals are er similar (see Fig. B). The small differences are proportional to measurement noise { }. In Fig.A the measured and filtered signals in local point are presented (blue and green cures respectiel). Filtered signal is much

7 smoother than the recorded one. As spectral analsis shows Kalman filter reduced the high frequenc components of the recorded signal. Of course the frequenc characteristics of the filter ar in time, because it depends on the reference signals. For the analsis window of length 48 samples the difference between recorded and filtered signals shown in Fig.B is zero mean, white noise, uncorrelated signal. For shorter analsis window these obserations are not alwas true. Electric components of the electromagnetic field are er similar because the are recorded close to each other (maimum distance is usuall less than few hundreds of meters). As can be seen in Fig.4A the time series differ in narrow spies and some bias (especiall seen in the first half of the series). Time series are mutuall uncorrelated if we compare signals in different channels. The Kalman filtering of the electric components gies as a result er similar cures (see Fig 4B). The filter remoes almost perfectl uncorrelated noise (small field ariations and narrow spies). It also reduces the bias components. If we compare measured and filtered signals we can see that coherent noise is not remoed from the signal. For eample artificial noise in time window from5 to 7 samples in Fig.4 (probabl generated b passing train) is recorded in the similar wa in each component in X direction. After filtering the amplitude of that noise is reduced but still measured and filtered signals are correlated (what is easil seen in Fig 5A). Onl the bias and narrow spies that appear in different times in different channels are remoed perfectl. The difference between measured and filtered signals in case of electric components is not a white noise in narrow window but is still a zero mean process (Fig. 5B). 7 A B Fig. Results of proposed Kalman filtration in case of local and remote magnetic components. Raw (A) and filtred (B) time series for channels H (blue), HR (green).

8 8 A B Fig. A) Comparison between original (blue) and filtered (green) H time series in case of two component Kalman filter. B) Difference between raw and filtered time series. A B Fig. 4 Results of proposed Kalman filtration in three component ariant. Raw (A) and filtred (B) magnetotelluric time series for channels E- (blue), E- (green)and E- (red).

9 9 A B Fig. 5 A) Comparison between original (blue) and filtered (green) E- time series in case of three component Kalman filtering. B) Amplitude difference between raw and filtered time series. 5. Discussion and conclusions The proposed method of noise remoal does not introduce an quantitatie information about the noise and useful signal. It also do not use an a priori parameters (such as filter tpe ore shape, cut-off frequenc of filters, or others). To perform Kalman filtration the measured signal is decomposed into the useful signal and noise. The construction of useful signal is based on principles of signals similarit in measurement and reference points. The correlated (similar) components of the signals are reconstructed as a result of filtering procedure discussed aboe. Because the Kalman filter is not a simple band pass filter it remoes noise without reducing the frequenc content of the signal. It is er important from point of iew further processing methods (e.a. calculation of the transfer function), because preseres the abilities of the magnetotelluric method to recognize both shallow and relatiel deep Earth structures. The filter implemented here adapt to local noise leel and to the local shape of the signal. The filter parameters (process and measurement noise coariances Q and l R ) l allows proper adjust the signals in different channels to each other in contrar to standard LMS adaptie filters. What is also important from point of iew of practical implementation the Kalman filter proposed in the article is er simple and eas for coding.

10 Article was written within the framewor of a research grant founded b the Polish Ministr of Science and Higher Education, no N55 9 /768. References [] Bieleca M., Dane T., Wojdła M., Leśnia A., Baran G.; Application of artificial intelligence methods to distortions reduction in magnetotelluric measurement data, 9th International worshop on Electromagnetic induction in the Earth, Beijing, China, 8, Vol., pp [] Dane T., Stefaniu M., Wojdła M.; Eamples of using singular spectrum analsis in magnetotelluric data processing, 8th International worshop on Electromagnetic induction in the Earth, El Vendrell, Spain, 6, pp. -. [] Gamble T., Goubau W.M., Clare J.; Magnetotellurics with a remote magnetic reference, Geophsics, 44, 979, pp5-68. [4] Kalman R.E.; A new approach to linear filtering and prediction problems, Transactions of the ASME Journal of Basic Engineering, 8 (Series D), 96,pp [5] Leśnia A., Dane T., Wojdła M.; Application of Kalman filter to noise reduction of magnetotelluric recordings, 9th international worshop on Electromagnetic induction in the Earth, Beijing, China, 8, Vol., pp. 45. [6] Simpson F., Bahr K.; Practical magnetotellurics, Cambridge Uniersit Press, UK, 5.