JIA-2019-11

2634 ZHANG Xi-wang et al. Journal of Integrative Agriculture 2019, 18(11): 2628–2643 and Y are each one of the points A, B, C, D, E, or F; a is the coefficient; and b is the constant residual term. Due to the impact of weather, planting habits and terrain, the times at which the key features occur are not necessarily identical. The NDVI values of points A, B, C, D, E and F are calculated as follows: = = = = = = + − + − + − + − + − + − ) NDVI , NDVI , Average(NDVI NDVI ) NDVI , NDVI , Average(NDVI NDVI ) NDVI , NDVI , Average(NDVI NDVI ) NDVI , NDVI , Average(NDVI NDVI ) NDVI , NDVI , Average(NDVI NDVI ) NDVI , NDVI , Average(NDVI NDVI 1 1 1 1 1 1 1 1 1 1 1 1 F F F F E E E E D D D D C C C C B B B B A A A A (3) where NDVI A , NDVI A-1 , and NDVI A+1 are, respectively, the corresponding NDVI values of point A and the points before and after A in the time series curve; and the other variables with subscripted letters B–F have corresponding meanings. The actual dates corresponding to points A, B, C, D, E and F are, respectively, November 1, 2009, December 27, 2009, January 25, 2010, March 22, 2010, April 23, 2010 and June 10, 2010. The equivalent stages are seedling, wintering, greening, booting, heading and harvesting in the winter wheat growing season. 3.3. Membership calculation based on Bayesian equations The calculation method for membership based on Bayesian equation adopted in this study is a typical, commonly used method (Schachtner et al. 2014). In the image, suppose n is the number of bands, x i is the value of a pixel in the i th band, and the pixel X T =[ x 1 , x 2 , ..., x n ] will be assigned into m classes, ( k i , i=1, 2, …, m ). According to the Bayesian equation, when the condition X appears, the membership for the class k i is defined as: ( ) ) () ( ) ( ) ( i m i =1 i i i i kPk|XP /kPk|XPX|kP ∑ × = (4) where P ( k i |X ) is the membership; P ( k i ) is the prior probability of class k i ; m is the number of classes; and P ( X|k i ) is the conditional probability that pixel X appears in class k i (probability density function). Assuming that the training samples are Gaussian normally distributed in the spectrum feature space, when the pixel X appears in the class k i , the conditional probability density function is expressed as follows: { } 1/2 /2 1 ) ( ) ( ) 0.5( ) ( ∑ ∑ − − − = − i n i i T i i 2 π μX μX exp k|XP (5) where n is the dimensionality of the feature space (i.e., the number of bands in the image); μ i is the mean vector of the training samples of the class k i ; and ∑ i –1 as well as |∑ i | are, respectively, the inverse matrix and determinant of the covariance matrix ∑ i of the class k i training samples. The covariance matrix ∑ i and the mean vector μ i of the class k i training samples can be expressed as: = = ∑ ) , , ... , ( 2 1 1 1 11 ni i i T i nni i n ni i i µ µ µ µ δ δ δ δ ... ... ... ... ... (6) where δ hli is the covariance of the class k i training samples between dimensions h and l in the spectrum feature space; and μ ij is the mean of the class k i training samples in dimension j . In the above equations, the a priori probability of each class can be determined based on prior experience and is generally considered as the acreage proportion of this 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 289 305 321 337 353 1 17 33 49 65 81 97 113 129 145 161 177 DOY NDVI Winter wheat Forest Shrub Grass Water Other A B C D E F Fig. 4 Normalized difference vegetation index (NDVI) time series curves of each land cover type during the 2009–2010 growing season. DOY, the day of a year. A–F, the key feature points in the time series curve of winter wheat;