To measure eroded and/or deposited surfaces in our study area, two orthophotos from 1950 and 1964 were used. They were produced by the French National Geographic Institute (IGN) and the Laboratoire Image, Ville, Environnement (LIVE) of the University of Strasbourg; they have a spatial resolution of 50 and 20 cm, respectively. Both are projected in RGF93/CC48 CRS (EPSG: 3948), which is the most accurate projection in this area. Despite the lack of hydrological data in the lower Bruche before 1965, we assume surveys were conducted during moderate to low water, according to the period of the year during which the photos were taken (13 September 1950 and 17 April 1964) and our inspection of the orthophotos.

Active channel is a widely used concept to objectively identify channel boundaries in aerial photographs, regardless of the river discharge. It basically refers to the unvegetated area . Here, active channel boundaries have been digitised by a single user in QGIS at a 1/300 scale. To reliably assess the SV error, we used a 2015 orthophoto as the base image; it was produced by the IGN with a resolution of 20 cm.

In both orthophotos (1950 and 1964) of our study area, spatial variations of geometric error are assessed by an approach similar to that used by . However, because we use orthophotos (which are already co-registered), we must rely on an independent set of ground control points (GCPs), as suggested by . We selected a total of 18 GCPs, including both hard (buildings and canal) and soft (pathway intersections and trees) features (Fig. a). After identification and manual plotting in the 2015 orthophoto, they are identified in both older orthophotos on a 1/200 computer-screen scale. The spatial distribution of GCPs in the study area is rather uniform, though hard edges are restricted to the northern sector (Fig. a).

Local root square error (RSE) is then measured for each of the 18 GCPs, in both orthophotos. Error in x or y corresponds to the Euclidean distance between the two points for x and y coordinates, respectively. SV error is calculated by interpolating local RSE in our whole study area with an inverse-distance weighting (IDW) technique at the original spatial resolution (Fig. ). IDW uses a linear combination of values at specific sampled points. It allocated weights proportional to the proximity of the sampled points to estimate values at unknown locations . We used the IDW interpolation method for two main reasons. First, based on a comparison of five interpolation methods, showed that linear and nearest-neighbour methods reduce the areal extent of large co-registration errors. These methods are thus discarded as they can strongly limit the influence of large co-registrations errors in the estimation of surficial changes. Then, in a comparative study of spatial interpolation methods for producing a digital elevation model from a small set of points that were not spatially uniform, showed that IDW provided better results than spline or kriging. Because of the difficulties of selecting a high number of independent control points spatially uniform over time in archival remotely sensed data, we argue that IDW is a reliable method for interpolating the registration error in our case.

To examine the implications of SV error in lateral migration measurements, we focus on four distinct sub-reaches (Fig. ). Their mean thalweg lengths amount to 530, 380, 700, and 890 m long (upstream–downstream order). They are (1) an extending and narrowing meander, (2) an almost straight (apparently inactive) sector, (3) two alternate meanders (the first one slightly extending and the second one displaying a small cut-off), and (4) a long meander extending at the downstream end of the curve. We selected geomorphologically distinct sub-reaches to evaluate the effect of both different magnitude changes and types of geomorphic processes.

Figure  shows the interpolated SV error in x and y, and the total SV error (ex2+ey2) for the year 1950. The mean value of total SV error in each sub-reach for both years (Fig. ) indicates that sub-reach 4 is affected by the highest (1.32 m) and the lowest (0.61 m) error in 1950 and in 1964, respectively. These values approximately correspond to the range of total SV error reached by the four sub-reaches. Whereas the total SV error was reduced by a factor of 2 between 1950 and 1964 in sub-reach 4, it remained fairly stable in the three other ones, ranging from 0.6 (sub-reach 3) to 1.2 m (sub-reach 1).

An example of variations in measurements of eroded surface through MC simulations is presented for sub-reach 1 in Fig. . The entirety of MC results are available in Appendix A. A large majority of the measurements appear to be randomly varying around and close to the mean value, inside the 95 % confidence interval. Note that having few outliers sometimes greatly extends the maximum range compared to the 95 % confidence interval, especially when very low values occur. For instance, MC simulations for deposited surfaces in sub-reach 1 include an outlier with a value (2.5×103 m2) corresponding to 38 % of the mean measured value (6.8×103 m2).

Mean changes inferred from MC simulations between 1950 and 1964 are presented in Fig. a. Comparison between sub-reaches is allowed by the normalisation of the surficial changes by the respective thalweg lengths of each sub-reach (expressed thus in m2m-1). Whatever the sub-reach, changes in eroded or deposited surfaces are much larger than those associated with erosion/deposition. The latter are either negligible (sub-reaches 1 and 3) or not recorded (sub-reaches 2 and 4). Sub-reach 1 shows the largest migration: eroded and deposited surfaces amount to 9.1m2m-1±4.9% and 12.8m2m-1±3.5%, respectively. By contrast, sub-reach 2 shows the lowest migration: eroded and deposited surfaces amount to 2.1m2m-1±24.7% and 1.8m2m-1±30.8%, respectively. Intermediate measurements are reported in sub-reaches 3 and 4, where they range between 1.4m2m-1±25.3% (deposition; sub-reach 4) and 4.5m2m-1±8.5% (erosion; sub-reach 4). Note that, in these two last sub-reaches, changes in eroded surfaces are at least twice as high as those in deposited surfaces.

The relative percentage of measurement uncertainty in surficial changes is presented both in the 95 % confidence interval (95 % uncertainty; Eq. (4); Fig. b) and in the whole range of measured values (total uncertainty; Eq. (3); Fig. c). As a reminder, the 95 % uncertainty does not take into account the presence of outliers, while the total uncertainty does. The 95 % uncertainty varies from 3.5 % to 43.4 %. These extreme values both occur in sub-reach 1, for the deposited and the eroded-then-deposited surfaces, respectively. The total uncertainty varies from 15.8 % to 52.9 %. These extreme values both occur in sub-reach 4 and 1, for the eroded and the eroded-then-deposited surfaces, respectively. Sub-reaches 2 and 4 both display the same pattern between the 95 % and the total uncertainty, with the uncertainty related to the deposited surface being higher than that related to the eroded one. In contrast, sub-reaches 1 and 3 do not display the same pattern between the 95 % and the total uncertainty. For sub-reach 1 and relative to the uncertainty of the eroded surface, the uncertainty of the deposited surface is higher than the latter only when taking into account the presence of outliers (total uncertainty). For sub-reach 3 and relative to the uncertainty of the eroded/deposited surface, the uncertainty of the eroded surface is higher than the latter only when taking into account the presence of outliers (total uncertainty).

Our results support the first hypothesis: they confirm that orthophotos are affected by a local significant SV error. Within our relatively small (∼6 km2) and flat study area, we interpolated a total SV error ranging from 0.26 to 1.89 m (Fig. ), while mean values of total SV error range from 0.61 to 1.32 m for the four sub-reaches (Fig. ). This emphasises the need to take the SV error into account and, importantly, to assess its impact on the uncertainty of the measured changes , even if the characteristics of the studied reach may appear unproblematic at first glance. Moreover, as orthophotos are used in this study, we draw particular attention to the relevance of this statement in the case of studies using co-registered aerial photographs for similar purposes e.g..

Our results also support the second hypothesis: the SV error greatly affects the variability of MC-simulated measurements of eroded and/or deposited surfaces. Variability of the surficial measurements has been assessed by calculating the relative percentages of uncertainty induced by the SV error through the MC simulations. Whereas the more conservative percentage of uncertainty (total uncertainty) ranges from 15.8 % to 52.9 %, depending on the metric and the sub-reach, the less conservative percentage of uncertainty (95 % uncertainty) still ranges from 3.5 % to 43.4 %. These results highlight the potentially high impact that SV error can have on variability of surficial measurements and consequently on their uncertainty.

When applying the more conservative threshold of significance (50 % of total uncertainty; cf Sect. ), it appears that only one surficial change has to be considered non-significant (eroded/deposited surface in sub-reach 1; Fig. c). However, it can be considered significant when applying the less conservative threshold of significance (Fig. b), because its uncertainty does not reach the 50 % threshold. While this contrast may call into question whether or not the presence of outliers should be taken into account, visual comparison of specific situations may help to unravel this issue. As illustrated in Fig. , only subtle areal and shape differences may be observed between an inlier and an outlier, the latter likely representing a geomorphologically plausible situation. When using MC simulations in this context, we thus strongly suggest inspecting outliers and not systematically rejecting them. When the geomorphological plausibility of outliers is doubtful, we recommend using the total percentage of uncertainty.

Our results partly validate the third hypothesis: the uncertainty of surficial changes depends not only on their magnitude but also possibly on their respective shapes. A contrasted pattern of uncertainty is observed in sub-reaches 2 and 4 versus sub-reaches 1 and 3. Whereas the former seemingly display uncertainties solely related to the magnitude of changes (i.e. higher uncertainties for lower surficial changes), the latter do not (i.e. in some cases, higher uncertainties for higher surficial changes; see Sect. ). It is the case for instance in sub-reach 3, which displays a higher total uncertainty for the eroded surface than for the eroded-then-deposited surface. Yet, as sub-reaches 1 and 3 display more complex geomorphological shapes and channel evolution than sub-reaches 2 and 4 (Fig. b), we suggest that uncertainty of surficial measurements might also be strongly influenced by channel morphology and its evolution through time.

In order to improve the generalisation of tools documenting fluvial planform changes and facilitate the implementation of our new methodological framework, we can summarise the complete workflow as follows (see Fig.  for more details): (1) interpolate the SV error in the study area (as recommended in ), (2) calibrate a normal distribution around nodes to randomly translate these (see Fig. 3 for more details), (3) choose a significance threshold, and visually check the outliers to eventually (4) assess the significance of the measured surficial planform changes. The key step (2) is achieved via MC simulation, which is well known for its simplicity, reliability, and transferability . Simulation outputs allow assessing both the total and the 95 % uncertainties (Fig. ).

We suggest a few practical recommendations when applying the proposed methodological framework. If orthophotos are employed, we strongly advise using an independent set of GCPs for co-registration, bearing in mind that orthophotos are affected by a significant SV error (see Sect. 5.1). As for GCPs, their amount must be high enough and their distribution over the entire study area as homogeneous as possible. As pointed out by , a location of these GCPs close to the river system is highly beneficial. As for the 50 % significance threshold, we recommend applying it in the total uncertainty, as outliers might represent geomorphologically plausible situations (Fig. ). Nevertheless, the few outliers (Fig. ) should still be treated in an empirical manner by visually determining if they should be rejected or not. Further work is required in the near future to deal with this issue in a more automatic way. More generally, when studying historical lateral migration of mid-sized rivers (active channel width <30 m) and/or low-magnitude changes (<1 m2m-1yr-1) with the channel polygon method, we emphasise the systematic need for assessing both SV error and uncertainty, as some of the measured changes might be non-significant (see Sect. 5.1).

This study, though focusing on short sub-reaches of a mid-sized (∼20 m) meandering (single-thread) channel using specific remotely sensed data on a short timescale (archival orthophotos), has great potential for transferability. Firstly, we assume that our methodological framework could be applied to any fluvial system, regardless of its size. Secondly, we likewise argue that it could be relatively easily extended onto an entire river reach by increasing the sub-reach database and/or onto a longer temporal scale by increasing the historical river channel database (Fig. ). As for the size/length of the sub-reaches, we recommend adapting it according to the complexity of the planform changes and/or the channel pattern (e.g. anastomosing and anabranching channel patterns). As for the river channel database, other remotely sensed data, such as co-registered aerial photographs and satellite imagery, or traditional planimetric data (maps) can be easily integrated as well. Thirdly, transferring this framework to other channel patterns represents a promising future research topic. In contrast to the centreline approach e.g., the channel polygon method would actually suit the study of lateral mobility of multi-threaded channels (including anastomosing rivers, which usually are characterised by low lateral mobility), with a robust assessment of the SV error. Unlike this present study, where planimetric changes associated with erosion/deposition are negligible, we might expect a higher proportion of these changes in this kind of fluvial setting. Overall, long-term landscape reconstruction studies could also greatly benefit from the methodology we propose. In particular, works combining multiple diachronic spatial sources (e.g. old aerial photographs, historical and cadastre maps, and satellite images) should draw particular attention because of the possible propagation of uncertainty in the assessment of landscape changes.

We conclude by stating that this study offers promising research prospects. Firstly, a key outcome is the ability of MC simulations to actually detect low-magnitude planform changes in mid-sized river channels. This positive achievement thus overcomes the main difficulty related to the use of classic planimetric methods in such settings , as recently highlighted by , who failed to detect noticeable changes in mid-sized active channels (width < 25 m). Secondly, as for river restoration, our methodological framework should help in constructing robust scenarios of future river management, especially those based on past planform changes e.g.. Thirdly, significance assessment of planform changes can strengthen the studies using surfaces of an active channel as input for sediment budgeting . Finally, while this study, together with and , focus specifically on the SV error and, more globally, uncertainties in planimetric studies in a wide range of fluvial settings, the proposed propagation of geometric error via MC simulations could be extended to other geomorphological contexts where surface extraction from remotely sensed data is involved.