Conceptually, we might expect the width of a river valley floor in an upland landscape to be controlled by the ratio of vertical to lateral erosion. suggested that lateral erosion will become more important than vertical erosion when sediment supply nears transport capacity. Through time, changes in slope, discharge, or sediment supply may result in increasing vertical erosion rates: in this case, the active channel will incise, leaving behind an abandoned bedrock surface with a veneer of alluvial sediment referred to as a strath terrace . suggest that, intuitively, valley floor widening can only occur when the channel is in contact with the valley wall. If we imagine a valley floor that is wider than the active channel, then we can hypothesise that the rate of valley floor widening will depend on the ratio of the active channel width to the valley floor width. If the active channel width is close to that of the valley floor width, there will be a greater probability of the channel impinging upon the valley wall, whereas if the channel width is small compared to the valley floor width, lateral erosion will occur less frequently. This ratio can change in one of two ways: either by widening/narrowing the valley or widening/narrowing the active channel.

Empirical studies of valley floor width in bedrock systems have suggested a simple power-law relationship between valley floor width and drainage area, as many workers have reported a relationship between valley floor width, lithology, and discharge e.g.. This relationship takes the form 1Wv=KvAcv, where Wv is valley floor width, Kv is a coefficient describing the influence of lithology on valley widening, A is drainage area, and cv is an exponent describing how quickly valley floor width changes with drainage area. The use of the term “bedrock systems” requires some explanation. In studies using Eq. (), authors did not necessarily look for a strath terrace: the valley floor away from the channel could be alluviated. However a valley flowing though hills must, at some point, have incised, so bedrock incision is inferred even if the valley contains alluvial sediment.

measured valley floor widths and drainage area in the western Alps and found that cv generally fell within the range of 0.3–0.4, and Kv varied between 8–160 m km-0.8 depending on lithology if cv was set to 0.4. measured valley floor widths for the Clearwater River, Washington, and found higher cv values of 0.76. This model suggests that increasing downstream discharge is the main control on valley widening, whereas lithology (through Kv) influences the width at any given discharge. found that in the western French Alps cv varied with lithology from 0.099 in the hardest lithologies to 0.42 in the softest lithologies. This suggests that in regions where the valley floor and valley walls are made of the same material, we may expect channel and valley floor width to increase at the same rate, whereas in regions where the valley walls are made up of a different material these rates may drastically differ. For example, in a valley with significant alluvial fill but bedrock walls, the channel may be able to widen rapidly with changes in discharge, whereas the valley floor widening rate will be set by the erodibility of the bedrock walls . developed a numerical model simulating lateral erosion in bedrock channels, which allows for two different mechanisms of valley floor widening: one in which the entire valley wall must be eroded for lateral migration to occur and one where undercutting and slumping can result in more rapid lateral erosion. Using the undercutting-slump mechanism, which may be suitable for regions with soft, easily erodible bedrock , they found a power-law relationship between valley floor width and drainage area with cv=0.46 and Kv=0.16.

A simple model of valley floor width based on drainage area and lithology alone lacks consideration of several factors which may influence valley floor width (Fig. ). Along with discharge, sediment supply is an important parameter that can either cause or hinder valley widening . If more sediment is supplied to the channel than can be transported (for example, through collapse of the valley side wall), this sediment can act to protect the valley walls and decrease the rate of lateral incision . Valley wall height is therefore an important parameter in setting lateral migration rates of the channel: sediment needs to be eroded from the entire height of the valley or the channel for the valley floor width to increase . Recent analytical models have suggested that in systems dominated by abrasion, the rate of particle impacts and the volume detached with each impact is important in setting lateral erosion rates . This suggests that variations in lithology between the valley walls and sediment within the channel should affect valley floor width. examined channel width and sediment lithology in the Rangitikei River, New Zealand, and found that channels containing sediment derived from resistant greywacke were up to an order of magnitude wider than those without. While this study looked at channel width rather than valley floor width, the same mechanism may be expected to control valley floor width over longer timescales than active channel adjustment. observed that valley 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. 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.. 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 of channel slope on valley floor width, 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 of , 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 post-glacial 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 e.g., as well as base-level changes which have been shown to influence upstream patterns of valley floor width e.g.. studied valley long and cross profiles in the Alpine Rhone River region, Switzerland, and found that valleys typically had a complex pattern of valley floor widths with a mix of narrow bedrock reaches and broad alluviated regions. They suggested that this pattern resulted from glacial sediment deposition as well as oversteepening of valley sides in the downstream direction, causing frequent landslides or debris flows and narrow, constricted valleys. Few models of fluvial erosion and lateral migration exist that have been 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 environmental factors. For example the models of suggest that different valley widening processes result in different values of the exponent cv, and the model of suggests that topographic gradient should influence the width of valley floors in a predictable manner. suggested that the width of valley floors should increase where rivers have higher curvature, on the basis of oblique particle collisions with valley walls. hypothesised that width will increase in areas with greater probabilities of debris flow sources in headwater 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 cv, which has been linked to lateral erosion process, in a landscape where tectonics and lithological heterogeneity are unlikely to play a role.

Most studies which have constrained valley floor width in mountain landscapes have used either field measurements e.g., hand-mapped valley floor widths from topographic maps e.g. or digital elevation models e.g.. 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 are generally taken hundreds of metres apart, meaning we lack a continuous dataset of valley floor width to compare to channel slope measurements, for example. Furthermore, different workers may have their own definition of what constitutes the edge of the valley, or it may be difficult to objectively identify. In the field, the presence of dense vegetation can make valley floor width hard to measure. Hand measurements from maps or shaded relief rasters can result in ambiguous results where the edge of the valley is not sharply defined or where there are significant anthropogenic modifications of the valley floor.

There are relatively few attempts to automate the extraction of valley floor widths from digital topography. presented a toolbox which allows the calculation of valley cross sections to identify topographic symmetry: their tool identifies the valley floor using a “flatness” threshold, where the valley floor is identified as regions of the cross section with a slope change of < 12.5 %. automated extraction of valley cross sections and calculated the valley width across either a user-defined elevation or the elevation of the lower of the two ridgelines but did not attempt to calculate the width of the valley floor. presented a novel method for identifying floodplains based on 2-D hydraulic modelling using high-resolution digital elevation models (DEMs), which could similarly be developed to calculate valley floor width. developed an automated technique for calculating valley floor width by taking the DEM-derived trunk channel and finding break-in slopes perpendicular to the flow direction. This algorithm is similar to the approach we employ here and is available as an ArcGIS plugin. However, their method relies on a sharp transition between the valley floor and surrounding hillslopes, as it is only based on slope rather than elevation compared to the channel. suggest their method is more effective in U-shaped valleys rather than V-shaped which are more common in fluvially carved landscapes. developed a semi-automated GIS tool that delineates valley bottom polygons and calculates valley floor width from valley cross sections. This tool needs manual processing and GIS analysis to extract valley floor widths. developed a valley finding component as part of a swath mapping tool, but this method is based on a topographic position index that includes the lower part of the hillslope and does not isolate the valley floor. 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 underpredicted compared to manual measurements.

In this contribution, we present a new method for continuously measuring valley floor width from DEMs. Our method is unique in that it is available as part of an open-source topographic analysis software package (LSDTopoTools), and it allows the extraction of the valley centreline to account for meandering systems. Our method allows measurements at every pixel downstream, giving a dataset of how valley floor width varies downstream along valley profiles and across landscapes. The method extracts the distance between valley walls (i.e. from hillslope to hillslope) where a valley floor is present, which could include either bedrock or alluvial channel reaches. We demonstrate the ability of our method to extract reliable valley floor widths by comparing with Quaternary superficial deposit maps from the UK and the US, which map out the valley floors in detail. We then explore downstream changes in valley floor width across the Cumberland and Allegheny plateaus to the west of the Appalachian Mountains to test the hypothesis that valley floor width should scale as a power-law function of drainage area.

The River Tweed is a gravel-bedded river with a well-developed valley floor located in southern Scotland, UK (Fig. ), originating in the Scottish Southern Uplands and draining to the North Sea. The underlying bedrock geology consists of Silurian and Ordovician greywackes, slates, and shales, with extensive glacial deposits . We extracted valley floor widths along the main course of the River Tweed automatically from the Ordnance Survey (OS) Terrain 5 DTM, which has a resolution of 5 m. We then compared these automatically extracted widths with those extracted using Quaternary maps (Fig. ). The mean valley floor width calculated by the automated method was 344 ± 281 m (1σ), and the mean width calculated from the Quaternary deposits was 312 ± 235 m. The mean difference in width between the two datasets was 17 m.

Figure  shows that the most significant difference between two datasets occurred in the uppermost 15 km of the study section, with the automated technique detecting a width ≈ 150 m wider than the Quaternary maps. This difference occurs where the automatically extracted valley occurs within glaciofluvial deposits which bound the alluvium on either side of the channel. Inspection of the hillshade (Fig. ) shows that the widths extracted by the automated method throughout this reach are more consistent with the break-in slope at the transition between the surrounding hillslopes and the valley floor, suggesting that they would be closer to a typical “field” definition of valley floor width compared to the Quaternary maps.

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 river terraces in the northeast 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 (Fig. ). 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.

Figure  shows the pattern of valley floor width downstream from both our automated technique and the BGS Quaternary deposits. We found that the pattern of valley floor widths was similar between the two datasets: the mean width from the automated method was 335 ± 224 m, compared to 390 ± 195 m. The mean difference between the two datasets was 27 m. However, the automated widths were ≈ 25 % wider than the BGS widths at the upper portion of the catchment, while they were up to twice as narrow as the BGS widths at 25–40 km downstream along the valley centreline. Inspection of the hillshade and Quaternary map reveals that terraces are the result of this difference. The upper part of the catchment contains upper terraces within the valley which are not identified as river terraces on the BGS Quaternary map but are flat surfaces close in elevation to the modern channel and are therefore picked up by the automated technique. Further downstream, the river terraces are high in elevation compared to the channel and are therefore missed by the automated technique despite being river terraces on the Quaternary map. This suggests users should carefully select the thresholds for elevation above the channel in areas with terraces of varying heights. We also include the option of ingesting Quaternary maps for width analysis in our software, so that users can use these as floodplain datasets for width analysis if preferred.

We then tested the sensitivity of our method to the resolution of the DEM for the Weardale site by resampling the Environment Agency lidar to 5 and 10 m, and downloading the 30 m SRTM data for the valley from OpenTopography. This represents the likely range of DEM resolutions available for the majority of the Earth's surface, as lidar data are increasingly available on regional and national scales and the SRTM 30 m data are freely available on a global scale between 60∘ N and 54∘ S. We found that the overall pattern of valley floor width remains consistent up to 30 m resolution, but there are some discrepancies in the valley floor widths calculated at each resolution (Fig. ). For example, at ≈ 36 km along the valley centreline, the 2 m dataset shows widths of up to 1 km, whereas the 10 m, 30 m, and BGS Quaternary deposits indicate valley floor widths of between 500–700 m. Furthermore, the 2 m lidar is able to calculate valley floor widths at the furthest upstream part of the valley, whereas datasets with coarsening DEM resolution can only identify the valleys further downstream. For example, the SRTM dataset only starts identifying valley floor widths at ≈ 4 km downstream along the valley centreline.

Table  shows the mean valley floor width and standard deviation calculated for Weardale from each DEM resolution as well as from the BGS Quaternary deposits for comparison. We found that the mean valley floor widths tend to increase with the DEM resolution, with the highest valley floor widths calculated from the SRTM 30 m dataset (416 ± 179 m). This is likely due to the edges of the valley being smoothed with coarsening DEM resolution, as well as the upper portions of the valley which have narrower widths being undetectable from the coarser resolution DEMs. The widths calculated from the SRTM dataset were closest to that of the Quaternary deposit mapping, with a mean width difference of 45 m between the two datasets. The Quaternary deposits are mapped at a scale of 1:50000, which corresponds to a DEM resolution of ≈ 25 m. This suggests that the automated technique performs well when compared to independent measurements mapped at a similar resolution.

We then tested the method in a downstream reach of the Russian River, northern California, where a 1:24000 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” (Fig. ). We then ran the algorithm using the automated valley extraction and by ingesting the USGS Quaternary maps. Figure  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. Figure  shows the downstream distribution of calculated valley floor widths. The automated widths in this site tended to be slightly narrower than the USGS-derived widths, with a mean percentage difference of 18 % between the two datasets. Close inspection of these maps (see left panels in Fig. ) suggest that the Quaternary units from the USGS maps include the toes of hillslopes.

Firstly, we focused on small tributary basins in the Cumberland Plateau, Kentucky. The Cumberland Plateau is a dissected plateau to the west of the Appalachian Mountains and to the south of the contiguous Allegheny Plateau. We focused on this region as we wished to examine controls on valley floor width in a relatively simple region: the Cumberland Plateau is tectonically inactive, has relatively homogeneous geology consisting of shallow dipping Carboniferous sediments, was unglaciated through the Quaternary , and has a humid, temperate climate . 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 Eq. (), and that the values of cv and Kv that can be identified from this power law should be reasonably consistent across different channels. To test this hypothesis, we focused on 10 valleys which are tributaries of the South Fork and Middle Fork Kentucky rivers, which form part of the Ohio River Basin (Fig. a). These valleys range in drainage area from approximately 7–79 km2. 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 descent flow trace for each valley and calculated the nearest point along the flow trace for each valley floor width to obtain a corresponding drainage area for each measurement. Following calculation of the width at 1 m, we filtered the dataset to remove width measurements that were more than 10 % wider than the mean width over a moving window of 500 m. This serves to remove widths at tributary junctions where the algorithm generally results in anomalously wide measurements if the floodplain extends into the tributary.

We fit a power-law relationship between width and area for each basin after Eq. () and calculated the widening coefficient Kv and the cv exponent (Table ). We found that there was generally a positive relationship observed between width and drainage area. In eight of the 10 basins a positive cv value was observed: the mean cv of these basins was 0.32 with a standard deviation of 0.15. However, in two basins, Short Creek and Stinnett Creek, there was an inverse relationship between valley floor width and drainage area, although these two basins had two out of the three lowest R2 values for this relationship amongst all valleys (Fig. ). For those basins which did show a positive relationship between width and area, the goodness of fit varied between basins from R2=0.78 in Flat Creek to R2 = 0.26 for Sugar Creek. Figure  shows examples of the relationship between valley floor width and drainage area for selected basins, demonstrating the variability between basins. Plots for all basins can be found in Fig. S1 in the Supplement. We normalised Kv by a reference cv of 0.21 which was the mean cv from the 10 basins.

Our results are consistent with the hypothesis that width can be well approximated as a power-law function of drainage area, and that the cv exponent is reasonably consistent across different channels. However, the lack of valley widening in two of the basins (Short Creek and Stinnett Creek), as well as the variation in the goodness of fit across all basins, suggests that there is heterogeneity in valley widening. Short Creek is, fittingly, the smallest catchment that we analysed, with a drainage area of 7.43 km2. The lack of widening signal in this basin may be a function of the smaller range of drainage areas available to perform the fit, although we note that Crane Creek is also a small catchment of 15 km2 which shows a clear widening signal. The valley in Stinnett Creek shows a relatively complex pattern of aggradation at the upstream portion of the valley, followed by a narrowing at around 6–8 km downstream along the valley centreline. From inspection of satellite imagery, it appears that there has been significant development of mountaintop removal mining in the lower part of the catchment since 1985. The mountaintop removal occurs up Knoblick Branch of Stinnett Creek: wide valleys occur upstream of Knoblick Branch, whereas there is valley narrowing just below the tributary junction. It is possible that enhanced sediment delivery to the channel through Knoblick Branch has caused valley aggradation upstream and altered the natural signal of valley widening in this catchment. showed that mountaintop removal mining has caused significant landsliding in central Appalachia: landslide dams and associated sediment delivery is also a potential control on valley floor width e.g..

Following on from this high-resolution, small-scale analysis, we extracted valley floor width and drainage area from large rivers draining the Cumberland and Allegheny plateaus in Kentucky and West Virginia to test whether larger catchments show the same trends of valley floor width and drainage area. For simplicity, we will refer to these regions together as the Appalachian Plateau. We analysed five major rivers which drain into the Ohio River basin: the Cumberland River, Kentucky River, Licking River, Guyandotte River, and Little Kanawha River (Fig. b). These channels drain the same tectonically inactive southern Appalachians as the smaller creeks, with the same underlying geology of dipping Carboniferous sandstones, siltstones and shales. However, their large extent means that they may drain across more units of varying hardness. We extracted valley floor width and drainage area from the 30 m SRTM dataset downloaded from OpenTopography, following the same approach as for the small creeks. Due to the coarser resolution, extracting a valley centreline was not necessary as the D8 flow routed channel was sufficiently smooth to avoid overestimation of valley floor width at meander bends. Figure  shows the width–area results for these large rivers. The relationship between Wv and A is remarkably consistent across these channels, with a mean cv of 0.3 and a standard deviation of 0.06. This result shows that our method is successful at identifying meaningful valley floor widths across large spatial scales (the largest basin is 24 019 km2), and that it is applicable on 30 m SRTM data available at a near-global scale. Furthermore, we conclude that in a tectonically inactive, non-glaciated region with relatively homogeneous lithology, reliable cv values can be extracted from topographic data.

Different cv values have been linked to different valley widening mechanisms, such as the undercutting-slump and total block erosion models put forward by . Our mean cv value of 0.3 for the Appalachian Plateau is similar to the values reported for intermediate hardness lithologies, mostly consisting of marly limestones, by . Their results of numerical modelling runs simulating end-member valley erosion mechanisms found cv≈0.25 for a resistant valley widening mechanism, where the entire valley wall height must be removed before widening can occur; while cv≈0.4 for an erodible mechanism where undercutting and slumping can occur. Our results for the Appalachian Plateau fall between these two end-members. This suggests that valley widening in the Appalachian Plateau takes place through a combination of these different end-member scenarios. Additional research into the size of material eroded from valley walls in the region and the grain size of sediment within the active channel and floodplain would help to constrain different valley widening mechanisms. Furthermore, a cv value of 0.3 is also similar to the analysis of for marine sedimentary units in Washington State (cv = 0.34) and for the western Alps, which found that cv generally fell between 0.3 and 0.4. found more variability in their normalised Kv values, which is unsurprising considering the lithological and tectonic variability present within the western Alps compared to the Appalachian Plateau.

constrained the vertical fluvial erodibility parameter, Kd, from across the Appalachians, and found that the Carboniferous sandstones, shales and siltstones of the Appalachian Plateau had intermediate Kd values of ≈ 1.0–1.5 × 10-6 m0.1 yr-1, compared to the more resistant rocks of the Blue Ridge (Kd ≈ 0.5 × 10-6 m0.1 yr-1) or more erodible carbonates, shales, and siltstones of the Valley and Ridge province (Kd ≈ 1.6–2.0 × 10-6 m0.1 yr-1). Our estimate of the normalised lateral erodibility parameter, Kv, ranges from 0.07–2.65 m km-0.6 for the Appalachian Plateau. Absolute values are not readily comparable because of differences in units between Kd and Kv, as well as between Kv values calculated with different values of cv, but importantly we find that the Appalachian Plateau has intermediate cv exponents compared to other studies, as well as the intermediate vertical erodibility values reported by .