We used the Trimble UX5 Unmanned Aircraft System, a fixed-wing RPAS manufactured by Trimble Navigation for photogrammetric applications. A single two-blade propeller, driven by a 700 W electric motor, propels the platform. Power is supplied from a 14.8 V, 6000 mAh lithium-polymer battery allowing a flight endurance of 50 min. Autonomous navigation is supported by a single-channel GPS receiver, which also provides approximate coordinates for each photo centre, while an accelerometer logs orientation data.

Imagery is captured by a large-sensor (APS-C) Sony NEX 5R mirrorless reflex digital camera providing a maximum imaging resolution of 4912 pixels by 3264 pixels or about 0.04 m GSD at 400 ft (122 m) a.g.l. The camera is fitted with a Voigtlander Super Wide-Heliar 15 mm f/4.5 Aspherical II lens, with focus fixed to infinity. Appropriate exposure to ensure suitable contrast on the range of imaged targets was achieved with maximum aperture, high shutter speeds between 1/1000 and 1/4000 s to minimize forward-motion blurring and automatic ISO sensitivity. Camera settings were checked prior to each flight to accommodate varying light conditions and the relative share of ground cover (vegetation vs. snow).

Three RPAS missions were undertaken with identical planning and differing states of snow cover in the catchment (Table 1). Flight planning was carried out using the Trimble Aerial Imaging software. All flights imaged 15 strips, aligned along the major axis of the study catchment (Fig. 2). The study area was imaged with 90/80 % forward/sideward overlap with respect to the lowest elevation to ensure that sufficient overlap was maintained when mapping rising ground. Exposure locations are determined automatically by the software to achieve the desired overlap, with the camera being triggered accordingly during flight using on-board Global Navigation Satellite System (GNSS) navigation. The duration of each flight was ∼35 min, with about 900 images being captured per flight. The average flying altitude of the flights was 1650 m a.s.l., with a standard deviation of less than 1.5 m during the mapping phase of the flight. For both the winter and spring flights, the snow surface had considerable texture, with a greater surface roughness overall for winter missions. Wind-affected recent fresh snow was present for the winter flight. It is recognised that homogeneous snow surfaces may represent particularly challenging targets for photogrammetry (Bühler et al., 2017). Nevertheless, the imaging quality and dynamic range of the camera used in this study provided sufficient contrast for all flights, across snow as well as when imaging mixed snow-bare ground conditions. Subsequently, full photogrammetric restitution could be completed without the need for image post-processing (e.g. Cimoli et al., 2017).

Achieving a robust constraint of exterior orientation parameters during aero-triangulation (AT) depends on the availability of a set of high-quality ground control points (GCPs). This is particularly true if the imaging platform lacks a precise-point-positioning capability (e.g. it carries only single-frequency GPS and is not capable of determining differentially corrected positions). Such code-only GPS navigation is accompanied by uncertainties 2 orders of magnitude greater than the expected accuracy of the models. Ground control networks were established for each RPAS flight mission using real-time kinematic (RTK) GNSS surveying with a Trimble R7 base station and R6 rover units. GCP locations were measured with accuracy of the order of ∼0.02–0.03 m. GCPs were signalled with 0.6×0.6 m mats painted with a high-contrast circular quadrant pattern for the autumn and winter flights. For the spring flight, chalk powder was used with a stencil to mark the target directly on the snow surface, using the same pattern as for previous flights. The use of chalk powder eliminated the need to retrieve GCP targets following RPAS flights. All survey work, as well as production of deliverables from photogrammetry, was carried out in terms of the New Zealand Transverse Mercator (NZTM) reference system. All RTK survey work utilised a base station established on a common benchmark, established for this project, the position of which was differentially corrected with respect to nearby continually operating reference stations. GNSS data were processed using Trimble Business Center (TBC) v3.40 software.

It is well established that photogrammetric control is best achieved within the bounds of the GCP network (Linder, 2016), while the uncertainty associated with the geolocation of resected points increases beyond the control network. To constrain the area within the study catchment for photogrammetric processing, the GCP network was distributed around the catchment perimeter, as well as through the central area of the catchment. Additionally, the placement of GCPs on the valley floor and at mid-elevation within the catchment ensured that the network also sampled the elevation range of the catchment. An extensive GCP network was established for the first flight with no snow on the ground, which permitted the robustness of AT to be tested under different GCP scenarios, as discussed further in Sect. 3.2.1. This allowed the network to be refined and reduced in size for subsequent missions, a matter of practical importance when working in alpine areas during the winter. Control point networks for each mission are illustrated in Fig. 2. A new GCP network was established for each survey campaign due to the inability to establish permanent markers (e.g. on poles) due to the conservation status of the study area. Although the layout of the network was similar for each mission, there were no common GCPs shared between different flights, with the only common setting being the set-up of the base station for each RTK survey.

To assess the quality of snow depth data derived from RPAS photogrammetry, independent measurements were collected by manual snow probing on 10 September 2016, the same day as the spring RPAS mission. This approach has been established as standard practice in similar studies (e.g. Nolan et al., 2015; De Michele et al., 2016). Aluminium avalanche probes with 0.01 m graduations, providing a nominal precision of 0.01 m, were used. The sampling strategy involved the measurement of snow depth every 50 m along three elevation contours within the study catchment, namely 1460, 1500 and 1540 m (Fig. 2). This strategy ensured that snow depth was measured across a representative sample of slope aspect and elevation, while optimizing navigation across the catchment. Snow depths were measured at each location by probing 5 times within arm's reach, and the location of the central measurement was surveyed with RTK GNSS, under the same protocol and achieving the same level of accuracy as the GCP survey. This provided 430 measurements of snow depth, with the mean snow depth at each of the 86 locations providing a sample for comparisons with RPAS-derived snow depth.

The goal of aero-triangulation (AT) in photogrammetry is to transform a set of images into a scene in which geometrically accurate measurements can be made in three-dimensional, often geographic, space. This georeferencing process requires a transformation from the inherent coordinate system of the device capturing imagery (a camera) to an appropriate geographic coordinate system (Vander Jagt et al., 2015; Linder, 2016). While traditional photogrammetry has long relied on metric (calibrated) cameras, the use of off-the-shelf non-metric cameras requires the simultaneous solving of both interior orientation (the camera model) and exterior orientation. This process, known as self-calibration, applies a bundle-block adjustment to solve the camera model describing the precise focal length (f), the offset between the principal point of autocollimation and the centre of the imaging sensor plane (x0, y0), and the departure between the image point coordinate (x, y) and the idealized linear projection due to lens distortion. Camera calibration parameterises radial and decentering distortion with models such as that of Brown (1971): x′‾=1+K1r3+K2r5+K3r7x‾+2T1xy‾+T2r2+2x‾2y′‾=1+K1r3+K2r5+K3r7y‾+2T2xy+T1r2+2y‾2, in which K terms are the radial distortion coefficients, T terms are the tangential distortion coefficients, and x‾=x-x0y‾=y-y0r=x‾2+y2. Image coordinates corrected for lens distortion are then used in the set of collinearity equations relating object point coordinates (XA, YA, ZA) to the corresponding image point coordinates (x′‾A, y′‾A) to solve for the exterior orientation (Vander Jagt et al., 2015; Linder, 2016): x′‾Ay′‾A=fr11XA-X0+r12YA-Y0+r13ZA-Z0r31XA-X0+r32YA-Y0+r33ZA-Z0fr21XA-X0+r22YA-Y0+r23ZA-Z0r31XA-X0+r32YA-Y0+r33ZA-Z0. (X0, Y0, Z0) are the coordinates of the perspective centre of the image frame in the ground coordinate system. The rij terms represent the 3×3 rotation matrix relating the sensor coordinate system orientation to the ground coordinate system. Since the UX5 camera is fixed with respect to the platform, the latter combines the roll, pitch and yaw (ω, φ, κ) of the platform at the time of exposure. The nature of bundle-block adjustment with camera self-calibration dictates that the quality of the final photogrammetric model is highly sensitive to errors in both sensor position and orientation as well as inaccurate refinements of the interior orientation parameters (Ebner and Fritz, 1980).

Initially, AT was carried out using the photogrammetry module of Trimble Business Center, v3.40 (TBC), which relies on a simplified implementation of the adjustment process from Inpho UAS Master. Deliverables produced using TBC, however, suffered from severe elevation artefacts which limited their usefulness for further analysis. This is discussed further in Sect. 5.3.

Following the identification of shortfalls in TBC, AT was carried out using Trimble Inpho UAS Master^® v8.0 (UAS Master). UAS Master is a feature-rich photogrammetry package that is targeted to RPAS applications (Trimble, 2015, 2016) and is a comprehensive alternative to software often used in similar studies such as Pix4D or Agisoft Photoscan. The AT solution is initialised by the positional parameters (X0, Y0, Z0) for each photo centre, as provided by the on-board GPS receiver. Relative adjustment is achieved after automatic tie point (TP) collection. TPs are common targets recognised in multiple overlapping images, which allow the relative position and orientation of images within the block to be determined. Subsequent measurement of GCP positions in images enables absolute adjustment. GCPs may be collected manually or automatically via feature recognition. In this case, targets marking GCPs were identified and selected manually. The absolute bundle-block adjustment then concurrently refines the exterior and interior orientation parameters.

The robustness of photogrammetric modelling was assessed by testing several alternate control scenarios, based on the autumn mission when 23 GCPs were placed and measured in the field. The following scenarios were evaluated:

all 23 control points as horizontal and vertical GCPs,

In each scenario, the balance of the control points was provided as check points (CPs). In retaining GCPs, we ensured that the perimeter of the study catchment remained fully constrained within the network. As the number of GCPs decreased, the root mean square error (RMSE) for CPs provided an indication of AT robustness. It was found that as few as 14 GCPs provided an acceptable triangulation across the study area, with some degradation apparent when only six GCPs were used, primarily in terms of z (Table 2). No spatial structure was evident in the distribution of GCP or CP error. This assessment aided the determination of an optimal number of GCPs to minimise the time required to place and survey control points when snow is present in the catchment. On this basis, 14 control points were placed and measured in the field for each of the winter and spring missions. For all missions, the AT from which deliverables were produced utilised all surveyed points as GCPs. A second AT was carried out using a subset of control points as CPs, as shown in Table 3. Thus, the RMSE provided by CPs is expected to be conservative compared to the quality of the deliverables obtained from the fully constrained AT.

Standard deliverables from the photogrammetric modelling included a dense point cloud; a digital surface model, interpolated to 0.15 m spatial resolution; and an ortho-mosaic, resampled to 0.05 m spatial resolution. The DSM and the ortho-mosaic are the principal products for further analysis. Each DSM provides the basis for determining snow depth, while the ortho-mosaics allow for assessment of the snow-covered area, and for snow-free areas to be identified when assessing the quality and repeatability of DSMs between flight missions.

Snow depth was derived by differencing DSM of flights 2 and 3 from the baseline obtained during flight 1 (ref) as follows: dDSMn=DSMn-DSMref. Equation (6) provides a map of difference between the two DSMs, henceforth referred to as the dDSM (after Nolan et al., 2015). Values of the dDSM are considered to represent snow depth, with the associated uncertainty considered in Sect. 3.3.

Since snow depth is determined via DSM differencing as a linear combination of two independently measured variables (Eq. 6), the uncertainty associated with snow depth (SD), measured in the vertical dimension, for each measurement date (n) can be obtained via Gaussian error propagation (James et al., 2012) as follows: εdDSM=εn2+εref2, where ε for each DSM is the elevation error determined from the AT as the RMSEZ value for the set of CPs. Inherent in this simple approach is the assumption that the planimetric accuracy of each constituent DSM has negligible contribution to εdDSM. Calculating εdDSM provides a single estimate of uncertainty assumed to apply equally throughout the map of RPAS-derived snow depth for each date. Under the assumption that errors are normally distributed and bias-free, the RMSEz derived from CPs identifies the standard deviation σz, allowing the 90 % confidence level of z to be determined as 1.65×σZ. In turn, inferences associated with uncertainties for elevation differences, εdDSM, also depend on the Gaussian assumption to provide the 90 % confidence level.

In reality, perfect co-registration between constituent DSMs and the Gaussian assumption are unwarranted. Subsequently, inferences associated with the evolution of snow depth may be compromised due to confidence intervals being conservative or immoderate. Therefore, we use dDSM for snow-free areas to characterise the experimental distribution of errors and assess the validity of the Gaussian assumption in this context.

The approach above provides a means to determine the expected accuracy of snow depth derived from RPAS photogrammetric surveys. In order to validate this estimate, a reference data set of in situ observations was sampled in the field using snow probes, with a nominal precision of ±0.01 m, as described in Sect. 3.1.4. De Michele et al. (2016) assessed the accuracy of RPAS-derived snow depths against snow depth surfaces interpolated from 12 point measurements. This approach, however, may be limited by an inability to accurately resolve the spatial variability of snow depth, as well as the compounding effects of uncertainty associated with the interpolation scheme, particularly beyond the domain defined by the measured points.

Here, 430 measurements of snow depth provided 86 mean reference values, with the standard deviation of each set of five measurements providing 95 % confidence intervals. The aim of this sampling strategy was to assess and account for co-location uncertainty and spatial variability between the RPAS and reference snow depth data sets. Reference snow depths were compared with those from the spatially coincident pixels from the map of RPAS-derived snow depth. RPAS-derived snow depth quality was assessed in terms of residuals and weighted linear regression between reference and RPAS-derived snow depths.

Emergence of snow-free areas at the time of the spring flight facilitated comparison between autumn and spring DSMs on those areas. As the same terrain surface mapped from two independent flights should yield identical DSMs, the residual between them provides a means to characterise the distribution of errors in the photogrammetric processing, which can be readily compared to the assessment made from CPs.

Snow-covered and snow-free areas were segmented using an unsupervised classification of the spring ortho-image using the Iso Cluster classification tool in ArcGIS v10.3. With five output classes, this approach enabled discrimination between illuminated snow pixels, shaded snow pixels, and vegetation and soil-dominated snow-free pixels. Snow-free pixel classes were then grouped to provide a mask within which the distribution of spring dDSM values could be characterized. While this approach relies on the characterisation of repeatability for snow areas, good image contrast and the high overall density of TPs generated across the image block, regardless of the presence or absence of snow, indicates that photogrammetric reconstruction performance should be comparable for both snow-free and snow-covered areas. This is a product of the camera properties, which maintain high dynamic range across scenes of mixed land cover and extensive snow cover. Therefore, this residual represents a measure of the repeatability of the technique for measuring surface height change, including derivation of snow depth.

A primary motivation for exploring the use of RPAS photogrammetry for mapping a snowpack is the ability to resolve fine-scale spatial variability in snow depth. This capability was assessed by computing and comparing the semi-variograms of reference and RPAS-derived snow depths from the autumn flight. While the sample size for reference snow depths remained fixed (n=86), the semi-variogram of RPAS-derived snow depths could be calculated from many more samples. Two random samples were extracted from the spring dDSM map (n=1000 and n=5000), each yielding a semi-variogram capturing the spatial variability of snow depth with increasing detail, which were compared to that of the in situ observations.

Since GCPs are used to solve the photogrammetric model, they do not provide an independent assessment of accuracy. Such an assessment is provided by the CPs, the RMSE of which was of the order of centimetres for all flights (Table 3). Planimetric RMSE (i.e. x and y) was always substantially less than the GSD. Vertical RMSE (z) tended to be about double that achieved planimetrically but never exceeded 0.05 m. The final models were produced from a second and more constrained AT with all surveyed points used as GCPs, thus making the assessment conservative relative to the final products.

While the RMSE of CPs increased for the winter and spring flights, possibly due to a less constrained model, the level of accuracy achieved is compatible with expectations for the determination of snow depth. Additionally, the more tightly constrained first AT reduced the error for the baseline model, in turn contributing a reduced uncertainty in derived snow depths, despite the reduced control for subsequent ATs. For all flight missions, the photogrammetric processing performed well in the correlation of images and the construction of the image block, as indicated in Table 3. Tie point (TP) generation relies on the successful match of unique targets across multiple images, which was achieved despite the complicated contrast over snow. For all flights > 80 000, TPs were generated across the imaged area.

Snow depth was found to be highly variable across the study catchment for both winter and spring (Fig. 3). The midwinter flight mapped near-complete snow cover across the study catchment, while large snow-free areas developed by the spring flight, where snow-covered area was reduced by about one-third (Fig. 3a and b). Where snow was present, depths ranged from less than 0.10 m, typically on exposed ridgelines and broad elevated slopes, to 2 m or more where cornices formed along ridgelines, as well as in gullies. Average snow depth was greater for winter, although maximum depths were comparable between winter and spring. Between winter and spring, considerable ablation was observed. Areas of deepest snow were spatially coincident between winter and spring, with the greatest retention of snow in cornices and gullies. Where shallow snow was present on ridgelines in winter, it was largely lost by spring.

Propagation of errors under the Gaussian assumption, based on the RMSE from each AT, yielded vertical uncertainties for snow depths at the 90 % confidence level of ±0.077 m for the winter flight and ±0.084 m for the spring flight. This one-dimensional approach to error propagation assumes that the planimetric geolocation of individual surfaces, and subsequently the co-registration of surface pairs, does not contribute significantly to the vertical uncertainty.

Comparison of RPAS-derived and reference snow depth yielded a mean residual of -0.069 m, indicating that, on average, reference depths were greater than RPAS-derived depths. Filtering the reference data set to exclude reference measurements that were made in areas occupied by tussock (Chionochloa rigida) vegetation, however, improved the mean residual to -0.01 m (Fig. 4a). The small residual is indicative of good agreement between the two data sets while also indicating that, overall, snow depths measured by probing may be overestimated. Limitations of probing and uncertainty introduced due to the presence of vegetation is discussed further in Sect. 5.2.1.

Good agreement between data sets is further demonstrated in Fig. 4b. Relatively large horizontal error bars accompanying the reference measurements (Fig. 4b) reflect the substantial spatial variability in snow depth measured by probing, even within arm's reach. Substantial departure occurs for reference snow depths between 0.20 and 0.60 m which tend to exceed RPAS measurements. Negative depths in the RPAS-derived data set is a product of co-registration uncertainty, particularly in areas where the surface model represents large vegetation or is influenced by rock outcrops, as well as spurious values from the constituent DSMs. Agreement between reference and RPAS-derived data sets improved with the removal of reference measurements made above tussocks. This filtering saw the R2 value improve by 22 %, while RMSE decreased by 23 % (Table 4). The 1 : 1 ratio line was contained within the 95 % confidence interval of the weighted (bi-square) regression between RPAS-derived and filtered reference snow depths. Some disagreement between RPAS derived and probed snow depths is likely due to the varying areas over which snow depth was sampled by the two techniques, and resulting spatial uncertainty in comparing the two data sets.

The emergence of snow-free areas for the September flight permitted a comparison of height derived on snow-free surfaces between the pre-winter and spring flights (Fig. 5). The small magnitude of the residuals, compatible with errors consistent with the uncertainty of the triangulation CPs, demonstrates the repeatability in the derivation of snow-free surfaces. Furthermore, the absence of any spatially structured trend in the distribution of the residual indicates robust photogrammetric modelling from the RPAS platform. At 0.15 m resolution, the snow-free pixels from the spring mission provided a large sample (n= 5 936 428). The mean residual (bias) detected with respect to the pre-winter DSM was 0.024 m (σ=0.239 m) (Fig. 5).

The set of residuals departed substantially from the Gaussian distribution and was better represented by the Student's t location-scale distribution (Fig. 6): f(x)=Γv+12σvπΓv2v+x-μσv-v+12, where μ, σ and v are the location, scale and shape parameters, respectively. Large kurtosis (calculated k=1956) associated with the histogram of residuals in Fig. 6 shows significant departure from a Gaussian law (for which k=3) of equal standard deviation, σ. The leptokurtic experimental distribution results in a narrower 90 % confidence interval than that estimated under the Gaussian assumption with σ=0.24 m, while the probability of large residuals is larger than predicted by a Gaussian distribution. Overall, the mean residual (μ=0.02 m) and the precision of ±0.14 m (90 % confidence level, calculated from the distribution 90th percentile; Fig. 6) exceeds the uncertainties estimated from error propagation alone (±0.08 m at 90 % confidence level; see Sect. 4.1.1) yet support the suitable repeatability of the photogrammetric modelling. Importantly, the significant departure from a normal distribution shows that assessing the variability from a Gaussian fit on stable targets (±0.39 m at the 90 % level) would significantly overestimate the confidence interval. On the other hand, the 90% confidence interval calculated from the fitted Student's t location-scale is ±0.10 m (Table 5). The significance of this result with respect to statistical inferences is discussed further in Sect. 5.2.2.

The non-Gaussian nature of the residual distribution deserves further scrutiny. Similar distributions have been identified for comparable repeatability assessments of photogrammetric dDSMs used for mapping snow depth (Nolan et al., 2015), but have not been explored in detail. Analysing the variability of the mean and standard deviation of the residual for discrete classes of slope, as well as the kurtosis of the residual distribution, provided insight into the role of terrain. For classes of slope up to 65∘ the mean residual remains within the standard error, before becoming increasingly negative for the remaining classes (Fig. 6). Standard deviation exhibits a similar trend, remaining largely within the overall standard error for slope classes up to 45∘, beyond which variability increases.

The observed pattern in the mean and standard deviation of the residual indicates that larger and more variable errors are associated with steeper slopes. Reduced kurtosis accompanying the error distribution on larger slopes (Fig. 6) reveals a tendency towards a Gaussian distribution of residuals as mean slope increases. Here, for slopes >50∘, kurtosis was reduced below 100, and for slopes >85∘, kurtosis was less than 10, approaching that of the normal distribution. Therefore, the statistical distribution of error, while non-normal, also varies significantly with terrain characteristics, as highlighted by the comparison of the residual histogram for discrete classes of slope (Fig. 7 and Table 6). Subsequently, the overall distribution of residuals (Fig. 5b) is the result of a convolution between non-normal distributions and the hypsometry of the area (i.e. area-elevation distribution).

The semi-variograms for RPAS-derived snow depth, compared to that from the reference measurements, are shown Fig. 8. They exemplify the new insight that high-resolution mapping provides into the spatial variability of snow depth. Both the 1000 and 5000 random point samples captured a comparable structure of spatial auto-correlation with a range of ca. 40 m. The 5000-point sample improved the resolution of the semi-variogram, with an improved signal-to-noise ratio. In contrast, the reference data, despite being demanding in fieldwork, performed poorly at capturing the spatial variability, as most measurements were separated by a minimum distance of 50 m. A lack of spatial auto-correlation in the reference data confirms a posteriori that probing samples could be assumed to be independent of each other, which is desirable for the accuracy assessment. Additionally, it also reveals that probing failed to capture most of the spatial structure of the snow depth field, thus stressing a limitation of this classical method to characterise the snowpack.

Vegetation contributes to uncertainty, particularly when validating RPAS-derived snow depths against reference snow depths. As described in Sect. 4.2.2, the agreement between RPAS-derived and probed snow depths improved substantially when areas of large tussock vegetation were excluded. It is likely that the presence of tussock introduces a bias into the snow depth measurement, whereby a probe may penetrate the tussock foliage, and possibly also a sub-vegetation void, before striking the ground surface. This is similar to the cavity effect highlighted for airborne lidar measurement of snow (Painter et al., 2016), and similar challenges have been documented by Vander Jagt et al. (2015). High-resolution dDSMs, on the other hand, resolve the vegetation surface, and so vegetation height is inherently better accounted for.

As identified by Nolan et al. (2015), photogrammetrically derived snow depths may also be affected by the compaction of vegetation below the snowpack, which may introduce an anomalous signal of surface height change, to the point of returning false negative snow depths. Correcting observed surface height change would not be straightforward, and is not possible with the data acquired within this study. The effects of vegetation compaction are likely to be greatest in the early winter. As grass typically does not rebound until after the complete removal of the winter snowpack, ongoing subsidence of vegetation below the snowpack through midwinter and spring is expected to be minimal. Ongoing future measurement of snow depth via surface differencing (regardless of the source of DSMs) will benefit from the development and incorporation of vegetation compaction and cavity models.

Ultimately, this study suggests that, for areas dominated by tussock vegetation, RPAS photogrammetry may provide a more reliable means of measurement than probing. A lack of knowledge regarding the specific location of sub-snow vegetation when making measurements by probing is likely to provide a systematic overestimation of snow depth (Fig. 4). In the New Zealand context, almost all seasonal snow occurs above the treeline, so the inability of photogrammetry to penetrate the forest canopy is a lesser concern than for the Northern Hemisphere.

In mapping snow depth across a catchment with relatively complex terrain, we have been able to characterise the influence of terrain on dDSM uncertainty. The assumption that error associated with physical measurements is normally distributed and often underpins subsequent statistical inferences. As demonstrated in Sect. 4.2.3, the error associated with the bias between independently acquired DSMs significantly departed from normal and was better approximated by the Student's t location-scale distribution. This extremely leptokurtic distribution of residuals reflects the influence of relatively low frequency, but high-magnitude residuals beyond the probability of the normal law, despite an overall dominance of residuals about and close to the mean. A possible source of large residuals between two DSMs is their relative planimetric accuracy and subsequent co-registration quality (Kääb, 2005). For steep terrain in particular, a horizontal displacement between DSMs could add a component to dDSM uncertainty beyond the vertical accuracy of constituent DSMs. The residual (Δh) between two surface profiles, which are identical but horizontally displaced by 0.5 m, is shown in Fig. 9a. The error introduced to DSM differencing resulting from co-registration uncertainty increases with steepening slope. Maximum residuals coincide with the steepest terrain (near-vertical areas associated with rock outcrops) and exceed 2 m. The sign of the error is aspect dependent, assuming a uniform horizontal displacement.

Consistent with Kääb (2005), the vertical error introduced by a uniform, one-dimensional (e.g. horizontal) offset, is given by the following: Δh=dxtan⁡θ, where dx is the offset between transects (i.e. 0.5 m in this case), and θ is the surface slope in degrees, as seen in Fig.  9b. It is clear from Fig. 9b that where the average slope of target surfaces is low and co-registration quality is good, the error introduced to a dDSM as a product of co-registration will be minimal. Increasing slope and/or co-registration uncertainty is accompanied by increased vertical uncertainty in the dDSM. This relationship is consistent with the findings of Sect. 4.2.3, resulting in the distribution of residuals departing substantially from the Gaussian distribution when the proportion of steep slopes is low. These findings provide context for the effect noted by Nolan et al. (2015), whereby a non-normal distribution of residuals associated with photogrammetric mapping of snow depth was found to narrow further when the area considered was restricted to a frozen lake (i.e. near planar) surface.

Complicating this effect is the fact that co-registration uncertainty exists in two dimensions. Subsequently, it will become dependent on aspect as well as slope (Nuth and Kääb, 2011), with neither possessing a uniform spatial distribution. This effect is expected to be more pronounced with very high-resolution (i.e. sub-metre) surface models due to a greater frequency and magnitude of breaks in surface slope being resolved compared to coarser models. The modification of surface slope for constituent DSMs (e.g. through the addition of snow) further convolves the propagation of vertical uncertainty. Despite this, the leptokurtic observed error distribution indicates that the reliance on statistics that assume a Gaussian distribution of errors will provide an overestimated characterisation of the expected accuracy. Overestimation of uncertainties may in turn affect statistical inferences and the computation of uncertainties on derived parameters.

The convolution of vertical and planimetric accuracy stresses the importance of ensuring a robust AT and the benefits of utilising high-quality ground control. With new photogrammetric platforms leveraging non-metric cameras and resulting image blocks prone to suboptimal photogrammetric modelling (Sirguey et al., 2016), there is a need to be wary of systematic bias or spatial structure in the distribution of errors, which may not be revealed readily by residuals from the AT alone. These considerations are especially important where a relatively high level of precision is required, and the signal-to-noise ratio may be low when assessing relatively subtle surface height and/or volume changes from dDSMs. Utilising independent ATs as the control of co-registration quality, rather than explicitly co-registering DSMs, has the further advantage of simplifying the processing chain from data acquisition to change detection, mitigating against the risk of introducing gross error when co-registering DSMs and avoiding the need for snow-free (or stable) reference areas within the analysis region.