# Harmonizing Multi-Source Remote Sensing Images for Summer Corn Growth Monitoring
## 2 Materials and Methods
### The Improved Kalman Filter  
The Kalman filter method is a linear fitting and prediction algorithm. It was published in the \"Journal of Basic Engineering Transactions\" by [PERSON] in 1960 [52]. It can combine information from observational systems and system modeling to minimize residual errors and produce more accurate predictive information [53]. The method consists of three hypotheses: (i) the posterior probability obeys a Gaussian distribution; (ii) the system is linear; and (iii) both the system noise and the measurement noise are subject to Gaussian distribution. The method is described in Equations (1) and (2):  
\[x_{k}=Ax_{k-1}+w_{k-1} \tag{1}\]  
\[z_{k}=Hx_{k}+v_{k}, \tag{2}\]  
where \(x_{k}\) and \(x_{k-1}\) are the estimated values of the current state \(k\) and the previous state \(k-1\) of the model, respectively. For image harmonizing, \(x_{k}\) and \(x_{k-1}\) are the reflectance images on time \(k\) and \(k-1\) respectively (Figure 3). The connection of time \(k\) and \(k-1\) is done by state-transition matrix \(\mathbf{A}\), and the observed value \(z_{k}\) is linearly related to the current state estimation \(x_{k}\). In addition, \(\mathbf{H}\) is a measurement  
Figure 2: Dates of the acquired GF-1, MODIS, Landsat 8, Grace and meteorological data for the study area in 2014.  
sensitivity matrix, \(w_{k-1},\ v_{k}\) are process noise _N(0,Q)_ and measurement noise _N(0,R)_, both of which obey a Gaussian distribution. The Kalman filter is applied in two steps: the time update and the measurement update. In the time update, the state-transition matrix **A** can be represented in linear regression. The linear regression is used to describe the phenological trajectory between two moments, which can be represented by linear equations. In this paper, 211 sampling plots were selected within the images. Based on these plots, a linear relationship between the \(k\)-1 state and the \(k\) state of MODIS data was established, as shown in Equation (3). The slope and intercept of the linear regression line modeled \(a\) and \(b\), respectively, which were used to transfer the prior estimation of the time change trend. The process noise \(w_{k-1}\) is calculated by weighted standard error, and the weight is residual error. The uncertainty \(P_{k}\) of the time update is calculated by Equation (4). \[x_{k}^{-}=a*\hat{x}_{k-1}+b \tag{3}\]  
\[P_{k}^{-}=AP_{k-1}^{-}A^{T}+w_{k-1} \tag{4}\]  
In the measurement update, a posteriori estimate value \(\hat{x}_{k}\) of the \(k\) state is calculated from the weighted sum of the observed value \(z_{k}\) and prior estimate \(x_{k}^{-}\), as shown in Equation (5). The weights are defined as Kalman gain \(\overline{K}_{k}\) and are updated by uncertainty \(P_{k}\). \(R_{k}\) is the measurement noise covariance matrix, and \(T\) represents the matrix transposition operation, as in Equations (6) and (7). \[\hat{x}_{k}=x_{k}^{-}+\overline{K}_{k}\Big{[}z_{k}-H_{k}x_{k}^{-}\Big{]} \tag{5}\]  
\[\overline{K}_{k}=P_{k}^{-}H_{k}{}^{T}\Big{[}H_{k}P_{k}^{-}H_{k}{}^{T}+R_{k} \Big{]}^{-1} \tag{6}\]  
\[P_{k}=\Big{(}1-\overline{K}_{k}H_{k}\Big{)}P_{k}^{-} \tag{7}\]  
For the Kalman filter algorithm, it is vital to update the _a priori_ estimate for the current point-in-time to obtain a _posteriori_ estimate. Unfortunately, there are generally insufficient observation data resulting from cloud, smog and raining, etc. To solve this problem, there are usually two solutions: one is to use the posterior estimation of the previous moment, which is suitable for the case of small interval between two moments; the other is to use the low spatial resolution image of the current moment instead. Obviously, the two methods will produce more uncertainty for image harmonization. Therefore, we developed the improved Kalman filter by carrying out _a priori_ estimate for the studied point-in-time using ESTARFM model for the NDVI images. NDVI is the combination of red band and NIR band reflectance, aiming at enhancing the difference between vegetation and non-vegetation [54; 55].