interpolation of wind fields (Zavala-Hidalgo et al., 2003). Synoptic-scale wind fields show propagating waves with east- or westward direction in mid-latitudes.
ON THE USE OF COMPLEX EMPIRICAL ORTHOGONAL FUNCTIONS FOR THE TEMPORAL INTERPOLATION OF NWP, RADAR AND SATELLITE DATA Andreas Wirth, Alexander Jann and Barbara Zeiner Zentralanstalt für Meteorologie und Geodynamik, Hohe Warte 38, A-1190 Wien, AUSTRIA
Abstract Before merging diverse data, it is regularly necessary to interpolate one data set temporally in order to obtain simultaneity (even if the data sources were available in the same temporal resolution, the problem is still relevant in cases of data outage). Simple interpolation approaches tend to fail in case of noticeably propagating features (the meteorological systems then do not actually move through the interpolated imagery, but slowly vanish at the initial location while gradually appearing at the target location). Therefore - following suggestions of earlier publications where such a technique was used for the temporal interpolation of NWP fields in order to get higher-frequency data - complex empirical orthogonal functions are extracted from a series of 2-dimensional arrays. After interpolating the phase and amplitude components of the resulting functions, fields of the variable under consideration (i.e. in the present context: satellite-measured brightness temperature, radar reflectivities, NWP forecasts) can be reconstructed for every intermediate point of time. The paper describes some experiments carried out to assess the strengths and limitations of this method.
1. INTRODUCTION Meteorological data like numerical model output, radar and satellite data are available at regular time steps. For some applications, it is desirable to have these data at higher frequencies or at time steps in-between the given intervals (in order to achieve simultaneity with other data, for example). To apply linear temporal interpolation for this purpose is mostly adequate when features behave like standing waves, i.e. they change their amplitude without moving. In case of fast-moving features, linear interpolation frequently leads to undesired results: the amplitude of the observed pattern changes while the pattern remains stationary. The interpolated fields exhibit a double appearance of the pattern at its start and end positions, without any actual movement. There is certain evidence that an alternative approach based on decomposing the fields into their complex empirical orthogonal functions (CEOF) is better suited for the time interpolation of moving wave-like features (cf. figure 1).
Figure 1: Comparison of linear with CEOF interpolation. While for linear interpolation the wave remains stationary and the amplitude changes, CEOF phase interpolation preserves the propagation of the wave and the amplitude stays constant. The solid blue line gives the initial position of the wave, the dotted blue line its end position. The red lines indicate the position of the interpolated wave. The image is taken from Yu et al. (2003).
2. MOTIVATION An interpolation method for NWP data was needed for the computation of the EUMETSAT Nowcasting-SAF product ASII (Automatic Satellite Image Interpretation). Based on NWP output and satellite images, ASII generates an analysis of satellite image features in terms of conceptual models. This analysis is done for MSG satellite images every 15 minutes. Since ECMWF forecast data with such high frequency are not available at meteorological services, a suitable temporal interpolation method was sought. 3. METHODOLOGY Complex Empirical Orthogonal Functions are used to decompose time dependent numerical fields into their Eigenmodes. Each Eigenmode consists of a vector pair and explains a certain amount of the variability of the field in space and time. A matrix with N points in space (measurements or grid points) and M points in time yields M
independent, orthogonal complex vector pairs, comprising a temporal (1) and a spatial function (2): T (t ) = At (t ) exp[iΘ(t )] S ( x ) = As ( x ) exp[iΦ( x )]
(1) (2)
where At (As) is the amplitude of the temporal (spatial) wave and Θ (φ) its phase. Equations (1) and (2) represent the Eigenmodes of the NxM matrix. The most significant modes (i.e. the modes with the highest Eigenvalues) are used to reconstruct the field. 4. INTERPOLATION CEOF interpolation can be applied either on the spatial or on the temporal function. In application to the spatial function, CEOF can be used to downscale a numerical field preserving its main characteristics. For the temporal interpolation, the amplitude At and the phase Θ are interpolated in-between the time steps under consideration. Maue (2004) recommends using cubic splines for this interpolation. Nevertheless, interpolation of amplitude and phase was made linearly in our experiments so far. This is justifiable when time steps are small compared to the phase speed. At the time of this writing, temporal interpolation has been applied to: • MSG-satellite images (15-minute interval → downscaled to 1-minute intervals) • Austrian radar imagery (5-minute interval → downscaled to 1-minute intervals) • ECMWF data (6-hours interval → downscaled to 15-minute intervals) 5. RESULTS 5.1 Application to NWP model data One of the most promising applications of CEOF analysis is the temporal interpolation of wind fields (Zavala-Hidalgo et al., 2003). Synoptic-scale wind fields show propagating waves with east- or westward direction in mid-latitudes. The decomposition of ECMWF wind fields with the CEOF method should allow analysing spatial structures that propagate in space and vary in time. In fact, Chu and Fang (2003) demonstrate how well-performing CEOF is in extracting the Rossby waves from geophysical data (satellite altimetry); Zavala-Hidalgo et al. (2003) discuss how the method handles moving atmospheric waves more properly than simpler techniques. The latter study deals with 12-hourly scatterometer wind products, and hence has a similar direction as the application to 6-hourly model wind output pursued here. Figure 2 shows ECMWF model winds interpolated in 1.5-hour time steps from 6hourly actual forecasts. While some of the plotted arrows show a nearly linear change of speed and/or direction, others clearly indicate the non-linear character of the temporal CEOF interpolation.
The improvement in quality has been verified by “leaving-out one” experiments, i.e. withholding a forecast field and comparing it with one resulting from CEOF decomposition/reconstruction. For other numerical fields based on wind fields, like vorticity, wind shear and advection fields, also good results were obtained, with a minimum requirement of M=4 input forecast fields for deriving sensible Eigenmodes. CEOF temporal interpolation has been applied also to other numerical fields with satisfactory results, e.g. the thermal front parameter (TFP). This smooth field reflects the changes of the temperature gradient at 850 hPa and is a good indicator for the position of a frontal zone (Huber-Pock and Kress, 1989). In the example shown below, the TFP-field is interpolated with the CEOF method. A broad frontal zone is located over the western Mediterranean Sea moving eastwards. The temporal interpolation has been done in-between model output 06 and 12 UTC; for the decomposition in Eigenmodes, 4 time steps (model output for 06, 12, 18 and 00 UTC) were taken. The interpolation result (figure 3) nicely captures the propagation of the cold front (Spain) and the displacement of a warm front over the Atlantic.
Figure 2: Example of temporal interpolation of the 925 hPa wind field by the CEOF method, 15 January 2008, 0000 to 0600 UTC (red arrows: 00:00 UTC forecast, green: 01:30 UTC (interpolated), blue: 03:00 UTC (interpolated), yellow: 04.30 UTC (interpolated), cyan: 06:00 UTC forecast.
Figure 3: Example of temporal interpolation of the thermal front parameter by the CEOF method. th ECMWF forecasts for 14 January 2008 (initialized at 12 UTC on 13 ); 0600 UTC forecast shown in red, interpolated field at 0900 UTC in green, 1200 UTC forecast in blue.
5.2 Application to image data The capabilities of the CEOF method to interpolate satellite or radar images shall be demonstrated on “simplified” images consisting of two superimposed sine waves, the first sine wave is oriented in east-west direction and propagates towards the west, the second sine wave is oriented in north-south direction and remains stationary (Fig. 4, left column). For further experiments, the sine functions have been slightly modified: The central column of figure 4 presents the results for a truncated sine wave (constructed as the first example but negative values substituted by 0), and the right column in that figure documents an experiment with square waves substituting the sine functions. The behaviour of CEOF interpolation for the pure sine waves is shown in the left column of figure 4 (rows b) and c)). The pattern is exactly reproduced and slightly shifted towards the left. This meets our expectations; the pattern is located inbetween start and end position without any changes in shape. For the truncated sine waves, the interpolated pattern gets blurred, when temporal interpolation is applied. The results of interpolation are degrading even more when the image patterns exhibit sharp edges (right column). Therefore we can state that CEOF temporal interpolation is best suited for sine patterns (for patterns which can easily be broken down to simple sine waves). In analogy to Fourier transformation, the interpolation results (b and c) are better when the temporal image evolution can be represented by fewer sine waves.
a)
b)
c)
d)
Figure 4: Comparison of an eastward moving double-sine structure with a truncated double-sine structure (negative values are set to zero) and a double square wave, all propagating in the same direction. a) and d) are the reconstructed fields based on 10 Eigenmodes, b) and c) are the interpolated images in regular intervals.
5.3 Application to radar data CEOF temporal interpolation has also been applied on 5 minutes radar imagery (temporal image interpolation to 1-minute interval). The interpolated images exhibited a somewhat blurred appearance, and “ghost echoes” around the main radar echoes could be seen. The latter were not visible in the reconstructed original images, yet appeared in the interpolated images. No satisfactory theoretical explanation of this behaviour can be given at this stage; this application requires more research. If working properly, an obvious idea of future implementation is to treat related data (potential candidates: the precipitation products provided by the Hydrology-SAF) by CEOF temporal interpolation as an alternative approach to previous attempts of filling gaps in infrequent precipitation data (such as Joyce et al. (2004), for example). 6. CONCLUSIONS The CEOF method is well-suited for the temporal interpolation of model fields, especially when sinusoidal pattern prevail in the data (e.g. wind fields). In these cases, this method clearly shows better results than linear interpolation. Temporal image interpolation of moving patterns shows better results with smooth structures. More Eigenmodes are needed when image structures show sharp edges – as for fronts or convective cells – to give acceptable results. 7. REFERENCES Chu, P.C. and C.-L. Fang (2003): Observed Rossby Waves in the South China Sea from Satellite Altimetry Data. Proceedings of SPIE, Conference on Remote Sensing of the Ocean and Sea Ice, Barcelona, Spain, September 8-12, 2003, 142-149. Huber-Pock, F. and Kress, C. (1989): An operational model of objective frontal analysis based on ECMWF products. Meteorol. Atmos. Phys., 40, 170-180. Joyce, R.J., J.E. Janowiak, P.A. Arkin, and P. Xie (2004): CMORPH: A method that produces global precipitation estimates from passive microwave and infrared data at high spatial and temporal resolution. J. Hydromet., 5, 487-503. Maue, R.N. (2004): Evolution of frontal structures associated with extratropical transitioning Hurricanes. Thesis, Florida State University. Yu, P., J. Zavala-Hidalgo, S.L. Morey, and J.J. O’Brien (2003): A new mapping method for propagating data. Proceedings of the Oceans 2003 MTS/IEEE Conference, San Diego, CA, September 22-26, 2003, Vol. 5, 2804-2807. Zavala-Hidalgo, J., M.A. Bourassa, S.L. Morey, and P. Yu (2003): A new temporal interpolation method for high-frequency vector fields. Proceedings of the Oceans 2003 MTS/IEEE Conference, San Diego, CA, September 22-26, 2003, Vol. 2, 10501053. Acknowledgements We thank A.A. Fernandes (National Institute of Oceanography, Goa, India) for freely providing the CEOF software which was the basis for our work. The temporal
interpolation of NWP wind fields has been investigated in the framework of the EUMETSAT Satellite Application Facility on Nowcasting and Very Short-Range Forecasting. Examinations of Eigenvector techniques for meteorological purposes are financially supported by the Austrian Federal Ministry for Transport, Innovation and Technology in the “Austrian Space Applications Programme” project REBECCA (project No. 815263).
