## 2 Materials and Methods
### Model Fitting  
#### 2.4.1 Cctm Model  
Generally, land surface changes are of four types [46]: (1) uniform variation whose variation rate is fixed; (2) decelerated variation whose absolute value of the variation rate is getting smaller and smaller; (3) accelerated variation whose absolute value of the change rate is getting smaller and smaller, which are always caused by deforestation, floods, fire, insects, urbanization, etc. ; and (4) periodic variation caused by seasonality, climate variability, vegetation growth, etc. Therefore, we developed the CCTM time series model, which includes components of gradual, periodic, and abrupt variations to capture all four types of surface change. A typical model can be expressed as per Equation (10), where \(f_{1}(y_{1}),f_{2}(y_{2}),f_{3}(y_{3}),f_{4}(y_{4}),y_{1},y_{2},\ y_{3},\ y_{4}\) are the functions of the Julian day, with ranges of [0, 365] or [0, 366]. In the above function-based model, \(f_{1}(y_{1})\) is used to fit the uniform variations, \(f_{2}(y_{2})\) is used to fit the accelerated variations, \(f_{3}(y_{3})\) is used to fit the decelerated variations, and \(f_{4}(y_{4})\) is used to fit the short-term periodic variations. \[\widehat{\rho}\left(i,k,x\right)_{simple}=f_{1}(y_{1})+f_{2}(y_{2})+f_{3}(y_{3 })+f_{4}(y_{4}) \tag{10}\]The typical mathematical functions can also be divided into three categories according to their second derivative (\(f^{\prime\prime}(x)\)): (1) the second derivative is larger than zero (\(f^{\prime\prime}(x)>0\)). This kind of basic function can be a polynomial function like \(ax^{k}+bx^{k-1}+cx^{k-2}+\ \ldots\ +d\), an exponential function like \(e^{x}\), or a negative logarithmic function like \(-\ln x\). (2) The second derivative is equal to zero (\(f^{\prime\prime}(x)=0\)), such as a linear function \(ax+b\). (3) The second derivative is lower than zero (\(f^{\prime\prime}(x)<0\)). This kind of basic function can be an exponential function like \(e^{-x}\) or positive logarithmic function like \(\ln x\) etc. To comprehensively represent the variations in yearly time series data, the final fitting function is composed of four parts. First, the uniform variation part can be fitted by a linear model, which can also capture the long-term change over a year: \(f_{1}(y_{1})=a_{1,k,i}\ y_{1}+\beta\). Second, the accelerated variation can be fitted by a positive exponential function model: \(f_{2}(y_{2})\)\(=b\ e^{y_{2}}\). Third, the decelerated variation can be fitted by a positive logarithmic function model: \(f_{3}(y_{3})\)\(=c\ \ln(y_{3})\). Finally, the unimodal variation or short-term periodic variation parts can be fitted by this harmonic model [47], where \(f_{4}(y_{4})=d_{1,k,i}\ cos(2\pi y_{4})+e_{1,k,i}\ sin(2\pi y_{4})\). The harmonic model is comprised of a sine function and a cosine function, and it can precisely represent periodic variations. Therefore, the final model can take the form of Equation (11). \[ \begin{array}{c}\widehat{\rho}\left(i,k,y\right)_{simple}=a_{1,k,i}\times y _{1}+\beta+b_{1,k,i}\times e^{y_{2}}+c_{1,k,i}\times\ln(y_{3})\\ +d_{1,k,i}\times cos(2\pi y_{4})+e_{1,k,i}\times sin(2\pi y_{4})\end{array}  \tag{11}\]  
To magnify the values of the coefficients (such as \(a_{1,k,i}\) and \(b_{1,k,i}\)), the Julian day values are normalized to the range of [0, 1] and \(y_{4}=\frac{max-x}{max-min}\). Due to the value of exponential function being persistently larger than 0 and rapidly increasing, the Julian day has also rescaled to [0, 1] and \(y_{2}=\frac{x-max}{10}\). For the logarithmic function model, the Julian days are limited to [1, e] and \(y_{3}=1+\frac{x-min}{max-min}*(e-1)\). Then, all of the function values are transformed to the range of [0, 1], which is beneficial for magnifying the coefficients of the fitting function. The final CCTM time series model is as follows:  
\[