Empirical orthogonal function

Peterson, J. T. The calculation of sulfur dioxide concentrations over a metropolitan area by using empirical orthogonal functions. American Institute of Aeronautics and Astronautics, Paper 70-113, January 1970. 7 pp. 

For a more detailed study of the intra-annual variability of the horizontal structure of the Black Sea main pycnocline, we decomposed climatic monthly salinity fields at a depth of 100 m over empirical orthogonal functions (EOFs). The results showed that 80% of the total dispersion of the intra-annual variability in the salinity fields are described by five EOFs three of them are presented in Fig. 9. 

Fig. 9 Empirical orthogonal functions (EOFs) of annual cycle of climatic salinity field variability in the Black Sea at a depth of 100 m a the first EOF, b the second EOF, c the fourth EOF, and d annual variations of its coefficients. 1 the first EOF, 2 the second EOF, 3 the fou rlh EOF...

Finnigan and Shaw [188] conducted an Empirical Orthogonal Function (EOF) analysis of an extensive wind tunnel data set obtained in a model wheat canopy. The same authors have recently performed an equivalent analysis based upon the output from the large-eddy simulation described above. 

Using empirical orthogonal functions (Dippner and Pohl, 2004) for a monitoring data set for the period 1993-2000 a positive trend in the order of one-third standard deviation appeared for dissolved Cd and Cu concentrations in surface waters of the B altic Proper. It was discussed that these trends were a result of a new stagnation period when the exchange with North Sea water was limited and the trace metal input via atmospheric and riverine sources resulted in the enrichment of trace metals in surface waters. This assumption was supported by the pollution loads entering the Baltic via rivers, industries, urban areas, and from the atmosphere (Table 13.2). 

Dissolved Cd concentrations increased slightly with a positive trend of 6.4%/year (0.0074nmol/(kg year)) between 1993 and 1999, which could be a result of riverine and atmospheric inputs (Fig. 13.5). This result was supported by the estimations of Dippner and Pohl (2004) using EOFs (empirical orthogonal functions). A positive trend on the order of one third of the standard deviation per year for Cdj.ss was calculated by using surface data for the whole Baltic Proper. 

Dippner, J. W., Pohl, C., 2004. Trends in heavy metal concentrations in the western and central Baltic Sea waters, detected by using empirical orthogonal functions analysis (EOF s). Journal of Marine Systems, 46, 69-83. 

Principal Components Analysis (PCA) is a multivariable statistical technique that can extract the strong correlations of a data set through a set of empirical orthogonal functions. Its historic origins may be traced back to the works of Beltrami in Italy (1873) and Jordan in Prance (1874) who independently formulated the singular value decomposition (SVD) of a square matrix. However, the first practical application of PCA may be attributed to Pearson s work in biology [226] following which it became a standard multivariate statistical technique [3, 121, 126, 128]. 

FIGURE 4 Indiviclvial impulse-response functions for the carbon cycle (response R. of the atmospheric COt concentration to a 5-function COi input at time t = 0, left) and the variables global mean near-surface temperature (center) and mean sea-level rise (right) for the physical ocean-atmosphere sy stem (response to a step-function increase in the CO2 concentration to a constant level at time t = 0). The units for temperature response Rj and sea-level response R< refer to the amplitudes of the first empirical orthogonal functions (EOFs) of the response patterns of the respective variables and are essentially arbitrary (see text). Adopted from Hooss ft /., 2001. 

Receptor models can also be used together with spatial distribution of measurements to estimate the spatial distribution of emission fluxes. The empirical orthogonal function (EOF) method is one of the most popular models for this. Henry et al. (1991) improved the EOF method by using wind direction in addition to spatially distributed concentration measurements as input. We describe this approach below. 

Gebhart, K. A., Lattimer, D. A., and Sisler, J. F. (1990) Empirical orthogonal function analysis of the particulate sulfate concentrations measured during WHITEX, in Visibility and Fine Particles, C. V. Mathai, ed., AWMATR-17, Air Waste Management Association, Pittsburgh, PA, pp. 860-871. 

Henry, R. C., Wang, Y. J., and Gebhart, K. A. (1991) The relationship between empirical orthogonal functions and sources of air pollution, Atmos. Environ. 24A, 503-509. 

Standing of source-receptor relationships for nonreactive species in an airshed. The.se methods include the chemical mass balance (CMB) used for. source apportionment, the principal component analysis (PCA) used for source identification, and the empirical orthogonal function (EOF) method used for identification of the location and strengths of emission sources. A detailed review of all the variations of these basic methods is outside the scope of this book. For more information the reader is referred to treatments by Watson (1984), Henry et al. (1984), Cooper and Watson (1980), Watson et al. (1981), Macias and Hopke <1981), Dattner and Hopke (1982), Pace (1986), Watson et al. (1989), Gordon (1980, 1988), Stevens and Pace (1984), Hopke (1985, 1991), and Javitz et al. (1988). 

RECEPTOR MODELING METHODS 1263 24.1.3 Empirical Orthogonal Function Receptor Models... 

E. Lorenz Empirical orthogonal functions and statistical weather prediction. Tech. Rep. 1, Statistical Forecasting Project, Department of Meteorology, Massachusetts Institute of Technology, Cambridge, MA, 49 pages (1956)... 

Repeating the experiment N times, the sample space of N equally likely outcomes is obtained. This sample can be analyzed by applying the empiric orthogonal functions (EOFs) technique as presented next. 

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