3.3.2. Tests with Simulated Data

Since it is very difficult to find appropriate snow-free vegetation pixels as a reference, we designed simulation experiments to evaluate the effect of snow on SOS detection. A linear spectral mixture model was used to simulate the pixel reflectance with different SCF values. For simplicity, it is assumed that each pixel is composed of soil and vegetation; the snow layer is covered above and only absorbs and reflects the incident light, i.e., the transmittance of the snow layer equals 0. Assuming that the vegetation and soil components are both homogeneous, which is reasonable for alpine grasslands, the presence of snow will not affect the areal compositions of soil and vegetation. As a result, the spectral reflectance of a snow-covered pixel can be computed as:

$$\mathbf{R\_{mixed}} = (1 - \mathbf{f\_{snow}}) \left[ \mathbf{f\_{veg}} \cdot \mathbf{R\_{veg}} + (1 - \mathbf{f\_{veg}}) \mathbf{R\_{soil}} \right] \left[ + \mathbf{f\_{snow}} \cdot \mathbf{R\_{snow}} \right] \tag{1}$$

where Rmixed denotes the simulated mixed pixel reflectance; fveg, fsoil, and fsnow are the coverage fractions of vegetation, soil, and snow, respectively; and Rveg, Rsoil, and Rsnow are the corresponding endmember reflectances.

To simulate the satellite-derived band reflectance, the spectral reflectance ρ(λ) was convolved with the spectral response function (SRF) of the MODIS sensor S(λ) as follows:

$$\mathbf{R} = \int\_{\lambda\_1}^{\lambda\_2} \rho(\lambda) \mathbf{S}(\lambda) d\lambda \tag{2}$$

where λ1 and λ2 are the minimum and maximum wavelengths of each band, respectively. The ρ(λ) for snow, soil, and vegetation were selected from the Johns Hopkins University Spectral Library [57], corresponding to medium granular snow, dark brown fine sandy loam, and green grass, respectively, as in Figure 3.

**Figure 3.** Schematic diagram of endmember reflectance spectra and the spectral response of the MODIS sensor.

Through Equations (1) and (2), the band reflectance and further VIs of a pixel can be computed using the endmember reflectance with varying FVC and SCF values. To investigate the effect of snow cover on SOS detection, simulation experiments were designed in the following two aspects.

First, we investigated the effect of snow cover on five VIs. We computed the VI values when FVC was varied from 0 to 1 for five cases of SCF = 0%, 25%, 50%, 75%, and 100%. The effect of snow cover on VI values can be described by the difference in VI values under specific snow scenarios and snow-free conditions (i.e., SCF = 0%), expressed as ΔVI = VISCF > 0 − VISCF = 0. As the values of ΔVI vary with SCF and FVC, we further define a quantitative indicator, the maximum impact of snow (MIS), to represent the maximum effect of snow on VI for a specific FVC, expressed as:

$$\text{MIS} = \frac{|\text{VI}\_{\text{SCF}} - \text{VI}\_{\text{SCF}} - \text{VI}\_{\text{SCF}}|}{\text{VI}\_{\text{max}} - \text{VI}\_{\text{min}}} \tag{3}$$

where VISCF = 100% and VISCF = 0% correspond to VI values when SCF = 100% and SCF = 0% for a given FVC, respectively; VImax and VImin are the maximum and minimum VI values, respectively. The denominator is the range of VI values and is used to eliminate the effect of different value ranges of VIs. Both VI and MIS vary with FVC. The numerator represents the absolute difference in VI between SCF = 100% and SCF = 0%, while the MIS represents the maximum percentage change in VI caused by snow relative to the range of VI values. The MIS provides a direct indication of the extent to which a VI is affected by snow and provides a basis to further investigate the effect of snow cover on SOS detection.

Second, to investigate the effect of snow cover on SOS detection, we designed a series of experiments to generate different VI time curves under different snow scenarios. We extracted the NDVI time curve of a typical snow-free vegetation pixel and converted the time series of NDVI to FVC using the dimidiate pixel model [58]. Using the derived time series of FVC, the time series of band reflectance and VI were calculated under different snow scenarios defined by snow parameters, including SCF, SCDc, and ESS. The time curves of five different Vis were then filtered, and the SOS was detected. As shown in Figure 2, given the same growth curve of FVC, the red and blue lines represent the VI trajectories for snow-free and preseason snow conditions, respectively. The preseason snow caused a bias in the detected SOS, expressed as ΔSOS = SOSsnow − SOSsnow-free, which is defined as the effect of snow on SOS detection.

Three sets of experiments were designed, and the corresponding settings of snow scenarios are shown in Table 2. In all experiments, as we were only concerned with the SOS, only the snow season from DOY 001 to DOY 208 was considered. Experiment I corresponded to a completely snow-free case with SCF = 0% during the period, which served as the baseline to assess the effect of snow cover on SOS detection (Figure 4a). Experiments II and III were snow cover conditions, where four cases of SCF = 25%, 50%, 75%, and 100% were considered, and in all cases, the SCF remained constant during the snow season. Experiment II referred to the cases of snow persisting from DOY 001 to ESS. Three cases of ESS at DOY 104, 136, and 168 were considered, which were the mean ESS plus or minus its standard deviation analyzed from snow cover data, as shown in Figure 4b. In experiment III, three cases of SCDc = 32, 64, and 96 days were considered. As the SCDc could be in any interval during the snow season, we simulated all cases by iterating the start of the snow season from DOY 001 at 16-day intervals while keeping the ESS no later than DOY 208. For example, a case with SCDc = 64 can generate 10 different time curves of a VI with a snow season ranging from DOY 1–64 to 145–208, as shown in Figure 4c.

**Table 2.** Experimental settings for the investigation of the effect of snow on spring vegetation phenology detection.


**Figure 4.** Temporal trajectories of coverage fractions of vegetation, snow, and soil for experiments I to III. (**a**) Experiment I with SCF = 0% constantly from DOY 001 to 208; (**b**) experiment II with snow persisting from DOY 001 to ESS with constant SCF; (**c**) experiment III with snow persisting from DOY *t* to ESS with constant SCF.
