Seetion 13.3: Empirieal Orthogonal Functions
241
The "patterns" (13.31) are markedly different from the eigenvectors of
the a ~ 1-covariance matrix calculated above. Thus, the result of the
EOF analyses of two random vectors with the same correlation structure
depends strongly on the allocation of the variance within the vector X.
This example demonstrates also the impact of using vectors X which
carry numbers subjective to different units. If air pressure from midlatitudes is put together with pressure from low latitudes then the EOFs
will favor the high-variance midlatitude areas. If a vector is made up of
temperatures in units of K and of precipitation in m/ sees, then patterns
of the EOFs will concentrate on the temperature entries .
• An EOF analysis deals with a veetor of observations X. This vector may
entertain physically very different entries, as outlined at the beginning
of this Section. The EOF analysis does not know what type of vector
it is analyzing. Instead all eomponents of X are considered as equally
relevant, independently if they represent a small or a large grid box in
case of a longitude x latitude grid, or a thin or a thick layer (in case of
ocean general circulation model output). If we study Scandinavian temperature as given by 10 stations in Denmark and one station in the other
Scandinavian states, then the first EOFs will invariably concentrate on
Denmark.
If we deal with a relatively uniform distribution of variance, and if we
know that the eharaeteristie spatial scale of the considered variable, such
as temperature, is comparable to the considered area, then the first EOF
will in most cases be a pattern with the same sign at all points - simply
because of the system 's tendency to create anomalies with the same sign
in the entire domain. The need to be orthogonal to the first EOF then
creates a second EOF with a dipole pattern (which is the largest-scale
pattern orthogonal to the uniform-sign first EOF). Our 2-dimensional
(a = 1, p > O)-case, discussed above, mimicks this situation. If, however,
the characteristic spatial scale is smaller than the analysis domain then
often the first EOF is not a monopole [see, for instance, the SST analysis
of Zorita et al. (1992)].
• Do EOFs represent modes or proeesses of the physical system from which
the data are sampled? In many cases the first EOF may be identified
with such a mode or process. For the second and higher indexed EOFs,
however, such an association is possible only under very special circumstances (see North, 1984). A severe limitation to this end is the imposed
spatial orthogonality of the patterns and the resulting temporal independence of the coefficients (13.22). Thus EOFs can represent only such
physical mo des which operate independently, and with orthogonal patterns. In most real-world cases, however, processes are interrelated.
8The unit mm/ sec is, admittedly, not widely used for precipitation. But precipitation
is arate, often given in mm/day - which is in standard units expressable as m/ sec.
241
The "patterns" (13.31) are markedly different from the eigenvectors of
the a ~ 1-covariance matrix calculated above. Thus, the result of the
EOF analyses of two random vectors with the same correlation structure
depends strongly on the allocation of the variance within the vector X.
This example demonstrates also the impact of using vectors X which
carry numbers subjective to different units. If air pressure from midlatitudes is put together with pressure from low latitudes then the EOFs
will favor the high-variance midlatitude areas. If a vector is made up of
temperatures in units of K and of precipitation in m/ sees, then patterns
of the EOFs will concentrate on the temperature entries .
• An EOF analysis deals with a veetor of observations X. This vector may
entertain physically very different entries, as outlined at the beginning
of this Section. The EOF analysis does not know what type of vector
it is analyzing. Instead all eomponents of X are considered as equally
relevant, independently if they represent a small or a large grid box in
case of a longitude x latitude grid, or a thin or a thick layer (in case of
ocean general circulation model output). If we study Scandinavian temperature as given by 10 stations in Denmark and one station in the other
Scandinavian states, then the first EOFs will invariably concentrate on
Denmark.
If we deal with a relatively uniform distribution of variance, and if we
know that the eharaeteristie spatial scale of the considered variable, such
as temperature, is comparable to the considered area, then the first EOF
will in most cases be a pattern with the same sign at all points - simply
because of the system 's tendency to create anomalies with the same sign
in the entire domain. The need to be orthogonal to the first EOF then
creates a second EOF with a dipole pattern (which is the largest-scale
pattern orthogonal to the uniform-sign first EOF). Our 2-dimensional
(a = 1, p > O)-case, discussed above, mimicks this situation. If, however,
the characteristic spatial scale is smaller than the analysis domain then
often the first EOF is not a monopole [see, for instance, the SST analysis
of Zorita et al. (1992)].
• Do EOFs represent modes or proeesses of the physical system from which
the data are sampled? In many cases the first EOF may be identified
with such a mode or process. For the second and higher indexed EOFs,
however, such an association is possible only under very special circumstances (see North, 1984). A severe limitation to this end is the imposed
spatial orthogonality of the patterns and the resulting temporal independence of the coefficients (13.22). Thus EOFs can represent only such
physical mo des which operate independently, and with orthogonal patterns. In most real-world cases, however, processes are interrelated.
8The unit mm/ sec is, admittedly, not widely used for precipitation. But precipitation
is arate, often given in mm/day - which is in standard units expressable as m/ sec.