Continuous measurements of valley ﬂoor width in mountainous landscapes

. Mountainous landscapes often feature alluviated valleys that control both ecosystem diversity and the distribution of human populations. Alluviated, ﬂat valley ﬂoors also play a key role in determining ﬂood hazard in these landscapes. Various mechanisms have been proposed to control the spatial distribution and width of valley ﬂoors, including climatic, tectonic and lithologic drivers. Attributing one of these drivers to observed valley ﬂoor widths has been hindered by a lack of reproducible, automated valley extraction methods that allow continuous measurements of valley ﬂoor width at regional scales. Here we 5 present a new method for measuring valley ﬂoor width in mountain landscapes from digital elevation models (DEMs). This method ﬁrst identiﬁes valley ﬂoors based on thresholds of slope and elevation compared to the modern channel, and uses these valley ﬂoors to extract valley centrelines. It then measures valley ﬂoor width orthogonal to the centreline at each pixel along the channel. The result is a continuous measurement of valley ﬂoor width at every pixel along the valley, allowing us to constrain how valley ﬂoor width changes downstream. We demonstrate the ability of our method to accurately extract valley ﬂoor widths 10 by comparing with independent Quaternary ﬂuvial deposit maps from sites in the UK and the USA. We ﬁnd that our method extracts similar downstream patterns of valley ﬂoor width to the independent datasets in each site. The method works best in conﬁned valley settings and will not work in unconﬁned valleys where the valley walls are not easily distinguished from the valley ﬂoor. We then test current models of lateral erosion by exploring the relationship between valley ﬂoor width and drainage area in the Appalachian Plateau, USA, selected because of its tectonic quiescence and relatively homogeneous lithology. We 15 ﬁnd that an exponent relating width and drainage area ( c v = 0 . 3 ± 0 . 06 ) is remarkably similar across the region and across spatial scales, suggesting that valley ﬂoor width evolution is driven by a combination of both valley wall undercutting and wall erosion in the Appalachian Plateau. Finally, we suggest that, similar to common metrics used to explore vertical incision, our method provides the potential to act as a network-scale metric of lateral ﬂuvial response to external forcing.

widening in an active reach in Taiwan was concentrated along reaches where the channel was curving, and limited where it was straight, suggesting that abrasive particles, directed at valley walls, were the main driver of the widening process.
The role of tectonics is not included within simple scaling relationships between valley floor width and drainage area, and yet rates and spatial patterns of uplift have been shown to correlate with valley floor width changes in tectonically active regions. 95 Non-uniform patterns of uplift affect channel slopes: faster flow in steeper channel reaches results in the channel occupying a smaller cross section, suggesting that channel slope and therefore uplift should be a key control on valley floor width (e.g. Finnegan et al., 2005). Whittaker et al. (2007) investigated the Rio Torto in the central Italian Apennines which crosses an active normal fault. They found that the ratio between the channel width to the valley floor width increased directly upstream of the fault strike as the wide, partly alluviated upstream valley transitioned to an incised gorge. To take into account the influence 100 of channel slope on valley floor width, Brocard and van der Beek (2006) suggested an alternative model of valley floor width evolution that assumes that valley floor width is set by the frequency of strath terrace erosion in transport-limited reaches.
They suggest that valley widening occurs when erosion is high enough to rework all the alluvial fill in the valley down to the bedrock strath underneath. The frequency of erosion of the strath is set by the ratio of current erosion in the reach, which is transport-limited, to a hypothetical maximum erosion which is detachment-limited. This model is consistent with the findings 105 of Cook et al. (2014), who showed erosion of valley walls in an active gorge in Taiwan slowed to almost zero subsequent to deposition of alluvium on the valley floor.
Alongside tectonics, the glacial history of a region is an important parameter that may control valley floor width. In postglacial landscapes, valley floor width may be preconditioned by prior glacial erosion leading to valleys which are much wider than the active channel width. Glacial landscapes may also have complex patterns of meltwater discharge and sediment supply 110 (e.g. Dadson and Church, 2005;Brardinoni et al., 2018), as well as base-level changes which have been shown to influence upstream patterns of valley floor width (e.g. Gran et al., 2013). Current models of fluvial erosion and lateral migration are not generally developed with high-latitude, post-glacial systems in mind, despite their prevalence over large regions of the Earth's surface.
Proposed models of valley widening have made various testable hypotheses about how the width might vary as a function of 115 environmental factors. For example the models of Langston and Tucker (2018) suggest that different valley widening processes result in different values of the exponent c v , and the model of Brocard and van der Beek (2006) suggests that topographic gradient should influence the width of valley floors in a predictable manner. Cook et al. (2014) suggested that the width of valley floors should increase where rivers have higher curvature, on the basis of oblique particle collisions with valley walls. Lancaster (2008) hypothesised that width will increase in areas with greater probabilities of debris flow sources in headwater 120 areas. In any of these cases, testing of hypotheses relies on accurate measurements of the width of the valley floor. Here we concentrate our efforts on a method that allows us to reproducibly measure the width of the valley floor, and use this method to explore the value of c v , which has been linked to lateral erosion process, in a landscape where tectonics and lithological heterogeneity are unlikely to play a role. Beek, 2006) or digital elevation models (e.g. Gran et al., 2013;Schanz and Montgomery, 2016;Langston and Temme, 2019).
These methods provide good constraint on valley floor width over small scales, but the time involved in either collecting field measurements or hand-mapping widths limits our ability to collect these measurements over large spatial scales. Measurements (2020) presented a curvature-based approach, in which valleys are identified using the scale at which the principal curvature is minimised across a valley cross section. They found this method can distinguish between V and U-shaped valleys, but noted widths extracted using their method are generally under-predicted compared to manual measurements.

Automated extraction of valley floor widths
We build on the technique for automatically identifying floodplains and terraces presented in Clubb et al. (2017). This method 160 identifies floodplain and terrace pixels by calculating two metrics for every pixel in the DEM: the elevation of the pixel compared to the nearest channel, and the local slope. The channel network for the DEM is defined by extracting channel heads using the techniques outlined in Clubb et al. (2014), and flow routing using a steepest descent algorithm from each channel head. For each pixel, these elevation and slope metrics must both be below a defined threshold to be identified as part of the floodplain or terrace (i.e. a pixel must be relatively flat and near to the elevation of the modern channel). Thresholds for 165 elevation above the channel and local slope can either be calculated statistically using quantile-quantile plots of the distribution of slope and elevation across the landscape, or they can be set manually by the user. The second option is recommended in lower-relief landscapes where there is less contrast in slope and elevation between surrounding hillslopes and the valley floor.
This leads to an integer classification of the raster into valley floor pixels (1), channel pixels (2), or non-fluvial pixels (0).
In some landscapes, particularly those with significant anthropogenic modification of the valleys floors or noise in the 170 topographic data, the initial extraction of the floodplain leads to isolated holes of non-valley floor pixels along the valley. This can confound the extraction of automated valley floor width. We therefore include an option to fill in these holes in the valley floor using a flood filling algorithm implemented within OpenCV (Bradski, 2000).
Following extraction of floodplain or terrace pixels, we extract the longitudinal profile along which valley floor width will be measured. This is done using one of two methods. For regions where the river does not significantly meander within its valley, 175 or for coarser resolution DEMs where flow paths tend to be less variable, we extract the steepest descent flow path from a userdefined upstream point on the channel to the outlet of the DEM. However, in some cases (such as where the river meanders within its valley), the steepest descent path may not align with the overall valley trend. In this case, valley floor widths can be over-estimated if extracted from the steepest descent trace. We therefore provide an option of extracting the valley centreline from which to determine valley floor width (Figure 2).

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Our method of determining the valley centreline is based on creating an artificial V-shaped valley using the topography of the identified floodplain. We begin by ingesting the valley mask and calculating the distance to the nearest valley edge pixel for every pixel identified as belonging to the valley. For brevity we will call these "bank" pixels. The algorithm does this by searching a widening radius until it finds the nearest pixel not in the mask. For every pixel in the valley mask, the elevation of the nearest bank pixel is recorded along with the distance to the bank. We then run a moving window over these valley pixels 185 to find the minimum bank elevation within the window. The moving window radius is set to be on the order of the width of the valley, so the resulting pixel values are entrenched relative to the original bank elevations. We then subtract elevation from this minimum bank elevation mask: the elevation subtracted is calculated by multiplying the distance from the bank with a scaling factor. This means the largest subtracted values are in the middle of the valley, and the valley forms a trough that is roughly triangular in cross section. We then overwrite the valley in the original DEM with this "trough" mask. This is then carved and 190 filled repeatedly, using the algorithm of Lindsay (2016), with the trough subtracted on each iteration until there is a single carved centreline. We favour this approach to, for example extracting a floodplain skeleton using computer vision techniques (e.g., Saha et al., 2016), because our method preserves flow routing information of the valley centreline pixels.
We then move down the centreline using D8 flow routing. We define a pixel spacing, n, so that for any given pixel we select a point n pixels upslope along the centreline and n pixels downslope. Two bearings are calculated: the bearing of the vector 195 starting at the upslope pixel and ending at the valley pixel in question (b u ), and the bearing of the vector starting at this valley pixel and the downslope ending pixel (b d ). The channel pixel bearing (b) is then calculated with: All bearings are calculated clockwise relative to North. We then calculate the vector orthogonal to this bearing. We follow the vector orthogonal to the valley centreline towards both the left valley wall and the right valley wall, and calculate the width

Weardale, England, UK
We then tested our algorithm on a more complicated system with preserved fluvial terrace deposits as well as modern alluvium.
We analysed valley floor width along the Upper River Wear in Weardale, a valley with extensive glacial deposits, alluvium and 245 river terraces in the North East of England. Weardale is notable for not being as nice as neighbouring Teesdale. The River Wear is sourced in the English North Pennines, and flows east to the North Sea. Alongside distinguishing between glacial or fluvial origin, the Quaternary deposit maps also separate alluvium and fluvial terrace deposits (Figure 3). We extracted valley floor widths automatically from the 2020 lidar DTM from Weardale compiled by the Environment Agency, which we resampled to 2 m resolution.

Russian River, California, USA
We then tested the method in a downstream reach of the Russian River, Northern California where a 1:24,000 geological map was available. This site allowed the evaluation of the algorithm's effectiveness in a non-glaciated landscape consisting of Quaternary alluvium and fluvial terrace deposits. We followed the same approach as for the BGS Quaternary maps, classifying alluvium and terrace deposits as 'fluvial' and any other Quaternary sediments as 'non-fluvial' (Figure 8). We then ran the 285 algorithm using the automated valley extraction and by ingesting the USGS Quaternary maps. Figure 8 shows that there was a very good agreement between the widths extracted from the automated and USGS-derived widths, with a mean width of 348 ± 136 m estimated from the automated technique and a mean width of 417 ± 192 m estimated from the USGS maps. ically inactive, has relatively homogeneous geology consisting of shallow dipping Carboniferous sediments, was unglaciated through the Quaternary (Sugden, 1977), and has a humid, temperate climate (Phillips et al., 2010). Our hypothesis is that in this relatively homogeneous landscape, we should find that valley floor width can be well-approximated as a power law of drainage area following Equation 1, and that the values of c v and K v that can be identified from this power law should be 305 reasonably consistent across different channels. To test this hypothesis we focused on ten valleys which are tributaries of the South Fork and Middle Fork Kentucky Rivers, which form part of the Ohio River Basin (Figure 10a). These valleys range in drainage area from approximately 7 -79 km 2 . We derived DEMs from the USGS 3DEP data at 1 m resolution for nine of the valleys, and at 2 m resolution for Bullskin Creek due to its larger catchment area. We extracted the valley centrelines and valley floor widths for each catchment using the same methods as for previous sites. We extracted drainage area along the steepest    Figure 10. cv and Kv are calculated from power law fits to width and drainage area along each channel.
Normalised Kv is calculated using a reference value of cv = 0.21 for valleys 1 -10 and cv = 0.3 for valleys 11 -15.