2.4.2. Normalized Logistic Model (N-Logistic Model)

The logistic model was originally developed for individual applications that may be inappropriate for spatial prediction of crop growth. The normalization method transforms the raw data into an interval of 0 to 1, which can eliminate the dimensional differences between plots. Therefore, the RDBA and relative effective accumulated temperature (*T*) were used to set up the growth model, lower data dispersion (from different plots), and form a regional model. Here, the logistic model with normalization is called N-logistic model, which has the same form as the logistic model but with different parameters:

$$\text{Y}\_{\text{D}} = A / \left(1 + B \exp(-KT)\right) \\ \text{Y}\_{\text{D}} = y\_{\text{D}} / y\_{\text{Dm}} \text{ } T = \text{t} / t\_{\text{m}} \tag{4}$$

where *Y*<sup>D</sup> is the RDBA, which is the ratio of yD (DBA in the maize growing season) to *y*Dm (DBA at harvest); *A* is the upper most asymptote implying the upper limit of RDBA; and *B* and *K* are model parameters; *t* is the same as in Equation (3); *t*m is equal to the value of *t* at harvest; *T* is relative effective accumulative temperature (*T*20, *T*40, *T*canopy, and *T*air mean the values in soil at 20 cm and 40 cm under surface, crop canopy, and air, respectively), which is the ratio of *t* to *t*m. Theoretically, the value of *Y*<sup>D</sup> equals *A* when *T* (0 ≤ *T* ≤ 1) reaches 1. Therefore, the value of *A* represents the theoretical upper limit of the RDBA.

#### 2.4.3. Revised Logistic Model (R-Logistic Model)

Like the simulation of LAI, the R-logistic model [28] was employed to verify the FBA growth pattern:

$$y\_{\text{F}} = c \left( 1 + \exp(gt^2 + et + f) \right) \tag{5}$$

where *y*<sup>F</sup> is the above-ground FBA (kg ha<sup>−</sup>1); *c*, *g* (>0), *e*, and *f* are the model parameters; *t* is the same as in Equation (3). When *t* = 0, *y*<sup>F</sup> = *c*/(1 + exp(*f*)) (the above-ground FBA in maize emergence); when *<sup>t</sup>* = (−*e*/2*g*), the value of (*gt*<sup>2</sup> + *et* + *<sup>f</sup>*) reaches a minimum, and the value of *y*<sup>F</sup> reaches a maximum; when *t* > (−*e*/2*g*), the value of *y*<sup>F</sup> begins to decline. These situations are consistent with the actual growth curve of FBA.

#### 2.4.4. Normalized Revised Logistic Model (NR-Logistic Model)

Similar to the logistic model, the R-logistic model was initially developed for individual plants. To scale up the simulation from a single plot to a region, the NR-logistic model was developed by the normalization method mentioned above. It takes the same form as the R-logistic model but with key parameters:

$$\mathcal{Y}\_{\mathcal{F}} = \mathbb{C} / \left( 1 + \exp(GT^2 + ET + F) \right) \\ \mathcal{Y}\_{\mathcal{F}} = \mathcal{Y}\_{\mathcal{F}} / \mathcal{Y}\_{\text{Fm}} \tag{6}$$

where *Y*<sup>F</sup> represents the relative fresh biomass accumulation (RFBA); *C*, *G* (>0), *E*, and *F* are parameters; *T* (0 ≤ *T* ≤ 1) is the same as in Equation (4); *y*Fm represents the maximum FBA (g m−2). When *<sup>T</sup>* = (−*E*/2*G*), the value of *GT*<sup>2</sup> + *ET* + *F* reaches a minimum, and *Y*<sup>F</sup> reaches a maximum; when (−*E*/2*G*) < *T* < 1, the value of *Y*<sup>F</sup> declines as the value of *T* increases.
