Theoretical Analysis
Overview of STARFM
In
STARFM
, a pair of
MODIS
and Landsat images at time
T
1
and
a
MODIS
image at
T
2
are used to produce a Landsat-like image
at
T
2
.
MODIS
images are first resized to the extent of Landsat
images with the same pixel size. A moving window (e.g., 50
× 50 pixels) is then used to search for pixels similar to the
central Landsat pixel (Figure 1) based on Landsat images on
T
1
. The corresponding
MODIS
pixels of those selected similar
Landsat pixels are used to estimate the reflectance change
from
T
1
to
T
2
. The basic assumption of
STARFM
is that similar
pixels share the same reflectance change with the central
pixel. Thus, the weighted average of the reflectance change
of all the selected similar
MODIS
pixels is used to estimate
the reflectance change of the central Landsat pixel. Adding
this estimated change to the observation at
T
1
, we obtain the
reflectance of the Landsat pixel at
T
2
:
L
ˆ (
T
2
) =
L
(
T
1
) +
∑
f
k
Δ
M
k
,
(1)
where
f
k
is the weight of the
k
th
similar pixel and the sum of
f
k
is equal to 1, and
Δ
M
k
is the reflectance change of the
k
th
similar pixel from
T
1
to
T
2
estimated from the corresponding
MODIS
pixel.
L
(
T
1
) is the reflectance of the Landsat pixel at
time
T
1
,
L
ˆ (
T
2
) is the estimated reflectance of the Landsat pixel
at time
T
2
. For
STARFM
, the weights are determined by three
measures including the spectral difference between the
MODIS
and Landsat pixels at
T
1
, the temporal difference between the
MODIS
pixel at
T
1
and
T
2
, and the spatial distance between the
central and neighboring pixels.
Error Analysis of BI
The
BI
strategy involves two steps. First,
STARFM
is applied
to estimate the reflectance of the near-infrared (
NIR
) and red
bands at
T
2
. Then, the
NDVI
at
T
2
is calculated by the blended
NIR
and red bands at
T
2
. The estimated increments of the
NIR
band (
Δ
N
ˆ ) and red band (
Δ
R
ˆ ) from
T
1
to
T
2
are expressed as
∆
∆
∆
∆
ˆ
ˆ
N f M
R f M
k k
k k
=
=
∑
∑
N
R
,
(2)
where
Δ
M
k
N
and
Δ
M
k
R
are the increments of the
NIR
and red
bands of the
k
th
selected
MODIS
pixel from
T
1
to
T
2
. As there
are inevitable errors in
Δ
N
ˆ and
Δ
R
ˆ compared to the actual
increments, these errors
are propagated into the
NDVI
calcula-
tion. Using the estimated increments of the
NIR
and red bands
in Eq. (2), the
NDVI
of
BI
at
T
2
[NDVI
BI
(
T
2
)] is written as
NDVI
BI
( )
[ ( )
] [ ( )
]
[ ( )
] [ ( )
]
T
N T N R T R
N T N R T R
2
1
1
1
1
=
+ −
+
+ +
+
∆
∆
∆
∆
ˆ
ˆ
ˆ
ˆ
,
(3)
where
N
(
T
1
) and
R
(
T
1
) are the reflectances of the
NIR
and red
bands at
T
1
, respectively. The increment of
NDVI
from time
T
1
to
T
2
is estimated using the increments of the
NIR
and red
bands based on second-order Taylor expansion (Buckdahn
et
al
., 2015). Therefore, the increment of
NDVI
in the
BI
process
(
Δ
NDVI
BI
) is written as
∆
∆
∆
∆
NDVI
NDVI
NDVI
NDVI
BI
= ∂
∂
+ ∂
∂
+ ∂
∂
N T
N
R T
R
N T
N
( )
( )
[ ( )]
1
1
2
1
2
1
2
2
2
1
2
2
2
1
1
1
2
+ ∂
∂
+ ∂
∂
∂
NDVI
NDVI
[ ( )]
( ) ( )
R T
R
N T R T
N R
∆
∆ ∆
ˆ
ˆ
ˆ
ˆ
ˆ ˆ
. (4)
The error in
BI
is the difference between the estimated and
the actual increment of
NDVI
from
T
1
to
T
2
, which can be writ-
ten as:
δ
BI
N
R
NDVI
NDVI
NDVI
= ∂
∂
+ ∂
∂
+ ∂
∂
∑
∑
N T
f M
R T
f M
N
k k
k k
( )
(
)
( )
(
)
[ (
1
1
2
1
2
∆
∆
T
f M
R T
f M
N T
k k
k k
1
2
2
2
1
2
2
2
1
1
2
)]
(
)
[ ( )]
(
)
( )
∆
∆
N
R
NDVI
+
NDVI
∑
∑
+ ∂
∂
∂
∂
∂
R T
f M f M L
k k
k k
( )
(
)(
)
1
∆
∆ ∆
N
R NDVI
∑ ∑
−
, (5)
where
Δ
L
NDVI
is the actual increment of
NDVI
of the current
Landsat pixel from time
T
1
to
T
2
.
Error Analysis of
IB
In the
IB
process,
NDVI
data are first calculated based on the
existing reflectance images of Landsat as well as
MODIS
. Fu-
sion methods are directly used to blend
NDVI
at the estimating
time. In the
STARFM
process, assuming that little error is pro-
duced in the
NDVI
calculations, the error in blending is caused
by the difference between the sum of the weighted increment
of the selected
MODIS
pixels and the actual increment of the
current Landsat pixels during the same period. The error in
the
IB
process (
δ
IB
) is expressed as
δ
IB
NDVI
NDVI
=
−
∑
f M L
k k
∆
∆
,
(6)
where
Δ
M
k
NDVI
is the increment of
NDVI
on the
k
th
similar
MODIS
pixel from
T
1
to
T
2
. Using second-order Taylor expansion to
estimate the
NDVI
increments of
MODIS
pixels in Equation 6,
the error in
IB
is expressed as:
δ
IB
N
N
R
R
N
NDVI
NDVI
NDVI
=
∂
∂
+
∂
∂
+
∂
∂
∑
f
M T
M
M T
M
M
k
k
k
k
k
k
[
( )
( )
[
(
1
1
2
1
2
∆
∆
T
M
M T
M
M T M
k
k
k
k
k
1
2
2
2
1
2
2
2
1
1
2
)]
(
)
[
( )]
(
)
( )
∆
∆
N
R
R
N
NDVI
NDVI
+
∂
∂
+
∂
∂
∂
R
N R NDVI
( )
]
T
M M L
k k
1
∆ ∆ ∆−
, (7)
Figure 1. Similar pixels within the window of 50 × 50 pixels.
66
February 2018
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
