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Empirical Orthogonal Function (EOF)


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Analysis in Terms of EOF's

EOF analysis has been applied to the atmospheric data for long time. Here, we make reference to "Physics of Climate" (Peixoto and Oort, 1992) for EOF analysis introduction:

"The empirical orthogonal function(EOF) analysis, sometimes referred to as eigenvector or principal components analysis, provides a convenient method for studing the spatial and temporal variability of long time series of data over large areas.This method splits the temporal variance of the data into orthogonal spatiaL patterns called empirical eigenvectors. "

"This approach enables one to identify a set of orthogonal spatial modes, such that, when ordered, each successive eigenvector explains the maximum amount possible of the remaining variance in the data.Each eigenvector pattern is associated with a series of time coefficients that described the time evolution ofthe particular spatial mode. The eigen vector patterns that account for a learge fraction of the variance are, in general, considered to be physically meaningful and connected with important centers of action." "The principal advantages of the eigenfunctions are that they provide the most efficient way of compressing geophysical data in both space and time and they may be regarded as uncorrelated (independent) modes of variability of the fields".

References: Peixoto, J.P., and A.H. Oort, 1992: Physics of Climate. American Institute of Physics New York.

EOF Analysis of TRMM Precipitation Data

TRMM TMI data--We use the monthly mean precipitation rate derived from TRMM3B31 algorithm.Data are of 5x5 degree resolution.(more description about the data needed).

TRMM 3A25 data-- The TRMM Precipitation Radar (PR), the first of its kind in space, is an electronically scanning radar, operating at 13.8 GHz that measures the 3-D rainfall distribution over both land and ocean, and defines the layer depth of the precipitation.The primary objective of algorithm 3A25 is to compute the monthly rainfall vertical structure and various statistics of rainfall and related variables measured by PR.In the following discussion we use monthly mean surface rain rate of TRMM 3A25 with 5x5 degree resolution.For the details of the TRMM 3A25 algorithm, please link here.

Reference:Meneghini, R., J.A., Jones, T. Iguchi, K. Okamoto, and J. Kwiatkowski,2001: Statistical Methods of Estimating Average Rainfall over Large Space-Time scales Using Data from the TRMM Precipitation Radar, J. Applied Meteorology .

The first four EOFs (figure below) and their respective time series (figure following) were derived from TRMM 3B31(TMI) precipitation data. The data used are the subset overthe region of 80oE-280oE and30oS-30oN. The percentage of explained totalvariance is labeled on the top of each panel. As described in the method description, you need to know the physics of the field to interpret the patterns.The number one EOF, which explains 23.5% of the total variances, depicts a dipole pattern extending zonally and symmetrically with respect to the equator. It mainly reflects the annual cycle of surface solar radiationenergy.The values shows the correlation field between precipitation value (that has been normalized in the process) andthe first EOF.The time series of the first EOF (the top panel of thenext Figure) shows a regular annual cycle.Combining the patterns with the time series, the precipitation annual variation is clear:over the Northern Hemisphere, precipitation amount is high during the summer and is low during the winter; the precipitation over the Southern Hemisphere is of the opposite phase, high during the Northern Hemisphere winter, and low during the Northern Hemisphere summer.

The second EOF shows a broad tongue extending from the equatorial eastern Pacific to the central Pacific.It is asymmetric with respect to the equator, and is accompanied by two centers with the opposite site directly over the two sides.The centers over the Australia and the maritime islands are of opposite sign too.The time seriesshows this pattern is of annual cycles too, but there is about 3 month phase lag between this annual cycle and the first one. This mode denotes the annual evolution of warm SST tongue which develops on about late spring and diminishes on later summer.

The physical explanation of the third EOF is not obvious, though the time series shows a clear annual cycle.The fourth EOF's time series shows a semi-annual cycle, in general, indicating the monsoon precipitation activity. Since this subset data does not includethe Southern America and the whole India the monsoon region did not come outclearly in the EOF pattern.

The ENSO signature is not clearly isolated out.However the abnormal amplitudes of the four time series during the early 1998 probably reflect the ENSO events.In general, TRMM data are too short for climatological signal analysis.

[First Four EOF patterns ] [Time series of First Four EOFs]

It is found that there are noticeable differences in the overall averaged precipitation estimation between TMI and PR.By using EOF to the difference field of TMI and PR, we decompose the difference into main spatial patterns which vary with time.Before the EOF computation, the seasonal mean comparison may provide the baseline for the later discussion on the EOF results. [seasonal mean of difference] [seasonal mean of difference] [seasonal mean of difference] [seasonal mean of difference]

Main features:

  • 1. Both TMI and PR precipitation estimation has similar spatial patterns. PR precipitation (TRMM 3A25) estimate is systematically lower than that of TMI over the high precipitation region, but higher than TMI data over the low precipitation region.
  • 2. For most of the areas, the difference is not dramatic.The differences are big over the heavy precipitation region, small over the light precipitation region.
[First Four EOFs] [Time series of First Four EOFs]

Main features:

  • 1. Both TMI and PR precipitation estimation has similar spatial patterns in terms of the first four prominent modes) and similar time scales.
  • 2. PR precipitation estimation is consistently lower than TMI over the high precipitation regions, and higher over the dry region.
  • 3. The time series of the second EOF of the difference field shows clearly an annual cycle as that of the TRMM3B31 reflecting the annual cycle of warm SST tongue.But the spatial pattern shows the difference occuring mainly at the north side of the tongue (see the second EOF of TRMM 3B31) and the southeastern side of the tongue.
  • 4. The difference field show large scale coherence in the first three EOF modes.The first three EOFs can explain 51% of the total variance,probably indicating the coherent spectral feature of the difference field.
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