Main

Valleys in mountain systems act as transient sinks for sediments that journey from sources on mountain hillslopes to their final resting place in forelands or ocean basins. This storage can buffer, shred or destroy propagating sedimentary signals1,2,3. Therefore, understanding the spatial distribution, volumes and longevity of valley sediment fills is essential to reconstruct landscape evolution from sedimentary archives. However, controls on the spatial distribution of valley fills across the Himalaya are currently unknown. Past efforts to map the volumes and residence times of valley fills at scale4 rely on the assumption that topography underneath the valley surface is similar to that of the exposed side slopes and therefore that little lateral erosion of the valley walls has taken place.

To explore valley widening, we consider a conceptual model where channels may either abrade or deposit sediment based on the ratio of sediment supply (Qs) to transport capacity (Qc) (Fig. 1). In channels with low Qs/Qc, little sediment will be deposited on the valley floor, resulting in bedrock incision, whereas channels with high Qs/Qc will deposit thick valley fills with subsequent valley widening5,6,7,8,9.

Fig. 1: End members of sediment-transport capacity model of valley-widening mechanisms and different factors that may control valley-width changes in each scenario.
figure 1

The photographs show examples of the two end-member valley types in the Upper Ganga basin (photo credit: R. Devrani). Wv, valley-floor width; Qw, water discharge; Qs, sediment supply.

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We can consider low Qs/Qc channels to behave similarly to the detachment-limited model for vertical incision, commonly used in mountain landscapes (for example, ref. 10). In this case, valley-floor width changes occur through lateral erosion of the valley walls and the balance between vertical incision and lateral erosion. Wall erosion is likely to occur when the channel is frequently in contact with the walls6,11, such as in narrow valleys. Valley-floor width Wv in this case has been suggested to scale with bank-full water discharge Qw, modulated by an erodibility coefficient K reflecting the impact of lithology (for example, refs. 12,13,14,15,16):

$${W}_\mathrm{v}=K{Q}_{\mathrm{w}}^{\mathrm{c}}.$$
(1)

In landscapes transiently adjusting to changes in rock-uplift rate, this relationship has been shown to break down (for example, refs. 17,18). An alternative formulation postulates that valley width is also dependent on valley slope (S)11,17 (Supplementary equations (1)–(5)).

Despite its common application, this low Qs/Qc case is contradicted by field observations, which show that mountain valleys are often infilled with sediment (Fig. 1). In valleys with a high Qs/Qc, widening through wall erosion will only occur if lateral erosion rates greatly exceed vertical incision, such that the channel regularly moves across the valley floor, impinging upon sidewalls6,11. However, Wv can also change purely through sediment deposition and/or erosion, without lateral wall erosion. If we imagine a roughly V-shaped valley infilled with sediment (Fig. 1), then increasing sediment fill would widen the valley, whereas incision into the fill would narrow it.

These end members of Qs/Qc represent contrasting mechanisms of valley-floor width changes, which are controlled by different factors (Fig. 1). In both cases, rock uplift is likely to be an important control on Wv, because highuplift rates elevate channel slopes, decreasing Qs/Qc through increased flow velocity, resulting in narrowing and bedrock incision17. Alternatively, increased frequency of landsliding in regions of high uplift (for example, ref. 19) could block channels, inducing upstream alluviation and widening.

The lithology of bedrock walls, K, is likely to be a more important control on Wv12,16 in the low Qs/Qc end member. In a valley that changes width primarily due to sediment erosion or deposition, variations in K are unlikely to play a dominant role, as width is not set by lateral bedrock erosion. In the high Qs/Qc end member, K may influence sediment delivery to the channel and thus Wv by changing the size and resistance of sediment from hillslope failures or upstream sediment transport20. However, the complex interplay of upstream and lateral sediment supply and downstream sediment transport means that it would be challenging to link variations in sediment erodibility to Wv at each point along the channel. Faulting may also increase rock fracturing and therefore erodibility (for example, ref. 21): we might therefore expect that valleys in fractured zones (such as near seismogenic faults) would be wider where lateral erosion is important, but not in the high Qs/Qc model.

Equation (1) suggests that water discharge is an important control on Wv: however, in our conceptual model, the ratio of sediment flux to water discharge, Qs/Qw, rather than Qw alone, is likely to influence Wv. Field studies22,23 and physical experiments7 have demonstrated that a decrease in Qs/Qw leads to incision and valley narrowing, whereas an increase in Qs/Qw leads to sediment deposition and widening. Over orogenic scales, we therefore hypothesize that the correlation between Wv and Qw would be complicated by spatial variations in sediment flux. Sediment-storage volume estimates across the Himalaya4 implicitly use the high Qs/Qc end member because they assume that little erosion of the valley walls occurs to modify the valley-floor topography.

In this contribution, we investigate dominant controls on Wv across the Himalaya and test these end-member models of valley widening and sediment storage. We generate a dataset of valley-floor widths across the Himalaya and investigate the relative importance of hypothesized controls on Wv through random forest regression. We also explore links between Wv, channel steepness (ksn) and exhumation rate using a compilation of thermochronometric cooling ages24.

We use an automated method25,26 to extract Wv from every major river basin in the Himalaya, resulting in 1,644,215 width measurements. We grid Wv into 10 km pixels to better reveal spatial trends: Fig. 2 shows the distribution of Wv across the orogen. We quantify each controlling factor that may affect Wv outlined in Fig. 1 (Methods).

Fig. 2: Spatial distribution of valley-floor width and channel steepness across the Himalaya.
figure 2

a, Map of the Himalayan orogen showing basins used for width analysis47. b, Topography across the region with main structural boundaries: MFT, MBT, MCT, and STD. c,d, distribution of valley-floor width (c) and distribution of normalized channel steepness (ksn) across the Himalaya (d). The data in c and d are gridded into cells with 10 km spatial resolution.

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Controls on valley-floor width

Figure 3a shows a bimodal distribution of Wv with elevation, where valleys are widest at elevations <1,000 m and >4,000 m. We would expect the southern, low-elevation region to have wider valleys as discharge increases towards the foreland. Although we remove areas affected by glaciation (Methods), widening at high elevations also results from past glaciations. We tested for this by removing valleys affected by Last Glacial Maximum glaciation, but this did not alter the results (Supplementary Figs. 1 and 2). High elevations also correlate with lower ksn (Extended Data Fig. 1) and erodible lithologies of the Tethyan Himalayan Sequence (THS), suggesting that increased Wv at high elevations may be explained by other co-varying factors.

Fig. 3: Box plots of valley-floor width (n = 7,414) against controlling variables.
figure 3

a, Elevation, z. b, Tectono-stratigraphic unit, where erodibility values (K, m1−2m yr−1) for each unit are labelled. GHS = Greater Himalayan Sequence; SHZ = Sub Himalayan Zone. c, Water discharge, Qw (m3 s−1). d,e, Normalized channel steepness, ksn (m0.9) (d) and distance from nearest major fault, df (km, MFT, MBT, MCT, or STD) (e). The solid black line shows the median of each distribution; the box represents the inter-quartile range; and the whiskers represent 1.5 times the inter-quartile range. Minima and maxima have been omitted to ensure readability. f, The normalized importance of each variable using random forest regression with two different methods for calculating importance: weighted impurity reduction (light grey) and permutation reduction (dark grey). Normalization is performed by dividing each variable importance by the most important variable (ksn in both cases).

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Figure 3b also shows that there is variation in median Wv among the main tectono-stratigraphic units. This is possibly due to lithological control on Wv, as the narrowest valleys are found in the high-grade gneisses and granites of the Greater Himalayan Sequence (GHS). The widest valleys are found in the sedimentary units of the Siwaliks in the Sub-Himalayan Zone (SHZ). However, these variations with tectono-stratigraphy co-correlate with elevation as discussed above, making it difficult to separate these two factors. Figure 3e shows there is little variation in Wv with distance from the major tectonic structures (MFT, MBT, MCT or STD), suggesting that increased erodibility through fracturing21 is not enhancing wall erosion.

Rock-uplift rates across the Himalaya since the middle Miocene have been controlled primarily by the geometry of the Main Himalayan Thrust (MHT)27, a northward-dipping décollement, which is the basal detachment for the MFT, MBT and MCT. The MHT is thought to be relatively flat under much of the Lesser Himalayan Sequence (LHS), steeper to the north over a mid-crustal ramp (for example, ref. 28) beneath the GHS, then flat again beneath the THS (Fig. 4). The ramp is associated with faster rock-uplift rates and steeper topography29, with a ‘physiographic transition’ (PT) marking the change from the southern (shallower) flat to the ramp. In central Nepal, we find a distinct area of wide valley floors within the LHS, with the transition to narrow valleys north of the PT coinciding with increased exhumation rate (Fig. 4). Considering that the PT cuts across the LHS in this region, the flat-ramp-flat structure of the MHT appears to influence Wv in central Nepal more strongly than the transitions across tectono-stratigraphic units.

Fig. 4: The impact of tectonics on valley-floor widths.
figure 4

a, Illustration of valley-floor width across part of the Narayani basin in central Nepal, where line width is scaled by valley-floor width (widths are scaled up for visibility), and line colour represents channel steepness (ksn). The dashed lines show the main structural boundaries. Note the presence of glacially widened valleys in the Greater Himalayan Sequence, and the distinct valley widening and flattening to the south of the physiographic transition (PT) within the LHS. M, Marsyandi river; BG, Budhi Gandaki River; T, Trishuli river. b, Median valley-floor width (black line, n = 81,208) and exhumation rate derived from thermochronometry24 (blue line, n = 218) binned by 0.1∘ latitude across the region shown in a, showing valley narrowing and rapid exhumation to the north of the PT at the location of the MHT mid-crustal ramp. The shaded areas show the range between the 25th and 75th percentiles. The points show the exhumation rate samples where the error bars represent the 1σ uncertainty in exhumation rate. c, Schematic cross section across the region in a showing the location of the mid-crustal ramp within the MHT (modified from ref. 48).

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Existing valley-widening models predict a monotonic relationship between Qw and Wv (equation (1)). Our results do not show this relationship (Fig. 3c). Although the widest valleys are found in regions with the highest Qw, the narrowest valleys (99 ± 280 m) tend to coincide with intermediate Qw of 0.2–1.0 m3 yr−1. At the lowest Qw of 0.01–0.05 m3 yr−1, median Wv increases to 139 ± 169 m. This lack of correlation suggests that in contrast to the commonly applied model of width evolution through lateral bedrock erosion, Qw is not the dominant control on Wv across the actively uplifting Himalayan orogen.

There is, however, a negative correlation between Wv and ksn (Fig. 3d). We tested this relationship across different tectono-stratigraphies and found it is consistent between lithologies (Extended Data Fig. 2). To account for the competing influence of Qw and S, we also calculated a discharge-weighted channel steepness, ksn−q 30. We found this did not alter the relationship between ksn and Wv (Supplementary Fig. 3). ksn is a widely accepted proxy for rock-uplift rate (for example, ref. 31), suggesting that Wv responds to spatial variations in rock-uplift rate. We also find no relationship between Wv and mean annual rainfall (Extended Data Fig. 3).

To further test tectonic control of Wv, we use a compilation of 1,148 thermochronometric ages24 (Fig. 5), from which we estimate exhumation rates (E) using a simple 1D thermal model (Methods). Figure 5b,c shows a correlation between Wv, E and ksn. The lowest E of 0.1–0.2 mm yr−1 corresponds to the widest valleys and lowest ksn. Intermediate E between 0.3 and 0.9 mm yr−1 show less variation in both Wv and ksn, whereas E ≥ 2 mm yr−1 correspond to narrow valley floors and steep channels. Variations in E in the Himalaya have been argued to be strongly tectonically controlled27,32,33. The correlation between Wv and E, along with the changes in Wv across the flat-ramp-flat geometry of the MHT (Fig. 4), indicate that Wv is probably controlled by tectonics.

Fig. 5: The relationship between valley-floor width, channel steepness and exhumation rate.
figure 5

a, Map of exhumation rate derived from thermochronometry data across the Himalaya: the colours represent the exhumation rate in mm yr−1, symbols represent the thermochronometric system. AHe: apatite (U-Th)/He; AFT: apatite fission track; ZHe: zircon (U-Th)/He; ZFT: zircon fission track; ArAr: 40Ar/39Ar. b, Box plots showing relationship between valley-floor width and exhumation rate: the numbers above each box show the number of samples in the corresponding bin (n = 1,148). c, Box plots showing the relationship between normalized channel steepness (ksn) and exhumation rate (n = 1,148). The solid black line shows the median of each distribution; the box represents the inter-quartile range; and the whiskers represent 1.5 times the inter-quartile range. Minima and maxima have been omitted to ensure readability.

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Thermochronologic cooling ages are representative of exhumation over long timescales (105 to 107 years)34. Patterns of exhumation across the Himalaya are likely to change through time with tectonic or climatic variations (for example, refs. 35,36,37), potentially disconnecting long-term exhumation measurements and valley-forming processes. We focus here on thermochronometry rather than cosmogenic radionuclide-derived (CRN) erosion rates because the spatial coverage of thermochronometric data is far greater than CRN and because the relationship between Wv and A makes it challenging to determine a representative Wv to compare with catchment-averaged erosion rates. Examining Wv and E separately by thermochronometer (Extended Data Fig. 4) shows that the relationship between Wv and E is strongest in chronometers with lower closure temperatures, representing more recent exhumation rates. Nevertheless, the correlations between Wv, ksn and E across the dataset (Fig. 5) indicate a tectonic control on Wv and ksn despite potential temporal variations.

Importance of valley-floor width controls

Figure 3a–e demonstrates that many factors may control Wv across the Himalayan orogen; we therefore take a data-driven approach to determine which has the strongest influence using random forest (RF) regression. To explore key controls on Wv, we focus on the following variables based on our conceptual model (Fig. 1): (1) elevation, z; (2) ksn; (3) Qw; (4) K; and (5) distance from the nearest fault, df (MFT, MBT, MCT or STD). We calculate K using CRN-derived erosion rates and ksn (Methods).

RF-regression estimates of variable importance (Methods) indicate that ksn is the most important predictor across all regression models (Fig. 3f), with K consistently the least important. z, Qw and df have relatively similar importance, although z tends to be more important among these three. There are distinct spatial trends in ksn with z, with highest ksn found at intermediate z and lower ksn at both low and high z (Extended Data Fig. 1). This co-variation may explain the high relative importance of z in the RF model.

Implications for valley-widening processes

Our results indicate moderate importance of Qw and low importance of K on Wv, contrasting with common valley-widening models (equation (1)). We propose that observed Wv are likely set by sediment accumulation, corresponding to the higher Qs/Qc end member in Fig. 1. This suggests little modification of topography under these fills, supporting a key assumption of Himalayan sediment volume estimates4. For a given Qs and Qw, the likelihood of a channel to incise or aggrade is set by S, dependent on uplift. The relationship between E, ksn and Wv indicates that high rock-uplift rates in rapidly exhuming regions, reflected by high values of ksn, are likely to increase Qc, mobilizing sediment that acts as tools for bedrock incision during peak Qw, with subsequent valley-floor narrowing. Therefore, rivers in high-uplift regions are likely to typify the low Qs/Qc end member, whereas slowly uplifting regions exemplify the higher Qs/Qc scenario. Nevertheless, the low importance of K suggests that sediment is important across the full range of E and that even under the highest rock-uplift rates, rivers are likely to contain substantial alluvial cover, with bedrock incision only during extreme transport events.

Damming behind landslides or uplifting structures increases Wv upstream. Considering that landslides occur more frequently in rapid exhumation regions19, a landslide-dam control on Wv at the orogen scale would generate wider valley floors in faster exhuming regions (Fig. 1), or at least highly variable widths. In contrast, if damming behind uplifting structures (for example, refs. 38,39) controlled Wv, wider valleys may be randomly distributed. We find that ksn is a first-order control on Wv and that ksn increases and Wv decreases with E. This implies that the distribution of valley fills is driven by tectonically controlled exhumation, rather than landsliding or structural damming. An exception is that at intermediate E of 0.3–0.9 mm yr−1, increased E does not lead to concomitant changes in ksn or Wv. If at these intermediate exhumation rates, channels are insufficiently steep to regularly flush aggraded sediment, the impact of landslide and structural damming could be enhanced.

Although our results point to Wv being set by the depth of sediment fill rather than wall erosion, valleys must experience lateral erosion during their evolutionary history. The Qs/Qc ratio may vary during climate oscillations5,6, leading to alternating periods of bedrock incision and widening through wall erosion and periods of sediment deposition and filling. However, valleys that are currently alluviated must also facilitate bedrock erosion to adjust to long-term uplift rates. The frequency of incision should be limited to the most extreme events that can remobilize valley fills40,41,42,43. Recent work shows that valleys regularly affected by glacial lake outburst floods (GLOFs) are generally narrower and contain less sediment, facilitating bedrock erosion, while valleys with less frequent GLOFs showed sediment trapping and lower incision rates44. Along the Bhote Koshi River, GLOFs were observed to mobilize the largest boulders41, indicating that they can effectively flush valleys and cause bedrock erosion.

Our findings raise questions about the residence times of valley-fill deposits compared to extreme event frequencies. The adjustment of Wv to E averaged over 105–107-year timescales indicates either that valley fills persist over geological timescales or that Wv adjusts relatively rapidly to the local exhumation rate. Residence times of Himalayan fills have been proposed to exceed 105 years for the largest valleys4. Recurrence intervals of extreme events are likely shorter, with the Bhote Koshi River affected by GLOFs with a return interval of ≈ 30 years45, although it is unlikely that every GLOF will strip all sediment from the valley floor. Dating of far-travelled boulders in the Trishuli and Sunkoshi rivers indicated a recurrence interval of ≈ 5 thousand years for the most extreme GLOFs46. Our results suggest that valley re-filling to adjust to local exhumation occurs on shorter timescales than valley-fill removal.

The link between E and Wv also has important implications for sediment routing systems and the transmission of sedimentary signals to basins. If slower exhumation rates lead to wider valleys, then sedimentary signals of external forcing in slowly exhuming areas are likely to spend more time in storage compared to rapidly exhuming areas, resulting in either buffering or shredding of the signal before it reaches its depositional sink (for example, refs. 2,3). Future work is needed to further explore (1) the timescales of Himalayan valley-fill preservation; (2) the impact of exhumation rate on the propagation of allogenic signals and (3) the sub-surface geometry of valley deposits to allow further investigation into valley-widening mechanisms.

Methods

Extraction of topographic metrics

First, we isolated our analysis to the extent of the orogen49,50, including the tectono-stratigraphic units of the SHZ, the LHS, the Greater Himalayan sequence (GHS) and the THS and excluding both the western and eastern syntaxial regions. We then split the DEM into major river catchments using catchment outlines from the Hindu Kush Himalayan region47 and limited our analysis to those draining to the southern edge of the orogen. We then analysed valley-floor width for every major river basin, using a method for reproducibly extracting valley-floor width from digital elevation models (DEMs)26. This method first identifies floodplains using a threshold of slope and elevation above the nearest channel25. These thresholds can either be set manually by the user or defined automatically; to ensure consistency across the orogen, we manually set a slope threshold of 0.15 and an elevation threshold of 100 m. The method then identifies the main flow direction of the channel and calculates valley-floor width orthogonal to this. The minimum possible width measurement is 60 m, which is set by the resolution of the DEM (2 DEM pixels).

Following extraction of width measurements for every channel, we removed any measurements that intersected each other (that is, at tributary junctions) as these measurements are unlikely to represent the true valley-floor width. We removed measurements from modern glaciers across the Himalayas using the glacier outline shapefiles from the Randolph Glacier Inventory51: we removed any measurements within the boundaries of each shapefile. Alongside modern glaciation, valleys that have been affected by glaciation through the Quaternary may have a topographic signature of glaciation rather than fluvial processes. We therefore performed a sensitivity analysis of our results to estimated glacial extents during the Last Glacial Maximum by estimating the minimum elevation of the LGM equilibrium line altitudes for glaciers across the orogen, using a regional compilation52. We found that removing the signature of Quaternary glaciations did not affect the results (Supplementary Figs. 1 and 2). After filtering, we gridded the valley-floor width data using a grid cell size of 10 km, taking the mean valley-floor width within each grid cell. We tested the sensitivity of the random forest regression to grid cell size (Supplementary Fig. 4) and found that the results were insensitive to gridding at cell sizes from 1 to 10 km.

We calculated the mean elevation of each 10 km valley-floor grid cell using the Copernicus 30 m DEM and determined the underlying tectono-stratigraphic unit using a geologic database50. We calculated normalized channel steepness (ksn (m0.9)) across each river basin using a segmentation approach53 as implemented in LSDTopoTools54. ksn is often used as a proxy for rock-uplift or erosion rates and has been shown to correlate with local relief and catchment-averaged erosion rate across the Himalaya (for example, refs. 55,56,57,58,59). We used a reference concavity value, θ = 0.45, which has previously been estimated for the Himalayan region (for example, ref. 60). We gridded the ksn data using the same approach as for valley-floor width (Fig. 2b).

To estimate water discharge, Qw, we use a simple proxy based on weighting upstream drainage area (A) by mean annual rainfall (P)30:

$${Q}_\mathrm{w}=PA,$$
(2)

We estimated P from 1981 to 2019 across the Himalaya using the Climate Hazards Group InfraRed Precipitation with Station (CHIRPS) dataset, which combines 0.05° resolution satellite imagery with ground-station data61. The advantage of using the CHIRPS dataset is that it has a near-global rainfall time series for more than 30 years, giving longer-term estimates of P that should be less sensitive to short-term temporal variations. We calculated P from this dataset using Google Earth Engine, then resampled P to a spatial resolution of 30 m to correspond to that of the topographic data. We test discharge rather than drainage area as the Himalaya have a strong orographic rainfall gradient resulting in an order-of-magnitude variation in P across strike and an ≈ sixfold increase in rainfall from west to east62,63. To test the ability of this simple model to reflect real variations in Qw, we compared the model predictions to measured Qw from gauging stations across major rivers in Nepal64,65. We found good agreement between modelled and measured Qw across a range of discharges (Supplementary Fig. 5).

To investigate the potential impact of fracturing on bedrock erodibility, we also calculated the Euclidean distance of each grid cell from the nearest major tectono-stratigraphic boundary (either the Main Frontal Thrust (MFT), Main Boundary Thrust (MBT), Main Central Thurst (MCT) or South Tibetan Detachment (STD))50.

Compilation of thermochronology data and calculation of exhumation rates

We updated an existing compilation of thermochronometric data from the Himalaya66 to include more recent publications up to July 2022, including all data falling within the basins outlined in Fig. 2a. We include results from five thermochronometric systems in our analysis: apatite and zircon (U-Th)/He (AHe, ZHe) and fission track (AFT, ZFT) and white mica 40Ar/39Ar (MAr). We removed any cooling ages ≥50 Ma, as these ages are pre-Himalayan49 and are therefore unrepresentative of valley-forming processes, and samples from the SHZ, as these are generally incompletely reset since deposition67. In some cases, multiple thermochronometric cooling ages were available for a single location: we filtered the dataset to keep only the youngest age for these samples, as these are more likely to be representative of the erosion rate shaping the modern topography. We also filtered the dataset based on uncertainty by removing any samples where the 1σ uncertainty in predicted exhumation rate was greater than the exhumation rate itself (Supplementary Fig. 6), and we removed any samples within the boundaries of modern glaciers51. The complete dataset and associated references can be found in ref. 24.

We use a 1D thermal model that assumes vertical exhumation and thermal steady state to estimate exhumation rates from the thermochronology data. The model (refer to ref. 24 for details) takes into account the advective perturbation of the geotherm by rapid exhumation68 and the control of cooling rate on closure temperature of each thermochronometric system69. We use the sample elevation to estimate the surface temperature using a linear atmospheric lapse rate (5 °C km−1) and a constant sea-level temperature (25 °C) and to estimate the vertical difference between the sample elevation and the average elevation smoothed within a radius that depends on the estimated closure depth of each thermochronometric system70. The latter is used to correct the estimated exhumation rate for relative sample elevation. For other model parameters, we assume the following: an initial linear geotherm of 25 °C km−1, a thermal diffusivity of 30 km2 Myr−1 and a model thickness of 30 km. We then mapped each exhumation rate sample to the corresponding valley-floor width cell in the gridded 10 km dataset and binned valley-floor width and ksn by exhumation rate.

Erodibility index

We calculated an erodibility index, K, for each of the main tectono-stratigraphic units across the Himalayan orogen using a compilation of catchment-averaged erosion rate data from cosmogenic radionuclides71, similar to the approach of ref. 72. The commonly used stream power incision model predicts a nonlinear relationship between channel slope and erosion rates:

$$E=K{A}^\mathrm{m}{S}^\mathrm{n},$$
(3)

which we can rearrange to find an expression for channel slope, S:

$$S={\frac{E}{K}}^{1/n}{A}^{-\theta },$$
(4)

where θ = m / n. We can simplify this equation to:

$$S={k}_\mathrm{sn}{A}^{-\theta },$$
(5)
$${k}_\mathrm{sn}=E/{K}^{1/n}.$$
(6)

We estimate ksn as described above and then assume that the CRN-derived erosion rates are representative of erosion across the entire basin, such that for each point on the network, we know ksn and set E as the catchment-averaged erosion rate. We can then rearrange equation (6) to solve for erodibility at each point on the channel network, Ki:

$${K}_\mathrm{i}=\frac{E}{{{{k}_\mathrm{sn}}_\mathrm{i}}^{n}}.$$
(7)

Many studies have suggested through both numerical modelling and field studies that n is likely to be > 1 (for example, refs. 72,73,74), with n ≈ 2 thought to be reasonable in most cases75. We therefore set n = 2 in equation (7): a similar approach was also taken by ref. 76. As we set m / n = 0.45 in our ksn calculation, this results in m = 0.9. We then separate the calculated erodibilities based on tectono-stratigraphic unit and calculate the median K for each. The median values of K for each unit can be found in Supplementary Table 1.

A similar approach to calculating K can be taken, which also accounts for the impact of climate, by back-calculating K from the relationship between erosion rates and a channel steepness calculated by weighting drainage area by precipitation, ksn−q77. We calculated ksn−q and found that the relationship between Wv and ksn−q was similar to that of ksn (Supplementary Fig. 3). Furthermore, we found no relationship between P and Wv, suggested that weighting K by P is unlikely to change the relationship between K and Wv. Other approaches to estimating erodibility have derived an erodibility index that incorporates (1) a rock strength index (LL), related to its composition and (2) an age index based on the stratigraphic age of the unit78,79. We also tested this method of determining erodibility and found that it did not alter the relative importance in the random forest analysis (Supplementary Fig. 7).

Random forest regression

Random forest (RF) regression is a form of supervised machine learning, which uses an ensemble of decision trees to predict a target variable (here Wv) from a high-dimensional dataset (for example, ref. 80). It allows the calculation of variable importance (VI) for each variable used to predict the target variable. It requires no assumptions about the structure of the underlying data and therefore is useful in cases where the relationship between the target variable and the predictors is unknown a priori81. We performed RF regression on the 10 km gridded dataset to isolate the key signals of valley widening and reduce dataset noise. Supplementary Fig. 8 shows the spatial distribution of additional metrics used in the RF regression across the Himalayan orogen (elevation, water discharge, distance from nearest fault and tectono-stratigraphy). Before running the regression model, we split the gridded dataset into 80% training and 20% testing to allow for validation.

The number of decision trees (NT) used to build the regression model has shown to be important when using RF regression, particularly when investigating VI80. We therefore performed a sensitivity analysis on the regression varying the number of decision trees from 10 to 2,000 (Supplementary Fig. 9). This analysis showed that the root mean square error of the regression model became relatively insensitive when the number of decision trees is greater than 1,000, with root mean square error 167 m. We therefore ran all RF-regression runs with 1,000 decision trees to ensure greatest computational efficiency.

VI in random forest regression can be determined through two approaches: average impurity reduction and permutation reduction (for example, refs. 82,83). Average impurity reduction80 states that the importance (Imp) of any variable Xj in predicting the target variable, Y, can be calculated by summing the weighted impurity decreases p(t)Δi(st, t), where t represents each node where Xj is used, and φm is tree m in the forest containing all trees m = 1, . . . , M:

$$\mathrm{Imp}({X}_{j})=\frac{1}{M}\mathop{\sum }\limits_{m=1}^{M}\mathop{\sum}\limits_{t\in {\varphi }_{m}}{\delta }_{{j}_{t},j}[p(t){{\Delta }}i({s}_{t},t)],$$
(8)

where:

$${\delta }_{{j}_{t},j}=\left\{\begin{array}{ll}1\quad &{{{\rm{if}}}}\,{j}_{t}=j\\ 0\quad &{{{\rm{otherwise,}}}}\end{array}\right.$$
(9)

p(t) is the proportion of samples reaching t, and jt is the variable used to split node t83. This approach gives the most importance to the variable that most decreases the mean impurity across all trees in the forest. However, the impurity reduction approach has been shown to be biased towards predictors that have a large number of values84. Therefore, an alternative approach to estimating variable importance called permutation reduction has been suggested80, which estimates the change in the mean standard error of the regression model when permuting a variable. The reader is referred to ref. 80 and ref. 83 for a full derivation and discussion of permutation reduction VI. We performed a sensitivity analysis of the variable importances derived for the valley-floor width regression model to choice of VI metric across a range of different decision trees (Supplementary Fig. 10). We find that the VIs are insensitive to the number of decision trees used in the regression model and that the order of VI is identical with our chosen model run of 1,000 trees.