2.2. Methodology
Empirical orthogonal function (EOF) analysis is a widely used statistical method for identifying dominant modes of variability in spatiotemporal datasets by decomposing the covariance structure of the data into orthogonal spatial patterns and their corresponding temporal coefficients ([7]; [10]; [9]; [3]). In this study, EOF analysis is applied to near-surface wind fields over Xinjiang to extract the leading modes of variability at multiple temporal scales.
Prior to the EOF decomposition, wind fields at each temporal scale (hourly, daily, and annual) are converted into anomaly fields by removing the corresponding long-term climatological mean. More specifically, for the hourly EOF, the sample dimension corresponds to the 24 hours of the day averaged over 1995-2024. For the daily and annual EOF, the sample dimensions correspond to the calendar day of the year and the 30 individual years from 1995 to 2024, respectively. Besides, to account for the decrease in grid-cell area with increasing latitude, a cosine-latitude weighting is applied to the anomaly fields. EOF analysis is then performed using the covariance matrix, allowing the dominant modes to retain the physical variance of the wind field.
In particular, for the wind vector analysis, a multivariate EOF (MV-EOF) framework is adopted to jointly analyze the zonal and meridional wind components ([16]). Specifically, for the study domain with M grid points, let x i ( i = 1 , 2 , ⋅ ⋅ ⋅ , M ) denote the i -th spatial grid point. The variable t ( t = 1 , 2 , ⋅ ⋅ ⋅ , T ) represents discrete time steps, where T is the total number of temporal samples at a given timescale. The anomaly zonal and meridional wind components at grid point x i and time t are denoted by u ′ ( x i , t ) and v ′ ( x i , t ) , respectively. For convenience in matrix operations, the spatial fields are reshaped into a state vector at each time step, denoted by X ( t ) , which is constructed by concatenating the wind components at all grid points:
X
(
t
)
=
[
u
′
1
(
t
)
,
u
′
2
(
t
)
,
⋅
⋅
⋅
,
u
′
M
(
t
)
,
v
′
1
(
t
)
,
v
′
2
(
t
)
,
⋅
⋅
⋅
,
v
′
M
(
t
)
]
T
where u ′ i ( t ) = u ′ ( x i , t ) and v i ′ ( t ) = v ′ ( x i , t ) . In this vector representation, the spatial index x i is implicitly encoded in the ordering of the vector elements. The EOF decomposition yields a set of orthogonal spatial patterns (EOF modes), denoted by E k , and their corresponding principal components (PCs), denoted by P C k ( t ) , where k = 1 , 2 , ⋅ ⋅ ⋅ , K indicates the mode number. At each grid point, the EOF mode provides a pair of values corresponding to the zonal and meridional components, which can be visualized together as a vector pattern, forming the spatial pattern of the mode. This approach preserves the physical consistency between wind direction and magnitude and enables the identification of coherent circulation structures ([17]).
For the wind speed analysis, the total wind speed is first calculated at each grid point x i and time step t as
W
S
(
x
i
,
t
)
=
u
(
x
i
,
t
)
2
+
v
(
x
i
,
t
)
2
where u ( x i , t ) and v ( x i , t ) denote the zonal and meridional wind components, respectively. Wind speed anomalies are then obtained by removing the climatological mean at each temporal scale, and EOF analysis is applied directly to the scalar wind speed field using the same covariance-based framework and latitude weighting as described above. Unlike the vector EOF, the resulting modes represent spatial variations in wind intensity only and do not explicitly contain directional information.
The temporal evolution of each EOF mode is represented by its principal component (PC), which is obtained by projecting the anomaly field onto the corresponding EOF pattern. The relative importance of each mode is quantified by the percentage of total variance explained. In this study, the leading three EOF modes are analyzed at each temporal scale, as they capture the majority of the wind variability and provide a robust representation of the dominant spatiotemporal structures.