# Offline-Online Change Detection for Sentinel-1 InSAR Time Series
## 2 Methods
### Changes in Displacement Time Series
#### 2.2.1 Offset Detection  
Offsets in time series take the form of an instantaneous jump (Figure 3). In order to detect these jumps we look for peaks in the displacement difference time series. For the difference series of order \(d\):  
\[\
abla^{d}Y_{t}^{k}=(1-B)^{d}\gamma_{t}^{k} \tag{1}\]  
where \(Y_{t}\) is the displacement at acquisition date \(t\) and \(k\) is the lag step size. \(B\) is the backshift operator [33]:  
\[BY_{t}=Y_{t-k} \tag{2}\]  
In the case of the first difference, the change between consecutive observations in the original series, \(k=1\) and \(d=1\) so that the first difference series is:  
\[\
abla Y_{t}=Y_{t}-Y_{t-1} \tag{3}\]  
While the original displacement time series may not be stationary due to the presence of linear trends and or seasonality, our assumption is that the first, second and third difference time series are stationary [33], meaning that they will have no predictable  
Figure 2: **(a)** The average spatial coherence of all initial interferograms represented as a bar centred on the middle of the primary and secondary acquisition dates. The coherence history shows a clear annual cyclicity between a maximum of about 0.8 and a minimum of about 0.4. **(b)** Baseline-time plot for the interferograms formed in this study. Each line represents an interferogram, colour coded with the average spatial coherence of that image. The dashed red lines are interferograms with average coherence below the threshold value of 0.45 and so were discarded unless they were needed to maintain network connectivity (solid red lines). patterns in the long-term. We test this assumption using an Augmented Dickey-Fuller (ADF) test [34] on each difference time series, which as the null hypothesis states that a unit root is present in the series, the principle cause behind non-stationarity. If the p-value from the ADF test is less than 5% we reject the null hypothesis, meaning the time series is stationary. In our tests we find that in all cases the first difference series is stationary. However, in cases of extreme seasonality the second and third difference series are not stationary. Differences with longer temporal separations are more likely to be affected by any strong gradients in the times series (e.g., seasonality), meaning they are less likely to be stationary. In such cases we take the second order difference, i.e. the difference of the difference with \(d=2\) in Equation (1), which results in stationarity (Figure 4). This is a routine procedure in time series analysis [33; 35]. While gradients, or changes that occur slowly over multiple time steps, are removed when differencing, offsets are retained in the difference series because they are an instantaneous jump in displacement. Offsets in the displacement time series manifest as a peak in the difference series. The challenge now becomes a question of how to statistically distinguish these peaks from the noise level. To do this we first need to characterise the noise level in the difference series using only 90% of the data (between the 5% and 95% quantiles), assuming that the remaining points may contain offsets that will bias the standard deviation calculation:  
\[y^{k}=\left\{y_{t}\in\
abla^{d}Y_{t}^{k}\mid Q(0.05)\leq y_{t}\leq Q(0.95)\right\} \tag{4}\]  
Figure 3: Two synthetic offset change examples. The top row are displacement time series with offsets at \(t=80\) and \(t=100\) respectively. There are no gradients before and after the offset in the first example, and different gradients before and after the offset in the second example. There are numerous change detections (shown in red circles) for each of the difference series. But these are reduced when we take the intersection of the 1 st, 2 nd and 3 rd difference detections (bottom row).