## 3 Data and Method
### Data Processing and Determination of Model Input  
The preprocessing procedures for HJ-1A/1B CCD images mainly include radiometric calibration and atmospheric and geometric corrections [49]. In case of HJ-1A/1B CCD images, the band 4 (near-infrared band) and the band 3 (red band) were used to calculate NDVI [58].  
\[\mathrm{NDVI}=\frac{\left(\varrho_{tA}-\varrho_{t,3}\right)}{\left(\varrho_{t,A} +\varrho_{t,3}\right)} \tag{1}\]  
To get daily NDVI series, we used an asymmetric logistic curve (Figure 2) [59] to fit the NDVI series of the available days, which was then used to calculate the daily NDVI values and extract the crop phenological characteristics [49]. The fitting curve equation is  
\[\mathrm{NDVI}=a\ +\left(b/k\right)\cdot\left(1+n\right)^{-\left(k+1\right)/k} \cdot n\cdot\left(k+1\right)^{\left(k+1\right)/k} \tag{2}\]  
\[n=\exp[\left(t+d\cdot\ln(k)-c\right)/d] \tag{3}\]  
where \(t\) is day of year (DOY); \(a\), \(b\), \(c\), \(d\) and \(k\) are curve parameters to be estimated from available NDVI series by the least-squares method.  
After curve fitting, three characteristic points in the fitted NDVI curve (Figure 2) were used to determine the crop phenological characteristics and corresponding NDVI values (Table 2). The equations for calculating the phenological characteristics could be found in [PERSON] and [PERSON] [49].  
Eight models were developed for the crop yield estimation with different predictors, each being a combination of NDVI series with a specified time interval and/or phenological characteristics (Table 3). Available studies showed that crop yield is closely related with NDVI [8; 13; 60], and [PERSON] et al. [7] used MODIS NDVI 16-day composite data to achieve an accurate estimation of maize yield in the United States. Moreover, [PERSON] et al. [37] estimated winter wheat yield accurately  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **NDVI Indexes** & **Phenological Characteristics** \\ \hline \multirow{2}{*}{1} & NDVI\_inf\_1, NDVI value of the left inflection point & t\_inf\_1, time corresponding to \\  & of the NDVI curve with maximum growth rate & NDVI\_inf\_1 \\ \hline \multirow{2}{*}{2} & NDVI\_inf\_2, NDVI value of the right inflection point & t\_inf\_2, time corresponding to \\  & of the NDVI curve with maximum whitering rate & NDVI\_inf\_2 \\ \hline \multirow{2}{*}{3} & NDVI\_max, maximum value of NDVI curve & t\_max, time corresponding to \\  & & NDVI\_max \\ \hline \hline \end{tabular}
\end{table}
Table 2: Selected phenological characteristics and corresponding NDVI indexes.  
Figure 2: Sketch of asymmetric logistic curve to fit the NDVI series and characteristic points.  
using MODIS NDVI data in Shandong province, China. Therefore, to compare the impact of time intervals on yield estimation precision, NDVI series in the crop growth period with time intervals of 5 days and 10 days were selected as the inputs for models 1 and 2, respectively. Considering that the phenological characteristics of crops also have close correlations with crop growth and yield, three phenological characteristics in Table 2 were added to the inputs of model 2 to obtain inputs for model 3. Since parameters \(d\) and \(k\) in the fitted NDVI curve also affect the crop growth period, \(d\) and \(k\) were added to the inputs of model 3 and used as the inputs for model 4. To test whether crop yield could be accurately estimated using the NDVI series before harvest, the NDVI series before the 210 th day (50 days before the harvest) instead of the entire growth period were used as the inputs for model 5. The phenological characteristic in the early growth period, t_inf_1, was added to the inputs of model 5 and used as the inputs for model 6. To further test if these three phenological characteristics and corresponding NDVI indexes in Table 3 can be substitutes of the NDVI series, these six indexes were used as the inputs for model 7. Combination of the inputs of model 7 and parameters \(d\) and \(k\) constitutes the input for model 8.