where
M
k
N
(
T
1
) and
M
k
R
(
T
1
) are the reflectances of the
NIR
and
red bands of the
k
th
MODIS
pixel at
T
1
.
Similar
MODIS
pixels have similar reflectance with the cur-
rent central Landsat pixel, suggesting that
N
(
T
1
)
≈
M
k
N
(
T
1
) and
R
(
T
1
)
≈
M
k
R
(
T
1
). Therefore, Equation 7 is approximated as the
following equation:
δ
IB
N
R
NDVI
NDVI
NDVI
=
∂
∂
+
∂
∂
+
∂
∂
∑
f
N T
M
R T
M
N T
k
k
k
[
( )
( )
[ ( )]
(
1
1
2
1
2
1
2
∆
∆
∆
∆
∆ ∆
M
R T
M
N T R T
M M
k
k
k k
N
R
N
NDVI
NDVI
)
[ ( )]
(
)
( ) ( )
2
2
1
2
2
2
1
1
1
2
+
∂
∂
+
∂
∂
∂
R NDVI
]
− ∆
L
. (8)
After introducing weight
f
k
, Equation 8 transforms to:
δ
IB
N
R
NDVI
NDVI
NDVI
=
∂
∂
+
∂
∂
+
∂
∂
∑
∑
N T
f M
R T
f M
N T
k k
k k
( )
( )
[ ( )]
1
1
2
1
1
2
∆
∆
2
2
2
1
2
2
2
1
1
1
2
f M
R T
f M
N T R T
k k
k k
(
)
[ ( )]
(
)
( ) (
∆
∆
N
R
NDVI
NDVI
∑
∑
+
∂
∂
+
∂
∂
∂
)
f M M L
k k k
∆ ∆ ∆
N R NDVI
∑
−
. (9)
Comparisons of Errors in BI and IB
The blending error is typically described by the squared error
(
SE
) (Namba, 2015). For each pixel, the squared error is
SE
= (NDVI
P
– NDVI
O
)
2
,
(10)
where NDVI
P
and NDVI
O
are the predicted and observed
NDVI
val-
ues of a Landsat pixel. The error difference between
BI
and
IB
is:
SE
BI
– SE
IB
=
δ
2
BI
–
δ
2
BI
= (
δ
BI
+
δ
IB
)(
δ
BI
–
δ
IB
).
(11)
First, we consider the sign of the term (
δ
BI
–
δ
IB
). Based on
Equations 5 and 9, the sign is given by:
(
)
[ ( )]
[(
)
(
) ]
δ δ
BI
IB
N
N
NDVI
NDV
− =
∂
∂
−
+
∂
∑ ∑
1
2
1
2
2
1
2
2
2
2
N T
f M f M
k k
k k
∆
∆
I
NDVI
R
R
[ ( )]
[(
)
(
) ]
( ) ( )
[(
∂
−
+
∂
∂
∂
∑ ∑
R T
f M f M
N T R T
f
k k
k k
k
1
2
2
2
2
1
1
∆
∆
∆
M f M f M M
k
k k
k k k
N
R
N R
)(
)
]
∑ ∑ ∑
− ∆
∆ ∆
.
(12)
Considering that the
NDVI
data are used to monitor vegeta-
tion, we only consider the vegetation pixels. It is reasonable
to assume that the increments of the
NIR
and red bands for the
vegetation pixels are linearly related:
Δ
N
=
β
Δ
R
,
(13)
where
Δ
N
and
Δ
R
are increments of the
NIR
and red bands of a
Landsat pixel from time
T
1
to
T
2
;
β
is the ratio between the in-
crements of the
NIR
and red bands, which should be less than
−1 for vegetation pixels because the increments of
NIR
and red
bands of vegetation are inversed correlated and the variation
of
NIR
band is larger than that of red band in usual.
STARFM
as-
sumes no land cover changes between
T
1
and
T
2
. The spectra
of non-vegetation pixels, such as soil and impervious surfac-
es, are relatively stable. Therefore, the relationship between
the reflectance increments of mixed
MODIS
pixels on the
NIR
and red bands is expressed as:
Δ
M
N
=
β
Δ
M
R
,
(14)
where
Δ
M
N
and
Δ
M
R
are increments of the
NIR
and red bands
of
MODIS
pixels from
T
1
to
T
2
. Combined with Equations 14,
Equation 12 transforms to
2 1
(
)
(
)[ ( )
( )]
[ ( ) ( )]
δ δ
β
β
BI
IB
term 1
− =
+
−
+
1
1
1
1
3
N T R T
N T R T
[(
)
(
) ]
f M f M
k k
k
k
∆
∆
R
R
term 2
2
2
∑ ∑
−
.(15)
Because
β
must be less than −1, term 1 in Equation 15
should be always negative. For
STARFM
, the weights are posi-
tive and sum to one; therefore, term 2 should be negative
according to the Jensen inequality (Bencze, 2001; Mićić et al.,
2013) because the quadratic function is a low convex func-
tion. Therefore, (
δ
BI
–
δ
IB
) is always positive.
The comparison of
SE
of the
BI
and
IB
methods clearly re-
lies on the sign of (
δ
BI
+
δ
IB
). When the
NDVI
of vegetation at
T
2
(prediction date) is higher than that at
T
1
(input date), for ex-
ample in the growth season of vegetation, the
NDVI
increments
of the
MODIS
pixels should be lower than those of the vegeta-
tion pixels in the Landsat image because the
MODIS
pixels are
mixed with non-vegetation cover with stable
NDVI
. Thus, the
following inequality applies:
0 <
Δ
M
k
NDVI
<
Δ
L
NDVI
,
(16)
where
Δ
M
k
NDVI
is the
NDVI
increment of the
k
th
similar
MODIS
pixel and
Δ
L
NDVI
is the
NDVI
increment of the vegetation pixel
of the Landsat image. Considering that the weights are posi-
tive and sum to one, the following relation applies:
0 <
∑
f
k
Δ
M
k
NDVI
<
Δ
L
NDVI
.
(17)
Therefore,
IB
underestimates
NDVI
(
δ
IB
<0) when
NDVI
at
T
2
is higher than that at
T
1
. Similarly, the changes in the
NIR
and
red bands of the vegetation pixels in the Landsat image are also
larger than those of the mixed
MODIS
pixel, which could lead to
NDVI
underestimation using the
BI
method (
δ
BI
< 0) when
NDVI
at
T
2
is higher than that at
T
1
. Therefore, the term (
δ
BI
+
δ
IB
) is
negative, suggesting that the error of
BI
is lower than that of
IB
.
In contrast, if the
NDVI
at
T
2
is lower than that at
T
1
, this
suggests that:
Δ
L
NDVI
<
Δ
M
k
NDVI
< 0.
(18)
Similarly, both
BI
and
IB
methods overestimate the
NDVI
values, leading to positive (
δ
BI
+
δ
IB
) term. Therefore, the error
of
BI
is higher than that of
IB
in this case.
In summary, it is proved that either
BI
or
IB
is superior for
fusing
NDVI
data for the vegetation landscape depends on the
vegetation growth stages during data blending, i.e.,
BI
has
smaller error than
IB
when the
NDVI
values at the prediction
date are higher than the input
NDVI
values and vice versa.
Experiments and Results
To verify the conducted theoretical analysis, three experi-
ments under different vegetation growth stages and land-
scapes were conducted. In each experiment, two Landsat
images were selected and the corresponding
MODIS
images
were simulated by the aggregation (16 × 16 pixels) of Landsat
images to avoid the radiometric and geometric inconsisten-
cies between the two sensors. One Landsat image and two
simulated
MODIS
images were used as input to blending,
whereas the other Landsat image was preserved for valida-
tion. Such strategy has been used in previous studies (Gevaert
and Garcia-Haro, 2015) and can help identify the error owing
to different data fusion processes.
Experiment with Images Captured in the Vegetation Growth Period
The first study area was the Coleambally irrigation area
(hereafter referred to as
CIA
) (Emelyanovaa
et al.
, 2013) of
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
February 2018
67
