GAO et al.: PREDICTING DAILY LANDSAT SURFACE REFLECTANCE 2209
The weight W
ijk
determines how much each neighboring
pixel contributes to the estimated reflectance of the central
pixel. It is very important and is determined by three measures
as follows.
1) Spectral difference between MODIS and ETM+ data at a
given location is
S
ijk
= |L(x
i
,y
j
,t
k
) − M(x
i
,y
j
,t
k
)|. (6)
This is an approximate measure to determine the ho-
mogeneity of an MODIS pixel. The direct measure of
homogeneity is limited by differences in projection and
geolocation errors, and the actual MODIS pixel footprint
is difficult to calculate. This approximation measures the
difference between the Landsat observation and averaged
neighboring pixels at coarse resolution. A smaller value
of S
ijk
implies that the fine spatial resolution pixel has
closer spectral features to the averaged surrounding pix-
els; thus, the change at fine resolution should be compara-
ble to that of the averaged surrounding pixels. Therefore,
the pixel’s reflectance should be assigned a higher weight
in (5).
An extended assumption is that if MODIS and Landsat
surface reflectances are equal at a given time, then these
values should be equal for the prediction date. A special
case is that of a homogeneous object at MODIS coarse
resolution. There may be some cases where the Land-
sat surface reflectance equals the averaged surrounding
values (MODIS surface reflectance) of a heterogeneous
area, but their changes through time diverge. In that case,
we need separate measures to minimize the bias. One
approach is to use differences of MODIS observations
during that period. The small changes of MODIS obser-
vations will ensure a better prediction on heterogeneous
areas.
2) Temporal difference between the input and the predicted
MODIS data is
T
ijk
= |M(x
i
,y
j
,t
k
) − M(x
i
,y
j
,t
0
)|. (7)
This metric measures changes occurring between the pre-
diction and the acquisition dates. A smaller T
ijk
means
less vegetation change between time t
k
and t
0
; thus, the
pixel should be assigned a higher weight.
An extended assumption is that if the MODIS surface
reflectance is constant over time, then the Landsat surface
reflectance should not change as well. This is a reasonable
assumption when we predict Landsat surface reflectance
on the date of a given input pair. Thus, we will see
the exact Landsat surface reflectance from predicted and
observed data over a cloud-free area when the prediction
date and the acquisition date are the same. This could be
used to replace clouds and gaps from Landsat images.
It will keep all cloud-clear Landsat observations from
the same day while only replacing bad observations with
predicted values.
However, if changes are too subtle to be detected
by the MODIS observation, this algorithm will not be
able to predict any change when synthesizing the fine
resolution imagery. Also, there may be situations where
the STARFM algorithm cannot detect changes when two
contradicting changes occur within a coarse-resolution
pixel simultaneously and compensate for each other.
3) Location distance between central pixel (x
w/2
,y
w/2
) and
candidate pixel (x
i
,y
j
) at date t
k
is
d
ijk
=
x
w/2
− x
i
2
+
y
w/2
− y
j
2
. (8)
This measures the spatial distance between the central
predicted pixel and the surrounding spectral similar can-
didate pixel. The spatial similarity is normally better for a
closer pixel; thus, the closer candidate should be assigned
a higher weight.
B. Implementation Considerations
The actual implementation of STARFM involves more de-
tailed considerations on how to weigh spatial information. The
weighting function needs to be adjusted depending on the
complexity and heterogeneity of the study area.
1) Spectrally Similar Neighbor Pixels: As noted above
(in Section II-A), the spectral similarity ensures that the correct
spectral information is used from fine-resolution neighboring
pixels. There are two ways to obtain spectrally similar pixels.
An unsupervised classification can be performed before data
blending, and pixels of same class are considered spectrally
similar. The unsupervised classification should be applied to
all fine-resolution (Landsat) images. These spectrally similar
neighbor pixels could be different from date to date, which is
useful in terms of capturing surface changes at fine resolution
and thus allowing us to predict changes between dates. Another
approach is to find spectrally similar pixels using thresholds
in surface reflectance directly. The search process can be in-
corporated within the STARFM algorithm. In this paper, we
used surface reflectance of red and NIR to determine the
spectrally similar pixels. Differences of spectrally similar pixels
are defined based on the standard deviation of fine-resolution
images and the number of classes used. Using a large number
of classes represents a stricter condition for selecting candidate
neighbor pixels from fine-resolution images. Although our
search process is similar to unsupervised classification, note
that the purpose of the search process is to find pixels within the
local moving window that are spectrally similar to the central
pixel. Each central pixel becomes the center of the class, and the
rules used to determine spectral similarity become local rules
and thus vary from pixel to pixel. In contrast to the traditional
classification, which applies the same classification rules over
the whole region, our search process (second approach) will
not be able to produce a unique classification map over the
study area.
2) Combined Weighting Function: We use three factors in
determining final weights for each spectrally similar pixel.
They are based on the assumptions that: 1) coarse-resolution
homogeneous pixels provide identical temporal changes as