Fig. 5. The VZA stratified DNB time series and high-resolution images of a riverside esplanade with new pedestrians built in 2016 near Charles River in Massachusetts, U.S. (a) DNB time series at the riverside esplana. The red, green, and blue dots indicate the DNB observations within different VZA intervals. (b) The high-resolution Google Earth image in June 2015 of the selected pixel in Fig. 5a. The red rectangle represents the location and size of the selected pixel. (c) The high-resolution Google Earth image in June 2017 of the selected pixel in Fig. 5a. The red rectangle represents the location and size of the selected pixel.
3.3. Continuous monitoring of NTL change
The VZA-COLD algorithms were applied to estimate time series models from DNB observations collected within different VZA intervals while collectively identifying NTL changes across all VZA strata. For DNB observations stratified within each VZA interval, an individual harmonic model was estimated to capture the seasonality and trend of the DNB time series, which could greatly reduce the impact from the intra-annual (e.g., vegetation phenology and snow) and inter-annual (e.g., gradual economic growth and vegetation long-term growth) changes. We tested the models with unimodal, bimodal, and trimodal seasonality (4, 6, and 8 coefficients) based on the Ordinary Least Square (OLS) regression, Least Absolute Shrinkage and Selection Operator (LASSO) regression (Tibshiranit, 1996), and robust regression (Hampel et al., 2011; Zhu et al., 2012) to explore the optimal combinations for modeling the daily DNB time series. We observed that models with a single-term harmonics model (Eq. 1) and based on robust regression had the best results for our calibration samples and were more robust to outliers and less likely to be overfitted. Therefore, the single-term harmonic model (Eq. 1) estimated based on robust regression was selected to predict the overall DNB magnitude, intra-annual seasonality, and inter-annual trends, which would be used in continuous monitoring of NTL changes.
\({\hat{\rho}}_{i,x}=\ a_{0}+\ a_{1}\cos\ \left(\frac{2\pi}{T}x\right)\ +\ b_{1}\sin\ \left(\frac{2\pi}{T}x\right)\ +\ c_{1}x\)(1)
where,
\({\hat{\rho}}_{i,x}\): Predicted DNB value for the i th VZA interval at Julian date x.
x : Julian date.
\(T\): Number of days per year (\(T\) = 365.25).
\(a_{0}\): Coefficient for overall value for the DNB.
\(a_{1},\ b_{1}\): Coefficient for intra-annual change for the DNB.
\(c_{1}\): Coefficient for inter-annual change (slope) for the DNB.
Continuous change detection was conducted based on the models estimated from each VZA stratification following the COLD algorithms (Zhu et al., 2020), by comparing the actual observations with the model predictions. Breakpoints were identified based on the number of consecutive anomaly observations beyond the applied change probability thresholds. The VZA-COLD made three major changes compared with the original COLD algorithm. First, due to the high temporal frequency and the large fluctuations observed with the daily DNB time series, VZA-COLD would tolerate one of the observations (except for the first one) not showing up as an anomaly in the consecutive anomaly test. Second, VZA-COLD detected changes based on a set of four time series models estimated from observations of different VZA intervals (instead of observations from different spectral bands), and when a breakpoint was identified by any of the VZA interval models, it would be applied to all the four VZA-stratified models, and thus, dividing their time segments with the same break time. Third, the optimal parameters for the change detection process were also different from the default COLD algorithm. We tested the number of consecutive anomaly observations to confirm a change from 12 to 16, along with the change probabilities of 70%, 75%, 80%, 85%, and 90%, and analyzed the performance metrics of omission rate, commission rate, and F1 score based on the calibration samples (Fig. 6). According to the results, the consecutive anomaly observation of 14 and 75% change probability were selected to detect NTL changes.