The first metric is rooted in a traditional approach for calibrating and estimating uncertainty of feature-tracked ice velocity. The central idea is to assume that adjacent ice-free terrain is static without horizontal or vertical movement, implying that any non-zero velocity values in these locations are measurement errors. This approach requires well-distributed control surfaces that cover a large enough area so that the velocity measurements are not heavily influenced by isolated hillslope movements e.g., or landslides (e.g., ), which are common in high-mountain environments. The central tendency (mean, median, etc.) of measured velocity components (Vx and Vy, where x and y are defined by the input image axes) over static terrain is traditionally used to calibrate the entire velocity product; see Sect.  for an example. The value of this central tendency can be regarded as the accuracy of the measured feature offset and is controlled by many factors, such as image misalignment, geolocation errors, and atmospheric disturbances (from clouds and ionosphere for optical and SAR images, respectively). In addition to the accuracy-based calibration, the residual variability (standard deviation or other similar metrics) of measured velocity components is conventionally used to assign velocity uncertainty (precision) for the entire product . However, this variability is practically an aggregated measurement of correct and incorrect matches and other terrain-dependent errors, and it is challenging to isolate these individual contributions to total observed uncertainty.

Here, we propose a way to better characterize the noise of only the correct matches for the first metric. We use nonparametric, multivariate kernel density estimation (KDE; ) to differentiate correct and incorrect matches and estimate the variability in the former population. Let Vx,i and Vy,i, i=1,…N be horizontal velocity components from the selected static area containing N measurements (i.e., N pixels). We calculate the kernel density distribution ρK at every possible velocity value using the following equation: 1ρK(u,v)=1Nh∑i=1NKu-Vx,ih,v-Vy,ih, where (u,v) indicates independent variables along the x and y directions (i.e., two axes in the velocity domain), K is the selected multivariate kernel, and h is the kernel bandwidth. In other words, ρK resembles the histogram of the measurements or the sum of individual density distributions centered at (Vx,i,Vy,i). Since the choice of kernel does not affect the density estimation as much as bandwidth does, we select the Epanechnikov (parabolic) kernel to achieve computational efficiency due to its finite support compared to the Gaussian kernel. We use the rule-of-thumb method to determine the bandwidth without prior knowledge about the velocity distribution. Assuming that all the correct matches have an identical, independent, and normally distributed noise term with the same variance in u and v, the rule-of-thumb bandwidth for a multivariate Epanechnikov kernel is calculated as follows: 2h=2.1991σVxσVyN-16, where σVx and σVy are the standard deviation of Vx and Vy, respectively. Once a kernel density distribution is found, we locate its main peak value and location on the u–v plane. This peak is assumed to be related to the distribution of the correct matches, and we can design a thresholding method based on peak value to differentiate correct and incorrect matches: 3ρKt=max⁡(ρK)ez22, where z is the preselected z score (always positive). Measurements with ρK(Vx,Vy)≥ρKt are considered correct matches, and measurements with ρK(Vx,Vy) lower than the same threshold value are classified as incorrect matches. The half ranges of Vx and Vy from correct matches, denoted as δu=zσu and δv=zσv, are thus considered zσ uncertainty. In this study, we set z=2 and report 2σ uncertainties for the convenience of comparing previously published results and our results. The variance of the correct matches is also defined as σu2=(δuz)2 and σv2=(δvz)2.

To summarize, while statistics computed over static terrain have previously been used to assess the uncertainty of glacier velocity maps, this study attempts to separate correct and incorrect feature matches over static terrain and to derive uncertainty only from the correct matches. The calculated uncertainty of correct matches can be compared with the theoretical uncertainty . Ideally, the former should approach the latter for optimized measurement precision. In addition, the number of incorrect matches over static terrain and their corresponding velocity distribution can serve as an auxiliary indicator of the accuracy of the velocity map, along with the correct-match uncertainty.

For computational simplicity, our previous metric overlooks the covariance of the correct matches. However, glacier motion is spatially coherent, and the covariance of neighboring correct matches controls the quality of the measured flow pattern. Hence, the second metric aims to estimate the co-variability in pixels on a glacier using the physics of glacier flow . We start by analyzing the horizontal (two-dimensional) strain rate tensor ϵ˙x′x′, ϵ˙y′y′, and ϵ˙x′y′, where the prime notations of x′ and y′ denote along-flow and across-flow directions, respectively. To obtain this tensor, we first calculate the strain rates along the image axes x and y: 4ϵ˙xx=∂Vx∂x,ϵ˙yy=∂Vy∂y,ϵ˙xy=12(∂Vx∂y+∂Vy∂x).

In GLAFT, these partial derivatives are computed using 3-by-3 Sobel operators. Next, we compute arctan2(Vx/Vy) and smooth the results with a two-dimensional median filter with a large window size of 10–35 pixels (depending on the pixel spacing) for the bulk flow direction angle θ (counterclockwise from the x axis). The strain rate tensor is then rotated by θ and projected into the along-flow direction as follows: 5ϵ˙x′x′=ϵ˙xxcos⁡2θ+ϵ˙yysin⁡2θ+ϵ˙xysin⁡2θ,ϵ˙y′y′=ϵ˙xxsin⁡2θ+ϵ˙yycos⁡2θ-ϵ˙xysin⁡2θ,ϵ˙x′y′=12(ϵ˙yy-ϵ˙xx)sin⁡2θ+ϵ˙xycos⁡2θ.

By replacing Vx and Vy with ϵ˙x′x′ and ϵ˙x′y′ in Eq. (), we can calculate the KDE in the strain rate domain and characterize the distribution. We follow the same steps in Eqs. ()–() and find the variance of the strain rate distribution under a preselected z value. To be consistent with the first metric, in this study we report the zσ uncertainty for ϵ˙x′x′ and ϵ˙x′y′ as δx′x′ and δx′y′, respectively.

Unlike the case of the previous metric, which indicates the variability from the zero ground truth, the strain rates along a flowing glacier are not zero, but we can still infer a reasonable range based on fundamental physical relationships. Consider a rectangular region on a glacier with one side running across the channel half-width Y and the other side along the flow direction (length X). The average driving stress (τ¯d) over the rectangular area is balanced with the basal drag (τ¯b), side drag, and longitudinal stress gradient (see Eq. 8.60 of , for details): 6τ¯d=τ¯b+1YΔ(Hτ¯x′y′)+1XΔ[H(2τ¯x′x′+τ¯y′y′)], where H is average ice thickness. τ¯x′y′, τ¯x′x′, and τ¯y′y′ represent average shear stress and normal stresses, respectively. Supposing half of the driving stress is balanced by basal drag, and the other half is balanced by side drag, and the longitudinal stress gradient is negligible (cf. Table 8.3 of ), Eq. () becomes 7τ¯b=1Yτ¯x′y′ΔH.

For the right-hand side of Eq. (), we let ΔH≈H, indicating the maximum possible ice thickness change within the rectangular region. Moreover, we can express the average side drag as a function of the average strain rate, assuming glacier ice can be modeled as a viscous non-Newtonian fluid with the following creep relation (Eq. 3.23 of ): 8τjk=2ηϵ˙jk, where η is ice viscosity, and the subscripted j and k denote any two of the three dimensions. Thus, Eq. () becomes 9τ¯b=1Y2ηϵ˙¯x′y′H, where ϵ˙¯x′y′ is the average shear strain rate.

The creep relation (Eq. ) can also be expressed inversely, known as Glen's flow law: 10ϵ˙jk=Aτjkn, where n and A are two empirical parameters. Combining Eqs. () and () leads to the expression of viscosity in terms of the flow-law parameters: 11η=12Aτjkn-1.

The along-flow ice speed at the surface (ux′) can be calculated by integrating Glen's flow law (Eq. ) along the vertical direction of the ice flow plus the basal slip speed (ub), assuming a linearly increasing shear stress with depth (see Eqs. 8.32 to 8.35 of , for details): 12ux′=ub+2An+1τbnH.

Combining Eqs. () and () with τjk set to τb leads to the expression of average basal drag as a function of average surface along-flow speed u¯x′ and average basal slip speed u¯b: 13τ¯b≈(u¯x′-u¯b)ηn+1H.

Finally, combining Eqs. () and () leads to the following equation: 14ϵ˙¯x′y′u¯x′-u¯b=(n+1)Y2H2, which can be further reduced to 2Y/H2 if we assume n=3.

Equation () provides a range of plausible ϵ˙x′y′ values based on glacier speed, channel width, and average ice thickness. Ideally, observed δx′y′ should be as close to ϵ˙¯x′y′ as possible. If the former is much larger, the glacier velocity likely contains errors, as the observed spatial variability is not physically achievable. On the other hand, if ϵ˙¯x′y′ is much larger than δx′y′, it is likely that the flow pattern is smoothed too much and has lost real dynamic signals.

Our 172 glacier velocity maps (six in Fig.  and the rest in Figs. S1–S8) are similar to the time-averaged speed from shown in Fig. : the average flow speed is around 0.3 m d-1 (100 m yr-1), with small local variations. While it is common to clip velocity maps to glacier outlines in publications and datasets, we show the full velocity map for each test so that readers can see the distribution of invalid and incorrect matches over adjacent terrain. For the examples in Fig. , the bad matches (empty and unrealistic values in each upper panel) roughly align with the changing illumination and corresponding shadow positions during the image acquisition period (March to April 2018). These six maps show δu and δv ranging from 0.06 to 0.64 m d-1. For every velocity map, δu and δv are close in value, likely because a square matching template was used to track features. Therefore, we use δu as Metric 1 to assess the velocity map quality in the rest of the study. Large δu values generally indicate a noisy velocity map, and small δu corresponds to a smoother velocity field (Fig. a–c). Besides the magnitude of Metric 1, velocity maps generated by different software packages show various clustering characteristics over static terrain. For example, maps derived using vmap and autoRIFT often have elongated, off-zero clusters (Fig. c–d), while maps from CARST and GIV contain artifacts due to the pixel locking effect (Fig. e–f), a biased tendency in which measurements, including incorrect matches, favor integer pixel offsets . Other results derived from the rest of the tests are available in Figs. S9–S16 and Table S2.

Unlike the static terrain velocities, the along-flow strain rate does not show characteristic spatial variability across most tests; these values tend to form a single cluster centered on zero (Fig.  bottom panels; also Figs. S17–S24). The variability in the normal and shear strain rates is similar, as indicated by similar δx′x′ and δx′y′ values. This suggests that random noise, not glacier physics, controls such variability. The magnitude of δx′y′ (Metric 2) does not show an obvious correlation to the velocity map (Fig.  upper panels), but a closer inspection by plotting the overall strain rate magnitude (ϵ˙x′x′2+ϵ˙x′y′2) shows that δx′y′ relates to the smoothness of the strain rate map (Fig.  middle panels). Metric 2 is insensitive to correlated error over long spatial scales (Fig. c) but is sensitive to high-frequency spatial variation with small amplitude (Fig. a). The δx′y′ and δx′x′ values range from 0.001 to 0.12 d-1 across the 172 velocity maps (Figs. S17–S28 and Table S2).

The metrics presented in this paper can be used to assess the quality of glacier velocity maps (Table ). We suggest that the correct-match uncertainty of static terrain velocity, δu (and δv if the map is derived using a non-square matching template, which is common for SAR feature tracking), should be as low as possible until the value reflects the inherent match uncertainty only. If the inherent match uncertainty (2σ) is 0.2 pixels , the desired range of δu is 15δu≤0.2×pixel size of source imagesduration of source image pair.

The recommended value for the variability in along-flow shear strain rate, δx′y′, depends on the flow-law parameter n and basal sliding velocity u¯b (Eq. ), which can be challenging to measure. However, based on the test results presented in this study, proposing an overestimated value by setting zero basal slip may be acceptable because it is more conservative on whether the observed strain rate field links to the actual ice flow dynamics. With the general assumption of n=3, we suggest the following guideline for setting a δx′y′ threshold, which relates to the average surface along-flow speed u¯x′, channel half-width Y, and average ice thickness H: 16δx′y′≈u¯x′2YH2.

We can apply these metrics and guidelines in various use cases. Users who run feature-tracking workflows can compute these metrics for their output velocity maps. If either metric deviates from the recommended range, they can try a different parameter combination, including prefilters, tracking parameters, subsampling, masking, and other postprocessing steps, until the metrics fall within the recommended values. In addition to these metrics, the users can analyze the number and spatial distribution of incorrect matches over the static terrain to identify directions of improvement; see Fig.  and discussion in Sect.  for an example. These metrics will also serve as a basis for comparing novel feature-tracking algorithms e.g., with traditional algorithms. Glacier modelers can use these metrics to select the best possible velocity maps to derive physical quantities (such as strain rate) with minimal error propagation. When a velocity map has a δx′y′ much larger than the suggested value, one can select an appropriate smoothing level for the velocity map and recalculate δx′y′ until it reaches the suggested value. Finally, we recommend that data producers calculate these metrics (with the help of the GLAFT package) and include them in the metadata associated with each velocity product. This will enable users to assess product quality and determine if customized velocity maps are required for their applications.

To estimate the uncertainty of ice flow velocity, Metric 1 (δu or δv) seems to be a plausible option and has been used in many studies (see Sect. ). However, on-ice velocities likely have a different noise distribution than off-ice velocities, and the use of Metric 1 as the flow uncertainty will be sub-optimal. Since Metric 2 (δx′y′) describes the spatial variability in the flow velocity, it can potentially offer an alternative uncertainty estimator. While the latter is one of the future goals of this project, at this time, we suggest that users should treat Metric 1 as a very conservative image-wide uncertainty of the ice flow velocity.

Finally, the strategy outlined here is based on our analysis using optical images for one glacier. We expect that these metrics will also work for SAR feature tracking and different glacier settings, but additional care might be necessary. For example, SAR images typically have different range and azimuth resolutions, which could result in a significant difference between δu and δv. Also, it can be difficult to calculate both metrics over an ice sheet where there is no static terrain and limited differential velocity within a single scene. We will continue to address these issues in future GLAFT applications.

The open-source GLAcier Feature Tracking testkit GLAFT; Python package accompanying this paper can be used to compute and evaluate these metrics and associated thresholds for arbitrary input velocity map data. GLAFT contains modules for deriving and visualizing the two metrics from velocity maps generated by most feature-tracking tools. GLAFT is available on Ghub (, https://theghub.org/resources/glaft, last access: 21 August 2020) for cloud access and can be installed locally via PyPI, Python's official third-party package repository and manager (https://pypi.org/project/glaft, last access: 21 August 2020). The GLAFT source code, Notebook examples, and documentation are hosted on GitHub (https://github.com/whyjz/GLAFT), with Binder-ready Jupyter Book pages at https://whyjz.github.io/GLAFT/ (last access: 21 August 2020) as the Supplement of this paper.