Integrating Landsat with MODIS Products for Vegetation Monitoring - Satellite-based Applications on Climate Change - page 243

However, due to the differences in the acquisition time, bandwidth, and geolocation
errors, bidirectional effect and small biases are expected. For a homogenous “pure”
pixel at the coarser MODIS resolution, surface reflectance measured by Landsat
data may be expressed as

Lðx; y; t k Þ¼Mðx; y; t k Þþε k

(16.1)

where ( x, y ) is a given pixel location for both Landsat ( L ) and MODIS ( M ) images,
t k is acquisition date for both MODIS and Landsat data, and ε k represents the
difference between the observed MODIS and Landsat surface reflectance (caused
by differing bandwidth and solar geometry).
Supposing land cover type and system errors at pixel ( x, y ) do not change
between prediction dates t 0 and t k , we will have ε 0 ¼ ε k , and thus

Lðx; y; t 0 Þ¼Lðx; y; t k ÞþðMðx; y; t 0 ÞMðx; y; t k ÞÞ (16.2)

However, this ideal situation cannot often be satisfied from MODIS and Landsat
observations. In most cases, the MODIS observation is not a homogeneous pixel
and may include mixed land cover types when considered at Landsat spatial
resolution. To consider mixed pixels in the prediction, we introduce additional
information from neighboring pixels and use spectrally similar pixels in the predic-
tion. The predicted surface reflectance for the central pixel at date t 0 is then
computed with a weighting function:

X

w

X

w

X

n

Lðw= 2 ; w= 2 ; t 0 Þ¼ W ijk ðLðx; y; t k ÞþðMðx; y; t 0 ÞMðx; y; t k ÞÞÞ (16.3)

x¼ 1 y¼ 1 k¼ 1

where w is the searching window size and ( w/ 2, w/ 2) is the central pixel of this
moving window. To ensure the correct information from neighboring pixels is used,
only spectrally similar (i.e., from the same spectral class) and cloud-free pixels
from Landsat surface reflectance within the moving window are used to compute
reflectance.
The weighting function W ijk determines how much each neighboring pixel
contributes to the estimated reflectance of the central pixel. It is determined by
three measures based on (1) spectral difference between MODIS and ETM+ data at
a given location, (2) temporal difference between input and the predicted MODIS
data, and (3) geographic distance between the central pixel and the candidate pixel.
These measures ensure that “pure” neighbor pixels get higher weights in the
prediction.
The STARFM approach was tested for simulated data and real satellite
observations (Gao et al. 2006 ). Figure 16.3 shows a simulation test for changing
reflectance and linear objects. Linear objects such as roads and small rivers are
normally visible in fine-resolution Landsat imagery but are not obvious in coarse-
resolution MODIS imagery. Figure 16.3a-c represent simulated Landsat-like