The study site is the Kamech catchment, located in Cap Bon, a peninsula in north-eastern Tunisia (Fig. a).

Kamech is one of the two catchments of the OMERE long-term hydro-meteorological research observatory (http://www.obs-omere.org, last access: 29 May 2018). A detailed description of the Kamech catchment can be found in , and . More than 70 % of the catchment area is ploughed and cultivated with rainfed crops. The climate is semi-arid to sub-humid with a mean interannual rainfall of 650 mm and a long dry summer season from May to October. The elevation ranges between 80 and 160 m. The slope can locally exceed 45∘. The substratum is mainly composed of marl and clay intercalated with sandstone layers. These layers have an average south-eastern dip of approximately 30∘ corresponding to the global anticline of Cape Bon. The right bank of the catchment shows a natural slope generally parallel to this dip and mainly presents marly layers. Hence, most gullies of the area have developed on this side. Sandstone outcrops are visible on the left bank of the catchment (Fig. b). The soils have a sandy-loam texture with depths ranging from zero to more than 2 m depending on the location within the catchment and local topography. The drainage network is composed of several kilometres of wadi and gully sections with decimetre to pluri-metre widths. The network drains intermittent flow discharge into a reservoir of 140 000 m3 built in 1994 that silts up at an annual rate of 15 t ha-1 because of water erosion . The gullies are permanent, and the gully heads are located at the edge of the agricultural fields. There is no significant ephemeral gully in the sense of or .

To ensure photogrammetric image acquisition of several square kilometres, the method is based on the following hypothesis: with a very stable kite as a payload carrier, the position of the camera remains stationary relative to the kite operator. With this hypothesis, the flight path (i.e. the kite coordinates) is then a translation of the operator's course. Moreover, to use the simplest and most reliable apparatus, image acquisition is automatically triggered at a pre-set time interval. The flight plan can hence be prepared prior to the survey itself and followed on the ground without any need for remote control of the platform or a radio link between the camera and the operator. Thus, this hypothesis and the conditions ensuring its validity have to be carefully verified. This verification has been done with two complementary approaches, namely field observations and numerical simulations, which are described in the two following subsections.

Image acquisition was performed in September 2013 after the dry season, when vegetation cover was minimal. The equipment used for photogrammetric acquisition is shown in Fig.  above. The Dyneema^® kite line was graduated every 10 m for the first 100 m and then every 50 m with a simple colour/thickness coding system with a comparable approach to that used by . Image acquisition was performed with the 10 m2 kite. A maximum flight altitude of 500 m was chosen to acquire images with decimetre ground sampling distance. The corresponding line length was estimated with the worst case for the flying angle (50∘) and resulted in a maximum line length of 600 m. The targeted area was covered with parallel flight lines. These lines were oriented north-east to south-west along the global orientation of the catchment. The corresponding ground path was walked from the right bank towards the left bank. To simplify the field work, the operator remained at first on the same path near the right bank crest and unrolled different line lengths (150, 360 and then 600 m) so that the kite was positioned at the right downwind distance from the operator. Then, the operator continued to walk the rest of the ground path towards the right bank and covered the targeted area as planned. Images were taken with a Sony NEX-5N camera (Fig. b), which has a 16 Mpix 23.4 × 15.6 mm sensor. This camera was used with a fixed 18 mm focal length, and the image stabiliser was disabled, which are two important settings for the lens auto-calibration step in SfM processing. This camera was chosen as the best compromise at the time of the experiment between mass, suitability for photogrammetric analysis and cost (see Table  in the Appendix). Automatic triggering was performed with a gentLED-Auto 05C intervalometer (Fig. a). A time interval of 5 s between each image was chosen to ensure sufficient overlap.

Complete coverage of the targeted area in the Kamech catchment was achieved within two flights of 3 h each. A total of 752 images were used to cover an area of 3.18 km2. The maximum flight altitude of 500 m led to a maximum estimated ground pixel size of 0.13 m (see Table  for a summary of all these data). The upstream part of the catchment being crossed by a power line, we avoided having the kite line near it for safety reasons. As a result, a small area of the catchment was not covered by multi-view imagery. However, more of the area downstream and outside the catchment was reached. As a result, the data set covered an area of 3.18 km2, which exceeded the area of the catchment itself (2.63 km2).

Finally, eight points (cross marks on Fig. c) that were clearly visible in the kite images were used as GCPs. Their position was measured with a Topcon GR-3 RTK DGPS with a theoretical altimetric and planimetric accuracy of 1.5 cm. These GCPs were used as a spatial reference in the photogrammetric processing.

Kite images were processed with MicMac open-source software . This software implements both a bundle block adjustment and a hierarchical, true multi-view dense matching algorithm that is also used by the French Institut Géographique National to produce 3-D cartography. MicMac hierarchically computes multi-view dense matching from coarse grids to the full resolution by gradually refining the results at successive scales. The full resolution of the DEM is the average ground resolution of the images, which is estimated from the average flying height. This average flying height itself is estimated from the mean flight altitude and the average altitude of key points computed with the SIFT algorithm . All images covering the same point of interest are taken into account in the same bundle adjustment for the calculation of each point in the DEM. This procedure results in an altimetric precision of 1 pixel on average. The MicMac process is typical of SfM + MVS algorithms (see Appendix) and is comparable with them (see for instance and ).

The SfM step (i.e. SIFT point recognition and matching plus bundle calibration) is completely automatic and followed by two manual steps. First, the area for dense image matching was selected. Second, the GCP positions were manually digitised in the images to give the project a cartographic reference. The automatic MVS dense image matching was finally run and resulted in a 0.20 m DEM. All image processing was performed on a laptop computer with an Intel Core i7-3840QM CPU at 2.80 GHz and 32 GB of RAM.

Similarly to , a gully is considered in this study to be a morphological object with a marked depression that is in the immediate proximity of a channel, the latter being determined by another algorithm. To delimit depressions, most recent studies use sliding windows. For example, use a sliding “normalised elevation” kernel. We chose another approach: we convolved the DEM with a Gaussian kernel by computing the inverse Fourier transform of the pointwise product of the Fourier transforms of the DEM and the Gaussian kernel. This method has two advantages. The first relates to computation time: with the Fourier transforms, the algorithm has a computational complexity of O(n log(n)), with n being the total number of pixels of the DEM. Sliding window algorithms have a computational complexity of O(n.m), with m being the window size in pixels. Hence, convolution with Fourier transforms is faster than filtering with sliding windows, except for very small windows. Above all, the processing time with convolution is independent of the kernel size. The second advantage is as follows: convolution by a Gaussian kernel simulates diffusive processes. Hence the DEM after convolution represents the hypothetical future shape of the ground surface after the processes involved in linear erosion have stopped and the processes leading to the healing of the gullies have begun.

For the delimitation of the channel network, the fully automated algorithm proposed by was tested at first (results not reported here). With this algorithm, the automated localisation of gully heads detected by high positive plan curvatures presented flaws. We observed that different threshold values – including the proposed default value – resulted either in an excessive number of missing gully heads or in categorising many anthropogenic depressions as gully heads, such as village streets or spaces between trees in orchards. As noted by , the automatic detection of channel heads is indeed most problematic for small-scale features such as some of those targeted by our work. Thus, gully heads were digitised from a shaded view of the DEM with the same type of expertise as one would use in the field. This approach was used by to produce their validation data set. The entire digitisation of the gully heads on the DEM was achieved within less than 2 h. Once the gully heads were digitised, the algorithm followed the flow chart in Fig. .

The raw DEM (Fig. 3a) was convoluted with a Gaussian kernel (b) of a standard deviation of 10 m, which resulted in the smoothed DEM (c). We chose this value so that twice the standard deviation of the kernel was equal to the width of the largest gullies to be detected (i.e. 20 m). The raw DEM (a) was subtracted from the smoothed DEM (c) to create a depth map (d), which was therefore the estimated depth of the natural surface below the smoothed surface. Step (e) consisted of applying a threshold to the depth map and cleaning the result up. The threshold was chosen as slightly larger than the pixel size considering that lower differences in elevation would probably be noise. Features that did not show depths greater than 25 cm were hence discarded. The cleaning consisted of pruning out patches with volumes less than 1 m3. This value allowed us to eliminate small-scale noise while keeping each detail of the gullies, even when they were made of discontinuous patches. Step (e) resulted in the (f) map. Steps (a) to (f) are illustrated with a section view in Fig. .

Steps (g) to (k) correspond to the extraction of the hydrological network. To map the hydrological network downstream of the previously digitised gully heads (g), a depression-free DEM (i) was generated from the raw DEM by filling gaps (h). The hydrological network (j) was generated by a steepest descent algorithm in (i) from gully heads (g). Considering the typical width of the gullies at the test site, a binary map (k) of the areas located less than 15 m from the network was computed. Intersecting the binary maps (f) and (k) resulted in the final gully map (m).

Figure  shows the results of kite line shape simulations with different wind speeds, line characteristics and physical processes taken into account.

This figure revealed the following three findings: (i) with light and thin lines, the kite line is almost straight, and the flying angle is maximal; (ii) when the kite is flown in sufficiently strong wind, wind speed variations cause only small effective flight angle variations; and (iii) the latter observation is all the more true when the kite line is thin and light. These conclusions corroborate the field observations, which made us choose a thin and light line for photogrammetric acquisitions. Using a thin and light kite line – and a kite adapted to the actual wind conditions at the time of image acquisition – is hence a key condition for obtaining a steady flight angle and the required stable position of the kite relative to the operator.

Figure  shows the simulated flight angle as a function of the line length for the Dyneema^® line and polyester line.

For both cases, the simulations showed that the flight angle dropped with increasing line length. The drop was slight for the Dyneema^® but critical for the polyester line due to the stronger “banana shape” of the line observed in Fig. . Hence, the use of thin and light kite lines such as Dyneema^® lines allows for kite flights with a steady flight angle at a given line length (Fig. ) but also with various line lengths (Fig. ). This steady flight angle makes the line length the only factor influencing the variation in the kite position relative to the operator. As a consequence, kite flights can effectively be planned and then properly realised. However, it is recommended to use a margin of security, considering the slight drop in flight angle for greatest line lengths. These findings were confirmed by the field experiments presented in the following section.

Figure  shows the measured effective flight angles for the two kites used with the Dyneema^® line for different line lengths and different wind conditions.

This figure corroborates the simulation results shown in Fig. : during field experiments, the flight angle dropped slightly but significantly with line length. This drop must hence be taken into account in preparation for image acquisition. We also noted that the smaller kite – which has a tail – flew at a significantly lower angle than the larger one. These experiments also included a flight (the leftmost blue box in Fig. ) with insufficient wind strength to fly the 10 m2 kite, which resulted in a wider range of flight angles and greater variability in the camera position relative to the operator. This result confirms that even with a thin and light line, the kite must fly within the appropriate wind conditions so that the flight angle remains steady.

The quantitative assessment of the DEM quality is reported in Table .

The error statistics demonstrated good agreement between the kite DEM and the independent DGPS-surveyed validation data set. In particular, the mean error and median error were smaller than the pixel size and the standard deviation of the error was of the order of the pixel size. These figures show that the DEM acquired by kite constitutes a reliable model of the catchment topography.

Moreover, a qualitative assessment of the kite DEM was carried out with a manual inspection of full-resolution DEM shaded views with three close-ups (Fig. ). The assessment showed that the kite DEM planimetric and altimetric resolutions allow for the visual detection of numerous landscape features, including most man-made structures (roads, tracks, buildings) and gully heads that were identified in the field (e.g. Fig. a and b). The plot locations and limits were also clearly depicted (Fig. a). Indeed, the boundaries between two separate adjacent plots are not exposed to tillage erosion and finally form small humps that are visible in the DEM. In the main thalweg (Fig. c), marks of regressive erosion were visible, and headcut locations could easily be identified. The potential of the kite DEM for use in extensive gully mapping within an area of several square kilometres is quantitatively evaluated in the next section.

An assessment of the gully network delineation was conducted at the scale of the whole channel network, and the 3-D restitution of the internal morphology of a gully was assessed at the scale of a single gully. Figure  shows the final gully map obtained by the proposed method superimposed on the shaded DEM and a map of the validation results. Statistics of the comparison between the network extracted from the DEM and the field reference network are presented in Table .

The analysis of the gully map and the statistics showed a very good agreement between the detected gullies and the field reference. The overall accuracy was 74 %, with 8 % overdetection and 26 % of the network length unmapped. An inspection of the comparison map showed that most gullies that were not detected by our algorithm were located on the left bank (i.e. the south-eastern half), where gullies are less incised than those located on the right bank. Furthermore, most overdetections (gullies found by our algorithm but not surveyed on the ground) consisted of small gully segments mainly located on the right bank of the catchment.

Next, validation of the 3-D gully map was performed at the local scale and at very high resolution. Figure  shows a 3-D comparison between a gully modelled by the kite DEM and dense measurements from a total station survey.

Comparison of the profiles extracted from the kite DEM and from the surveyed DEM showed good agreement along the whole profile, except for areas covered by vegetation. The influence of vegetation was clearly detected (Fig. c), with elevation differences significantly differing from the surrounding noise. These observations were supported by the associated error statistics (Table ), with a mean error of +0.08 m, which decreases to +0.002 m when not taking into account zones covered by vegetation (parts of the profiles surrounded by vertical dashed lines in Fig. b and c). These results indicate that the kite DEM constitutes a reliable source of topographic data for the description of gully erosion forms. These findings seem likely to be extended for other gullies considering the good accordance between error statistics shown in Tables  and .

Moreover, a comparison of the quartiles estimated from the whole gully and from the non-vegetated part of the gully showed that vegetation mainly resulted in larger positive extrema, whereas the first, second and third quartiles remained comparable. The results at this scale were very similar to those computed with the 469 ground points sampled near the lake (Table  above), with a standard deviation of the error for the DEM statistics being closer to that for the non-vegetated case. This result may indicate that vegetation is more likely to result in local errors rather than in a global deviation.

Our study showed that achieving coverage of several square kilometres with decimetre resolution and accuracy was possible with basic equipment for the acquisition of a photogrammetric data set. These results represent an improvement over those presented in , where the same acquisition method was also used successfully over an area of only one-tenth of that covered in this study. In other works that obtained DEMs with kites, the maximum areas covered were also of the order of several hectares (; ; ; ; ). Our results clearly constitute an extension of the kite's capability. In particular, our work presents novel findings on the conditions that must be met to make kite photogrammetric acquisition successful at this scale. A correct realisation of a planned flight is hence a critical issue for tethered platforms, as has been noted by others (; ). Numerical and field experiments have revealed that the choice of kite line was a key factor in the success of our workflow. To the best of our knowledge, the importance of the kite line for a proper photogrammetric acquisition has rarely been considered with both numerical and field experiments in previous works. stressed out the importance of a thin and light line on the basis of personal and external e.g. empirical observations and advised the use of Dyneema^® to mitigate line sag. Our study hence corroborated previous empirical observations and provided new insights on the importance of the kite line, with original numerical experiments.

If kites are proven to be valuable platforms for photogrammetric acquisition, they have some limitations. The two main limitations are (i) the fact that the line must be clear of obstacles and (ii) the need for a minimal wind speed. We faced the first issue in the most upstream part of the catchment because of a power line. Obstacles can also be found in densely vegetated or densely urbanised areas. Cases of obstruction have also been discussed by , who concluded that not every place is suitable for performing image acquisition from tethered platforms. The second issue, also noted by , can be approached as in , who used a kite to which a small helium blimp was added. used kites and balloons in alternation. In our opinion, in most cases, when the use of RPASs is not hampered by local regulations, kites associated with small-format multi-rotor RPASs represent a relevant all-weather solution. Indeed, the great advantage of small RPAS systems, in addition to being fairly inexpensive platforms, is the fact that they can provide very high-resolution spatio-temporal data with reduced response times in varied conditions . However, typical small-format RPASs may remain grounded during windy periods, thus preventing the requested rapid response. The other main niche for kites is related to local regulations, either for the flight itself or due to regulations regarding crossing borders with the equipment.

On another note, kites can fly for hours when weather conditions are appropriate. This autonomy represents a completely different paradigm to that for most RPASs. With the equipment presented above, the overall autonomy was only limited by the internal power supply of the camera. Within a single flight of 3 h, several thousand overlapping images can be acquired. This figure gives an idea of the mapping potential of this method, which produces DEMs in the gigapixel range.

Beyond the ability to acquire 3-D data over several square kilometres with kites, our aim was to demonstrate that these topographic data were reliable. We showed that the estimated altimetric bias was lower than the pixel size and that the estimated deviations were in the order of magnitude of pixel size. In other words, validation with an independent ground survey showed that the produced DEM had a decimetre resolution and accuracy.

Few works using kites have assessed the DEM elevation error. Our results compare quite well with these works. achieved a +0.13 m mean error, 0.36 m standard deviation of the error and 0.75 m maximal error on a 0.25 m resolution DEM (one thousand validation points). acquired images with an estimated 0.01–0.02 m ground sampling distance. The authors obtained a -0.01 m mean error and 0.065 m standard deviation error estimated with 399 independent validation points. acquired images with a 0.03–0.08 m ground sampling distance and computed a 0.04 m resolution DEM of a test site of roughly 10 ha. They estimated x, y and z errors with 61 independent validation points and found a mean error of 0.04 m and RMSE of 0.16 m on the altitudes. computed a DEM with a ground sampling distance of 0.06 m. A quality check with 176 independent validation points resulted in a mean error of +0.04 m and a standard deviation of 0.07 m. Finally, with 0.004 m resolution images acquired on a 50 by 150 m area and a final DEM ground sampling distance of 0.05, estimated the mean error at -0.019 m and the standard deviation at 0.055 m with 86 validation points. Our work confirmed that the range of kites can be extended to several square kilometres with decimetre resolution while maintaining the accuracy in the pixel size range.

Although the overall accuracy of 74 % proved that our method was effective, the very process of validating the gully maps obtained from high-resolution DEM processing raises issues. To begin, validation methods are quite varied in the literature. Most authors (; ; ; ) have used manual digitisation of the gullies on the DEM as validation data and focused on different gully characteristics: width and depth , visual comparison (; ), and areal and volume difference (; ). Some authors (e.g. ) did not even validate the gully mapping results. Infrequently, studies such as have used field surveys as validation data for gullies that were automatically mapped from a DEM. Similarly to their study, our validation data were in the form of a channel network, and we used the same indicators as they did. We obtained a false positive rate (overdetection) of 8 % and a false negative rate (underdetection) of 26 %. Our results compare favourably to those of , who had false positive rates ranging from 5 to 16 % and false negative rates ranging from 29 to 55 %. Moreover, our results follow the same tendency, with false negatives rates being higher than false positives. This result may be explained by the fact that all gully mapping algorithms, including the one presented here, are based on the morphological characteristics of gullies (i.e. what a gully is) but do not benefit from characteristics that are known not to be shown by gullies (i.e. what a gully is not). In our opinion, this approach would be especially useful for avoiding confusion between gullies and man-made structures, which may be among the most delicate features to handle. This confusion may indeed explain some of the remaining inaccuracies we observed, and more generally, these issues have also been faced by others .

The detection of gullies in DEMs faces the difficulty of determining an unambiguous and generic definition of what a gully is. indicated that to their knowledge, no one has yet assessed where gullies “begin” in the transverse direction. Conversely, stated that “gully edges are the critical features for gully mapping”. noted that the assumptions usually used in channel-like extraction techniques do not apply to the environment of alluvial fans in which they propose an ad hoc gully mapping method. In brief, due to the variety of gully shapes and the fuzzy definition of their extent, each gully mapping algorithm in the literature so far requires the manual tuning of parameters and/or thresholds and is preferably applied to specific landscape types.

A possible workaround would be the use of multi-scale analysis, which was still seen by as a future research direction for high-resolution topography analysis. For future work in this direction, our algorithm has the advantage of being based on Fourier transforms instead of sliding windows, which makes the computation time independent of the characteristic size of the kernel and hence opens the door to multi-scale filtering with controlled computation times. Computing time is indeed one issue for DEM processing: for instance have been unable to process the full resolution of their largest DEM. Considering that upcoming topographic data sets will probably be more extensive and will have higher resolutions, this may still be an issue that will have to be mitigated by algorithmic improvements such as the one we have proposed.

The interest in multi-scale approaches is as strong as the 3-D information of such DEMs is rich, which is the case for the data obtained in our study. The comparison of dense elevation profiles between the kite DEM and ground reference hence showed good agreement. These findings thus confirmed, at a very local scale, the results on DEM accuracy found at the scale of the whole DEM. However, this detailed analysis raised the issue of vegetation cover. This issue is present in several classical cases where image-based approaches have limits that lidar does not have. However, with big image data sets, SfM + MVS DEMs can reach densities that are comparable to or even exceed those of aerial lidar point clouds. This property would allow for the development of vegetation filtering algorithms tailored to these dense and multi-view image data. Second, our results can be compared to the work of , who used kites and balloons with a focus on two gullies. They determined that gully morphology and even gully changes could be assessed. In our case, with comparable sensibility and data available at the scale of the whole catchment, this morphological information would enable the description of different processes that occurred in different gullies or even at different times in the same gully, such as renewed erosion in older gully systems. Further work may then repeat our experiments to monitor ongoing gully erosion processes. These experiments would indeed be of great help, for instance, in understanding the source of the sediments responsible for reservoir siltation.