Detection of Intermittent Turbulence In Stable Boundary Layer Using Empirical Mode Decomposition Xiaoning Gilliam, Christina Ho, and Sukanta Basu Texas Tech University

Overview Introduction The Description of the Data The Description of the EMD Method Hypothesis Testing The Threshold Technique for Intermittency An Example Resulting Intermittency Rate and Stability Concluding Remarks

Introduction The EMD method often applied to signal that is non-linear and non-stationary. In this work, we use EMD to detect intermittent turbulence. By clearly defined the incoherency, a hypothesis test is used to extract intermittent coherent structures from an incoherent signal using a threshold min runs (time series) are used to study the intermittent rate associated with various stability conditions.

The Description of the Data The data used in this study is classified as Cases-99. The data were collected at four elevations: 0.5m (1.5m), 5m, 10m, and 20m and consist of U, V, W components at each elevation. 40 runs at each elevation of the W-components with a total of 160 runs are used in this study. The next slide is an example of the W- component of a time series. The series is divided into high- and low frequency components.

The low- and high- freq component

IMF (Intrinsic Mode Function) We first define the IMF: (1) In the time series, the number of extrema and the number of zero crossing must be equal or differ by at most 1. (2) At any point, the mean value of the envelope defined by the local maximum and the envelope defined by the local minima must be zero. (No positive minima or negative maxima is allowed).

The Description of EMD Method EMD is a method that extracts the IMF’s from the time series using the so-called sifting process based on certain stopping criterion. A cubic spline is used during the sifting process. It breaks down time series without leaving the time domain. The first IMF contains the short oscillation of the time series. Here we may need to focus on the sums of IMFs because individual components may not carry well-defined physical meaning. In our case, we consider the first 4 IMF modes due to the high frequency components.

Sifting Process 1.) Initialize r 0 (t)=x(t) 2.) Extract the i th IMF a.) initialize h 0 (t)=r i (t) where k=1 b.) extract the local maxima and minima of h k-1 (t) c.) interpolate the local maxima and minima by a cubic spline to form upper and lower envelope of h k-1 (t) d.) calculate the mean m k-1 (t) of the upper and lower envelope of h k-1 (t) e.) create h k (t)=h k-1 (t)-m k-1 (t) f.) if stopping criteria is satisfied (IMF mode with small standard deviation), then set h k (t)=ImF i (t) or else go to part (b) and set k=k+1

Sifting Process (continuous) 3.) Define r i (t)=r i-1 (t)–ImF i (t) 4.) If r i (t) still has at least two extrema, then go to step (2) with i=i+1, else the decomposition is done and r i (t) is the residue of the data set. The last mode will be a trend, which is a monotonic function. x(t)=∑ ImF j + r n

Extrema & Cubic Spline

Resulting 1 st IMF

Resulting 4 th IMF

Incoherent signal Incoherent Signal: consider a discretely sampled signal x=(x 1,x 2, …,x N ). Using X for the discrete Fourier transform of the signal, we write, for each frequency component: X=A i e jøi, for i=1,2,…, N/2. We call a signal incoherent if the ø i are uniform [0,2π], and independent of each other and the A i. An exemplar is a signal that has the same spectrum as the signal but is known to be incoherent.

Hypothesis Testing Ho: The signal is incoherent. H1: The signal contains intermittent turbulent. Based on the EMD method, the test statistic IMF4 is obtained by the combinations of the first four IMF modes. In this case, we will conclude H1 (detection at time t) if the test statistic is greater than the threshold th. We will choose th so that the probability of detection is maximized and the probability of false alarm is less than or equal to the size of the test a.

The Threshold Technique We use the n incoherent exemplars to determine the threshold. In this case, each exemplar IMF4 k e for k=1,2, … n, will have the same spectrum shape as the test stat and are know to be incoherent. Thus, the threshold is defined by th={sub t : (1/n∑ | IMF4 k e | >t >a)}

IMF4 series and the threshold

Time-series and Detection

Intermittency Rate and Stability Intermittency Rate for each of the runs = (Total Detection)/(Length of the Time Series) For the example, Intermittency Rate= Based on the stability, the 160 runs in this study are divided into five class intervals: Column 1: Stability <=1 Column 2: 1< Stability <=2 Column 3: 2< Stability <=3 Column 4: 3< Stability <=4 Column 5: Stability > 4

Stability VS. Intermittency Rate

Concluding Remarks EMD method provided a useful signal processing technique for non-stationary and non-linear data analysis. Based on the one-way one factor Analysis of Variance (ANOVA), the means of the columns are different (p-value =1.36*10 -8 ). This result can also be found by using a Boxplot. A comparison study of EMD method and wavelet method may be needed to explore the uniqueness of the method. More data analysis is needed to concluded the relationship between the intermittency and the stability.