is the Paci¬c/North American (PNA) pattern

EOFs used in the study were computed separately

(Figure 13.7). The PNA is characterized by two

for each month of the year from correlation ma-

centres of the same sign over the Aleutian Islands

trices derived from a 35-year data set of monthly

and the southeastern United States that ¬‚ank a

mean 700 hPa heights analysed on a 358-point

centre of opposite sign located over western North

grid. The data set itself was carefully screened

America. The PNA is evident in winter (December

to remove known analysis biases. Rotation was

to April) and again in September and October.

performed on the ¬rst 10 EOFs in each month.

It is strongest in February when Barnston and

They represent about 80% of the total variance in

Livezey estimate that it represents 13.2% of the

winter and 70% in summer.

total variance.

The result of the exercise is an extensive

collection of NH circulation patterns. Barnston Even though the rotated EOFs appear to be

and Livezey identi¬ed 13 patterns: nine cold less prone to ˜mixing™ than ordinary EOFs, a

season patterns, two warm season patterns, and great deal of sampling variability still clouds the

two transition season patterns. Only one pattern, patterns that are produced, and a considerable

13: Empirical Orthogonal Functions

310

middle and right hand columns) results in little

change.

• Figure 13.8 (middle column) displays the

result of the rotation using K = 5 normalized

EOFs as input vectors. The rotated EOFs

represent 38%, 24% and 10% of the total

variance. Similar results are obtained when

K = 5 non-normalized EOFs are used (not

shown).

• The result of the rotation using the ¬rst

K = 10 non-normalized EOFs is shown in

the right hand column of Figure 13.8. The

patterns represent 26%, 15%, and 13% of

the total variance, respectively. These patterns

deviate somewhat from those in the left hand

and middle columns of the diagram. They

are noisier than the other sets of patterns,

Figure 13.7: As Figure 13.6, except the February including the rotated patterns derived from

K = 10 normalized EOFs (not shown).

Paci¬c/North American pattern is displayed.

Courtesy R. Livezey.

Intuitively this is what we expect since

the non-normalized patterns enter the

minimization functional with equal weights.

amount of skill and subjective judgement are

Thus the poorly estimated EOFs are as

needed to classify and name the patterns. This is

in¬‚uential as the well-estimated EOFs in

amply illustrated by Barnston and Livezey [27],

the determination of matrix R. In contrast,

who discuss the types of latitude they permitted

normalization gives the well-estimated EOFs

themselves in developing their classi¬cation. Their

relatively more in¬‚uence on the form of R.

illustration of six renditions of the NAO obtained

for different times of the year (we show two of

The ¬rst rotated pattern represents less variance

these in Figure 13.6; see Barnston and Livezey [27,

Figure 2]) demonstrates the kind of variability than the ¬rst EOF, simply because the ¬rst

that the analyst must be able to penetrate when EOF was constructed to maximize the variance.

Higher-order rotated EOFs typically represent

classifying estimated patterns.

more variance than the respective EOFs (see, e.g.,

Table 1 of Barnston and Livezey [27]).

13.5.6 Example: Atlantic Sea-Level Air In this example, little is gained by processing the

Pressure. In this subsection we consider the original EOF patterns with the varimax machinery.

EOFs and varimax-rotated EOFs of North Atlantic The rotated EOFs become noisy when too many

monthly mean SLP in DJF.24 non-normalized EOFs were used as input.

The ¬rst three EOFs of SLP (Figure 13.8, left

hand column) represent 41%, 26% and 9% of

the total variance, respectively. The ¬rst EOF has 13.5.7 Example: North Atlantic Sea-surface

almost uniform sign and exhibits one large feature. Temperature. The ¬rst three EOFs of the

The second and third EOFs have dipole structures monthly mean SST in DJF represent 26%, 17%

that re¬‚ect the constraint that the higher-order and 10% of the total variance, respectively

EOFs must be orthogonal to the ¬rst EOF. (Figure 13.9, left hand column). These EOFs do

The EOFs of North Atlantic SLP have simple not have simple structure. The ¬rst contains three

structure, even without rotation. It is therefore well-separated centres of location located in the

not surprising that the application of the varimax West Atlantic off the North American coast, south

rotation technique to these EOFs (Figure 13.8, of Greenland, and in the upwelling region off the

west coast of Africa.

24 The analysis presented here and in [13.5.7] were provided

Varimax rotation leads to a substantially

by V. Kharin (personal communication). Note that all

eigenvalues, EOFs, and rotated EOFs presented here are different distribution of variance between patterns

(Figure 13.9, middle and right hand columns).

estimates.

13.5: Rotation of EOFs 311

Figure 13.8: First three rotated and unrotated EOFs of North Atlantic SLP in winter. From top to bottom

j = 1, j = 2, j = 3. Courtesy V. Kharin.

+

Left column: Normalized EOFs e j .

Middle column: Rotated EOFs derived from K = 5 normalized EOFs.

Right column: Rotated EOFs derived from K = 10 non-normalized EOFs.

• When the input is K = 5 normalized obtained with the normalized EOFs (which

EOFs (Figure 13.9, middle column) the three have unequal lengths). When more input

centres of action in the ¬rst EOF are separated vectors are used, the rotated patterns become

and distributed to the ¬rst three rotated noisier and represent less variance (not

patterns (which represent 21%, 16% and 15% shown).

of the variance, respectively).

• When the input is K = 5 non-normalized We will revisit the analysis of North Atlantic

EOFs (i.e., all EOFs have unit length; SST and SLP in [14.3.1]. There we will see that

Figure 13.9, right hand column) the three the ¬rst two conventional EOFs of the North

rotated EOFs represent about the same Atlantic SST re¬‚ect two forcing mechanisms,

percentage of variance, namely 15%, 15% two characteristic variations in the large-scale

and 13%, respectively.25 Note that the atmospheric state that are encoded in the ¬rst

sequence of patterns is changed from that two SLP EOFs shown in Figure 13.8. Thus, in

this case, the rotation makes interpretation more

25 Note that the concept of degeneracy is irrelevant for rotated

dif¬cult by masking the underlying physics (see

EOFs, since degeneracy is immaterial for the minimization of

[14.3.2]).

the functional V .

13: Empirical Orthogonal Functions

312

Figure 13.9: First three unrotated (left hand column) and rotated (middle and right hand columns)

EOFs of North Atlantic monthly mean SST in DJF. From top to bottom: j = 1, j = 2, j = 3. Courtesy

V. Kharin.

+

Left column: Normalized EOFs e j .

Middle column: Rotated EOFs derived from K = 5 normalized EOFs.

Right column: Rotated EOFs derived from K = 5 non-normalized EOFs.

13.5.8 Rotation: a Postscript. EOF rotation 13.6 Singular Systems Analysis and

is often useful, but it is not meant to be a Multichannel SSA