Developing a Historical Precipitation Record - Satellite-based Applications on Climate Change - page 94

X

N

Fðx ; tÞ E i ðxÞa i ðtÞ (7.1)

i¼ 1

As more modes are used in the approximation, it approaches the true field. The
modes are ordered such that the first mode explains the most variance, the second
explains the second most variance, etc. For an EOF-based reconstruction, the
spatial modes, E i ( x ), may be computed from a satellite-based analysis of modern
data. The historical time-series weights are estimated from the available historical
data. Since historical data are typically sparse, statistical testing is done to ensure
that there is enough sampling to reliably sample the set of modes used, with poorly
sampled modes omitted from the analysis. The historical time-series weights are
computed to minimize the error of the reconstruction fit at locations where histori-
cal data are available. For each time, the best-fit weights are obtained by solving

" #

X

a i X

X

N K K

;

E i ðxÞE j ðxÞ δ ðxÞAðxÞ ¼ DðxÞE j ðxÞ δ ðxÞAðxÞ j ¼ 1 2

; ...; N

;

i¼ 1 x¼ 1 x¼ 1

(7.2)

where K is the number of spatial regions, the delta function

δ ðxÞ¼

1 if data are

defined for region x and 0 otherwise, and A ( x ) is the relative area of region x .

The partly sampled data used for the reconstruction are represented by D ( x ).
The EOF method can reconstruct variations described by modes with large
enough scales to be sampled by the available historical data. Small-scale variations
are filtered out of the analysis. Variations not represented in the set of reconstruc-
tion modes will also be filtered out of the analysis. This filtering gives the analysis
spatially smooth features. The goal of a reconstruction is to represent large-scale
historical variations associated with climate. These reconstructions cannot repre-
sent fine-scale features, which require much more data to resolve.
The EOFmethodwas used to reconstruct precipitation (Xie et al. 2001 ; Efthymiadis
et al. 2005 ). They used satellite-based data to define modes and gauge data to compute
weights. Both studies found that oceanic precipitation variations associated with El
Ni˜o/Southern Oscillation (ENSO) variations are best reconstructed. That is because
ENSO variations have very large scales over both land and sea, and they are usually
described by the first fewmodes. Such large-scale variations are easily sampled by the
available gauge sampling. Outside of the tropics, both analyses had less skill, with
lowest extratropical skill in mid-oceanic regions far from gauge sampling.

δ ðxÞ¼

7.3 An Improved EOF-Based Reconstruction

In an attempt to improve on the ocean-area reconstructions noted above, data and
methods for precipitation reconstructions were reviewed and reevaluated (Smith
et al. 2008b ). As part of this effort, several satellite-based analyses were evaluated