Fig. 10

Comparison between the predicted values with different fitting periods. a Spatial distribution of the residuals of the short-period logarithmic term in Eq. (1) for the spatial functions with the fitting period of 2.0 years and the time constants of the 3.9-year model [S2.0T3.9] (green line in Fig. 9). As a reference value, we use the 3.9-year model (the spatial functions with the fitting period of 3.9 years and the time constants of the 3.9-year model [S3.9T3.9]). b Same as a but for the spatial functions with the fitting period of 3.9 years and the time constants of the 2.0-year model [S3.9T2.0] (red line in Fig. 9). c Histogram of the ratio of the spatial function for the different time constants of the 2.0-year model [S3.9T2.0] to that of the 3.9-year model [S3.9T3.9]. Green color depicts the short-period logarithmic term (a), red color indicates the long-period logarithmic term (d), and blue color represents the exponential term (f) in Eq. (1) (see Additional file 1: Fig. S6 for more information)