Rainfall attractors and predictability
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1 Rainfall attractors and predictability Aitor Atencia, Isztar Zawadzki and Frédéric Fabry J.S. Marshall Observatory, Department of Atmospheric and Oceanic Sciences, McGill University McGill Department of Atmospheric and Oceanic Sciences.
2 MOTIVATION Predictability Degree to which a correct prediction or forecast of a system's state can be made either qualitatively or quantitatively.
3 MOTIVATION Predictability Degree to which a correct prediction or forecast of a system's state can be made either qualitatively or quantitatively. Atmosphere Actually, we focus on rainfall fields
4 MOTIVATION Predictability Degree to which a correct prediction or forecast of a system's state can be made either qualitatively or quantitatively. Atmosphere Actually, we focus on rainfall fields NWP model Model Approximation of the real system NWC algorithm
5 MOTIVATION Predictability Degree to which a correct prediction or forecast of a system's state can be made either qualitatively or quantitatively. Atmosphere Actually, we focus on rainfall fields NWP model Model Approximation of the real system NWC algorithm Observations
6 MOTIVATION Predictability Degree to which a correct prediction or forecast of a system's state can be made either qualitatively or quantitatively. Atmosphere Actually, we focus on rainfall fields NWP model Model Approximation of the real system NWC algorithm Observations Predictability of the model Inherent predictability of atmosphere
7 MOTIVATION Predictability Degree to which a correct prediction or forecast of a system's state can be made either qualitatively or quantitatively. Atmosphere Actually, we focus on rainfall fields NWP model Model Approximation of the real system NWC algorithm Observations Predictability of the model Inherent predictability of atmosphere
8 INFORMATION ABOUT DATASET Temporal resolution : 15 minutes Spatial resolution: ~5 km x 5km Grid: 512 x 512 points Length data period: 10/1995 to 03/ ,480 fields in a total of 15 years of data
9 PHASE SPACE OF RAINFALL FIELDS The domains has 512 x 512 pixels but only 212,394 of them are rainfall data because of the radar coverage. Phase space is a space in which all possible states of a system are represented.
10 PHASE SPACE OF RAINFALL FIELDS The domains has 512 x 512 pixels but only of them are rainfall data because of the radar coverage. There are ~ Phase space is a space in which all possible states of a system are represented. possible combinations (states) of rainfall fields. (We have only 420,480 rainfall images ~ 0.001% )
11 PHASE SPACE OF RAINFALL FIELDS The domains has 512 x 512 pixels but only of them are rainfall data because of the radar coverage. There are ~ Phase space is a space in which all possible states of a system are represented. possible combinations (states) of rainfall fields. (We have only rainfall images ~ 0.001% ) Statistical properties of the rainfall field will be used to reduce the number of dimensions of the phase space
12 PHASE SPACE OF RAINFALL FIELDS Statistical properties of rainfall field: Mean 27.8 Std dev 71.7 Skewness 0.62 Kurtosis Area [# pix.] # cells 12 Area biggest storm Decor distance 120 Eccentricity 0.98 Orientation 67 Slope PS 2.54 Marginal distr. Spatial autocor.
13 PHASE SPACE OF RAINFALL FIELDS Removing of correlated statistical parameters.
14 PHASE SPACE OF RAINFALL FIELDS Removing of correlated statistical parameters.
15 PHASE SPACE OF RAINFALL FIELDS Keeping the uncorrelated statistical parameters.
16 PHASE SPACE OF RAINFALL FIELDS Keeping the uncorrelated statistical parameters. The final phase space for the statistical properties of the rainfall patterns is represented by Marginal mean, Eccentricity of the correlation ellipse and Area or decorrelation distance.
17 ATTRACTOR IN PHASE SPACE An attractors is a set towards which a variable evolves over time, moving according to the dictates of a dynamical system.
18 ATTRACTOR IN PHASE SPACE An attractors is a set towards which a variable evolves over time, moving according to the dictates of a dynamical system. LORENZ SYSTEM RAINFALL FIELDS IP (x,y,x) [ -40,40 ; -40,40 ; -20,80 ] &''''% !"#$% 50 stochastic perturbations of I.P. &'''% &''% &'%
19 CHARAC. CHAOS: DIMENSION ESTIMATION Correlation dimension: Cr = 1 N(N!1) N $ i=1 N $ j=1; j#i "( r! X i! X ) j Small-scale effects Large-scale effects Scaling region Addison, P. S., 1997: Fractals and Chaos: An Illustrated Course.
20 CHARAC. CHAOS: DIMENSION ESTIMATION Correlation dimension: Cr = 1 N(N!1) N $ i=1 N $ j=1; j#i "( r! X i! X ) j Small-scale effects Fractal have non-integer dimensions. Scaling region Large-scale effects Dc gives information about complexity of the system Attractor with fractal structure is called strange Addison, P. S., 1997: Fractals and Chaos: An Illustrated Course.
21 CHARAC. CHAOS: DIMENSION ESTIMATION LORENZ SYSTEM Correlation dimension RAINFALL FIELDS Correlation dimension Small-scale effects Scaling region Slope=2.06 Large-scale effects Scaling region Small-scale effects Slope=1.94 Large-scale effects
22 CHARAC. CHAOS: DIMENSION ESTIMATION It seems to be a clusterization of rainfall pattern statistics. Smaller than 2 because of partial correlation between parameters in the phase space Slope=1.94 These effects are caused by the noise in the radar fields.
23 CHARAC. CHAOS: ERROR GROWTH Unpredictable (Stochastic) Exponential Growth Lead-time (Predictability) 12 h
24 LOCAL PREDICTABILITY Highly predictable Less predictable 17.5 h 7 h
25 LOCAL PREDICTABILITY RAINFALL FIELDS Although the boundaries are irregular, a pattern or structure can be observed in the interior of the predictability cloud. Large mean - high eccentricity - large coverage highly predictable
26 LOCAL PREDICTABILITY (VARIANCE) Map of the standard deviation of predictability time Correlation as a measurement of the variance explained 24 Predictability [h] Decor. Distance
27 CONCLUSIONS A 15 years length dataset has been used in this study: Three statistical properties of rainfall fields have been chosen (uncorrelated) to represent the phase space. Rainfall fields have a strange attractor (chaotic system) with fractal structure with correlation dimension of A inherent predictability of 12 hours is obtained. Inherent predictability can be determined by the initial statistical properties of the rainfall field. ±4 hours around the prediction time explains 95% of the variance.
28 Acknowledgements Thanks to my department colleagues for all their help, especially to Madalina Surcel. And to Dr. Alan Seed for his advices/comments during this conference. Thank you for your attention. McGill
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