Glacier Terminus Estimation from Landsat Image Intensity Profiles Joseph Usset1, Arnab Maity2, Ana-Maria Staicu3 and Armin Schwartzman4 Abstract Mountain glacier retreat is an important problem related to temperature increase caused by global climate change. The retreat of mountain glaciers has been studied from the ground, but there exists a need for automated methods to catalog glacial change with a wider scope. A viable approach is to extract intensity profiles from Landsat images along the glacial flowline and follow the terminus location over time. We propose a new robust and accurate statistical algorithm to estimate the movement of glacial termini over time from these extracted image intensity profiles. The method we propose first uses regression splines to smooth the image intensity profiles. For each profile the glacial terminus location is assumed to lie near a point of high negative change in the smoothed profiles. An approximate path of termini locations over time is obtained by an algorithm that seeks to minimize the cumulative first derivative value across the profiles. Spline smoothing is applied to this pilot path for estimation of long-term terminus movement. The predictions from the method are evaluated on simu- lated data and compared to available ground measurements for the Nigardsbreen, Gorner, Rhone, and Franz Josef glaciers. Keywords: Change-Point Estimation; Cross-Validation; Non-parametric Regression; Satellite Imagery; Spline smoothing. 1Department of Biostatistics, University of Kansas, Kansas city, USA (E-mail: [email protected]). 2Department of Statistics, North Carolina State University, Raleigh, USA (E-mail: [email protected]). 3Department of Statistics, North Carolina State University, Raleigh, USA (E-mail: [email protected]). 4Department of Statistics, North Carolina State University, Raleigh, USA (E-mail: [email protected]). 1 1 Introduction Mountain glacier retreat is an important problem for several reasons. Glaciers can be used as a proxy indicator for global climate change, and in particular temperature change [17]. Also, glacial retreat affects water supply in communities where glacial melt is a source of freshwater [5, 21]. Over the past several decades, there has been a widely reported retreat of mountain glaciers, and this retreat has accelerated in the past decade [18, 6, 17, 21, 5]. For these reasons, there is an interest in tracking mountain glacier retreat worldwide. One method for tracking glacial retreat has been ground measurements, but these can be difficult to obtain. Instead, [12] and [11] have proposed to locate and track glacier termini over time using Landsat images, which have been available to the public since 2009. This approach is based on image intensity profiles extracted from the Landsat images along the glacier flowline. In this paper we present an improved statistical algorithm used to locate and track glacial termini over time from the extracted image intensity profiles. Figure 1 demonstrates the basic elements of our analysis, using the Nigardsbreen mountain glacier as an example. The top left plot shows a 2-dimensional (2-D) false color composite image of the glacier cropped from the Landsat archive at a given time frame. Drawn on it in yellow is a 1-dimensional (1-D) outline of the glacier flowline, produced semi-manually by the method of [12]. The red arrow indicates the terminus location, at the boundary of the ice and a sub-glacial lake. The top right plot displays the extracted intensity profiles along the flowline from 15 spatially registered Landsat images (thermal band B62) between 2000 to 2012. The terminus location corresponds to the high-to-low transition in each profile. The bottom plots present the image profiles over time, where the terminus location corresponds to the blue-to-brown transition. The estimated glacier terminus path from our proposed statistical algorithm is shown in yellow, with confidence bands shown in black and compared to ground measurements represented by the red dots. Note that the ground measurements were not used at all in the estimation, indicating how remarkably accurate the proposed method can be based on the Landsat images alone. In this paper we describe the statistical algorithm that produced the estimated glacier terminus path and confidence bands. The algorithm we propose receives as input the time series of 1-D spatial intensity profiles and gives as output the estimated terminus path over time. The proposed algorithm contains three main steps: 1. Smooth the intensity profiles; and obtain the first derivative estimates of the smoothed 2 250 200 150 Profile Intensity 100 50 0 500 1000 1500 2000 2500 Space Along Glacier Path 250 250 2010 2010 MIAE = 7.7 MAD = 12.9 200 200 2008 2008 COV = .90 2006 150 2006 150 Year Year 2004 2004 100 100 2002 2002 0 500 1000 1500 2000 2500 1300 1400 1500 1600 1700 Space along the Glacial Path (Meters) Space along the Glacial Path (Meters) Figure 1: Top left: 2-D cropped Landsat image of Nigardsbreen glacier with marked flow- line (yellow) and terminus (red arrow). Top right: Extracted 1-D intensity profiles along the flowline. Bottom left: Extracted 1-D intensity profiles laid out over time; along with the estimated terminus path (yellow), confidence intervals (black), and ground measure- ments (red dots), with evaluated metrics in the legend. Obstructed profiles appear as light blue stripes. Bottom right: Zoomed in display of estimated path, confidence intervals, and ground measurements. Time points where the image profiles were observed are marked by purple hash marks on the left of the image. 3 profiles. 2. Input the estimated first derivative profiles into an algorithm that outputs a pilot path of the terminus location over time by minimizing the integrated first derivative. 3. Globally smooth the pilot path to estimate the long-term advance or retreat of the terminus. The above algorithm, beginning with Step 1, is motivated by the fact that the terminus of the glacier is often marked by a sharp high-to-low transition of the intensity profile, which corresponds to a highly negative local minimum of the first derivative after smooth- ing (Figure 1). However, as we shall see in other glaciers, sharp local minima can also occur due to obstructive elements such as shadows from nearby mountains and debris. In addition, the glacier terminus itself may be occluded by clouds and seasonal snowfall, producing flat profiles such as those observed in Figure 1 (top right panel) and as blue horizontal stripes in Figure 1 (bottom panels). These sources of error, which we call sys- tematic, are mostly deterministic but hard to model because not enough information about them is available. Step 2 overcomes the systematic error by finding a controlled sequence of locations over time that captures persistent local minima. Step 3 removes local noise. The proposed algorithm follows roughly the same framework as [11], although there are some important differences in the approaches. In Step 1 of [11], first derivatives of the mean intensity profiles are estimated directly with local kernel smoothing. A drawback of this approach is the difficulty of determining a fixed local smoothing bandwidth that will provide a good fit globally over the entire spatial domain of the intensity profiles. We correct this by using global spline smoothing instead, fitted by penalized weighted least squares with generalized cross-validation. In Step 2 of [11], a tracking algorithm is applied to connect local minima of the first derivative between consecutive time points. However, because it forces the pilot path to pass through those local minima, it is sensitive to systematic noise. We fix this by globally minimizing the integrated first derivative over time. The proposed pilot path algorithm robustly finds the terminus path despite systematic error in the derivative profiles. Finally, in Step 3, [11] estimate the long-term terminus location change by local kernel smoothing of the pilot path. In addition to the bandwidth selection problem mentioned before, local smoothing gives imprecise or non- existent estimates over time periods where the image data is sparse. Instead, we gain more precise estimates of long-term glacial retreat by smoothing the pilot path with global spline smoothing. 4 From a broader statistical viewpoint, the work presented here relates to the problem of change point detection, which has been well studied in the context of nonparametric regression [15, 13, 10]. However, the specific data analysis problem treated here presents a unique methodological challenge; as opposed to identifying change points in an individual continuous curve, we seek a path of change points from a time series of continuous curves. Moreover, while multiple change points may exist in each intensity profile (e.g. due to shadows in the original images), we are interested in a sequence of change points whose location changes smoothly over time. The 3-step algorithm presented above solves this problem. The paper is organized as follows. Section 2 provides a description of the data and how it was obtained. Section 3 discusses smoothing of the intensity profiles with penal- ized regression splines (Step 1), and shows how this facilitates smooth derivative estimates. Section 4 describes the pilot path algorithm that gives an initial estimate of the terminus path (Step 2), and compares this method to the tracking algorithm of [11]. Section 5 dis- cusses temporal spline smoothing of the pilot path (Step 3), and evaluates several weight- ing schemes that can be used to estimate the long-term trend of the terminus location. Sections 6 and 7 include a numerical study and real data analyses of the Nigardsbreen, Gorner, Rhone, and Franz Josef glaciers. Section 8 summarizes our method and presents future directions for research. 2 Data Description For a given mountain glacier, the proposed method takes as inputs Landsat image intensity profiles extracted along the glacier flowline. The image profiles used here were obtained using a Matlab graphical user interface (GUI) designed for this purpose; see [12].