Evaluating river morphology change with a geomorphic form variation approach

Geomorphic river design strives for natural resilience by encouraging geomorphic form complexity and morphological processes linked to greater habitat diversity. Increasing availability of high‐resolution topographic data and spatial feature mapping methods provide advantages for morphological analysis and river restoration planning. We propose and evaluate an approach to quantifying topographic variability of geomorphic form and pixel‐level surface roughness resulting from channel planform geometry differences using spatially continuous variety computation applied to component metrics including flow direction, aspect and planform curvature. We define this as the geomorphic form variation (GFV) approach and found it scalable, repeatable and a multi‐stage analytical metric for quantifying physical aspects of river‐bed topographic variability. GFV may complement process‐based morphological feature mapping applications, hydraulic assessment indices and spatial habitat heterogeneity metrics commonly used for ecological quality evaluation and river restoration. The GFV was tested on controlled synthetic channels derived from River Builder software and quasi‐controlled sinuous planform flume experiment channels. Component variety metrics respond independently to specific geometric surface changes and are sensitive to multi‐scaled morphology change, including coarser‐grained sediment distributions of pixel‐level surface roughness. GFV showed systematic patterns of change related to the effects of channel geometry, vertical bed feature (pool‐bar) frequency and amplitude, and bar size, shape and orientation. Hotspot analysis found that bar margins were major components of topographic complexity, whereas grain‐scale variety class maps further supported the multi‐stage analytical capability and scalability of the GFV approach. The GFV can provide an overall variety value that may support river restoration decision‐making and planning, particularly when geomorphic complexity enhancement is a design objective. Analysing metric variety values with statistically significant hotspot cluster maps and complementary process‐based software and mapping applications allows variety correspondence to systematic feature changes to be assessed, providing an analytical approach for river morphology change comparison, channel design and geomorphic process restoration.


| INTRODUCTION
Existing geomorphic river design in stream restoration strives for greater naturalness in form and process (Beechie et al., 2010;Fryirs et al., 2018;Johnson et al., 2019).Dynamic river systems often show complex feedbacks between reciprocal biotic-geomorphic interactions (Balaguer et al., 2014;Corenblit et al., 2007), and geomorphic form complexity is generally linked to fluvial processes that enhance habitat diversity (Kovalenko et al., 2012;Williams et al., 2020;Wohl, 2016) and system resilience to external and internal disturbances (Lorang & Hauer, 2017;Modrak et al., 2017;Tullos et al., 2021).Several spatial and topographic metrics can quantify physical aspects of surface variability (Kowarik, 2011;Polvi et al., 2014;Tranmer et al., 2022) and contribute to the concept of geomorphic complexity.Although 'complexity' may not have a consensus definition, the term will be used in this paper as a concept generally opposing simplicity, including reference to variability, irregularity, roughness and variety of geomorphic form.Understanding topographic surface conditions requires a range of analytical methods and benefits from a greater diversity of approaches capable of such investigation, including geomorphic metrics that are repeatable, scalable and spatially linked to morphological processes for river restoration design.
Near-census data, for example, is a spatially explicit, process-based characterization using 1-m to sub-metre resolution for morphological feature mapping and flow field assessment (Pasternack & Wyrick, 2017;Wyrick et al., 2014).Spatial mapping approaches in fluvial geomorphology apply nearcensus data of high-resolution topography to quantify fluvial forms.
For example, the Geomorphic Unit Tool (GUT) objectively divides complex 3D river topography into surface shapes and forms that can be further associated with morphological features of geomorphic and ecological importance, such as pools and riffles, and may be measured for dominant unit types and diversity (Wheaton et al., 2015).The Morphological Quality Index (MQI) was developed to assess ecological stream habitat conditions relating to these feature types (Belletti et al., 2017), whereas metrics like the Shannon's Diversity Index have been used to quantify GUT-derived maps and compare fluvial landform diversity (Williams et al., 2020;Woodworth & Pasternack, 2022).Although advances in high-resolution automated mapping applications have progressed, morphological feature type classification systems remain largely descriptive because feature types are separated according to a set of defined parameters (Kondolf et al., 2016) prior to quantification.Comparatively, variety raster cell computation provides statistical values derived from multiple input metrics and high-resolution maps of form variety.This may be especially useful for mapping local morphological complexity and spatial patterns to identify features and forms that influence variety values related to aspects of channel morphology and do not require prior feature type classification for variety computation.
River channel morphology variation and geomorphic form more generally have not been explicitly thought of in this way, but variety computation has been applied to large-scale landform type diversity analysis (Burnelli et al., 2023).This may provide a fundamental approach to comparing channel forms of all types.Geomorphic form variety may also support decision-making processes as a tool for river restoration planning and design, particularly when geomorphic complexity enhancement is a design objective and may complement quantitative spatial heterogeneity metrics often applied for ecological research (Iskin & Wohl, 2023).Example project objectives may include better understanding of hydrodynamic processes (Wyrick et al., 2014), modelling ecological patterns (Dilts et al., 2023) or understanding the impacts of stormwater runoff (Walsh et al., 2012).
Here, we use high-resolution river channel digital elevation models (DEMs) to propose an approach to quantifying geomorphic form variation (GFV) and grain-scale surface roughness of river beds.
The application of spatially continuous surface variety computation of multiple input metric rasters was used to derive output variety rasters from neighbourhood processing of input metrics including flow direction, aspect and planform curvature.Variety raster computation provides an objective approach for evaluating morphological feature assemblages by quantifying both form and surface roughness, and we refer to this as GFV.We evaluate a new GFV approach as a measure of river-bed surface-form variety with selected component surfaceform metrics in ArcGIS Pro.The approach was developed with reference to the Compound Terrain Complexity Index (CTCI) developed by Huaxing (2008), where component surface metrics were synthesized as a single value of topographic complexity (Huaxing, 2008), showing a suitable method for evaluating surface variation differences.
Three different sets of DEMs were compared to investigate GFV responses to changes in morphology and planform channel geometry differences.River Builder software (Brown et al., 2014) was first used to investigate the response of this type of metric to controlled adjustments of pools using artificial channels of systematically varying morphology of increased pool frequency and amplitude to which GFV was applied as a measure of variety.The second and third datasets included topographic data of physical model flume experiments of sinuous, semi-alluvial gravel-bed channels completed for other studies for which we had access to digital channel surfaces (Papangelakis et al., 2021;Peirce et al., 2021) provided as GeoTIFF file formats.The GFV was applied to data sources representing quasi-controlled adjustments of morphological features such as bars and riffles with different extents and amplitudes of bar development resulting from DEMs derived from the flume experiments.One flume dataset is from a symmetrical meander bend channel, defined as regular, whereas the other is from an irregularly sinuous planform pattern with varying channel widths.These represent increasing complexity of morphology on which the response of GFV can be assessed.
As a starting point for testing surface-form metric variety computation, the GFV is applied to these channel DEMs to understand (1) GFV response to changes in bed shape resulting from pool frequency and amplitude adjustments, (2) GFV response to differences in morphological details including changes in feature form geometry and orientation, (3) GFV response to feature amplitude changes resulting from different planform channel geometries (regular vs irregular), (4) how hotspot variety cluster maps can support analysis of GFV responses, and (5) if the GFV also responds to pixel-level surface roughness of mixed grain size distributions.

| The variety statistic and GFV processing
The proposed GFV approach is derived from geoprocessing Spatial Analyst Tools in ArcGIS Pro software.Component metrics are applied to DEMs with Focal Statistics as a tool for generating output rasters based on a neighbourhood operation.A neighbourhood window is a defined spatial extent for processing raster cells, where output cell values are a function of input cell values.The algorithm scans input rasters and calculates a statistic for the output raster of each processing cell within a selected neighbourhood shape (Esri, 2023a).
GFV uses the Focal Statistic-type 'Variety' to process multiple input metric rasters and will be referenced as the variety statistic here.
The variety statistic calculates the number of unique value cells within a neighbourhood window and will generate a variety class value for each processing cell of the input metric raster (Esri, 2023b).
The default 3 Â 3 cell rectangular neighbourhood shape was used and results in a value range of 1 to 9 based on the number of unique cells within the window.A variety class of 1 shows no variety, or unique values, whereas a class of 9 shows the maximum possible variation of input raster cell values.Esri (2023a) further describes the neighbourhood window approach and illustrates the computation method for input processing cells that are near corners or edges of raster surfaces.
The variety statistic has been applied to large-scale (e.g., cell size 1000 m Â 1000 m) geodiversity analysis of abiotic surface elements (Nikolova & Zareva, 2022) and more recently proposed as a geomorphodiversity index for landform type classification (Burnelli et al., 2023).The novelty here is in its processing of component surface-form metrics applied to river morphology and GFV analysis at finer scales (e.g., particle and bed roughness, bar morphology and reach-scale topography).Input rasters used for GFV calculations were derived from three surface-form metrics, including flow direction, aspect and planform curvature, which are processed from a singlesource DEM.These are well-known GIS-based surface-form variables, and computation methods are provided in the Supporting Information (see File 1).The variety statistic is then applied to each input metric raster to generate output raster cells calculated with the neighbourhood window algorithm.

| Parameters and assumptions
The variety statistic requires input rasters to be of integer format, meaning each cell is given a unique whole number value for neighbourhood processing (Esri, 2023c).Initial testing of candidate surface-form metrics was completed on high-resolution channel DEMs in ArcGIS Pro and showed that the above metrics responded to morphological irregularities of slope, bedform shape configurations and surface variation resulting from grain-scale roughness.Metrics also were selected for their geomorphic and ecological significance, providing insight to potential conditions for habitat heterogeneity that may contribute to, for example, microclimate and micro-habitat development (Bennie et al., 2008;Higgs et al., 2018).
Other metrics were tested on DEMs during GFV development, including local relief (Aili, 2008), bed relief index (Liébault et al., 2012), standard deviation of elevation (Scown et al., 2015), coefficient of variation (Wohl, 2016), rugosity (Jenness, 2013) and local slope (Esri, 2021).Statistical indices displayed processing issues when converting DEM data to integer format, and rugosity showed limited variety difference between channels, whereas aspects of slope were analysed from component metrics selected for the GFV.Depending on research objectives, the variety statistic may be applied to additional or alternative topographic metrics than those selected in our study but should consider the computation of variety and source DEM characteristics.For example, if channel bed relief values (Aili, 2008) statistically range within 1 unit (e.g., 0-1 m), converting the input relief raster from float to integer would result in a single whole number value (e.g., 0) showing no variability for variety computation.Therefore, component metrics must generate a sufficient statistical range for a given DEM to classify integer values.
Classification of component metric rasters can also influence the sensitivity of variety calculations applied.The default minimum metric cell value difference required for a unique variety classification is a difference of 1 unit.This minimum variety value difference can be adjusted by (re)classifying the input metric raster with a defined interval size or value range (Esri, 2023d).For example, setting an interval size to 2 units would group metric values in new class bins and result in a minimum variety difference of 2, compared with the default minimum of 1 unit.Manual, equal or defined intervals may be adjusted (Esri, 2023d) for specific research objectives and will reveal locations of more exaggerated topographic variety, or greater value ranges, relative to interval size selected for analysis.For example, aspect is a common surface metric applied for habitat structure assessment (Bennie et al., 2008;Bouchet et al., 2015;McGarigal et al., 2009) and for ecological microclimate analysis and, in particular, an aspect value difference of 1 degree may not warrant a variety value increase.Considering the 3 Â 3 neighbourhood window, if the 9 cells are within 1 degree from each other, the resulting variety statistic would show the maximum value class of 9 while only showing an aspect range of 9 degrees.However, (re)classifying the aspect raster with a defined interval size of 10, for example, could result in the minimum variety class value of 1 for the same window location.This provides flexibility of investigation methods and criteria, highlighting the potential for multi-stage and scale approaches to GFV investigation.
GFV results and variety class values are sensitive to choice of neighbourhood window size and also pixel size relative to channel and feature scale.Although a standard 3 Â 3 square window was applied in this study, larger windows (e.g., 5 Â 5) and/or neighbourhood shapes (e.g., circle, annular and wedge) could be applied (see Burnelli et al., 2023, for circular window shape).More investigation is needed of neighbourhood window size and shape effects to guide selection of appropriate values for a given analysis and pixel size selection relates to this decision-making.Raster-based metrics are inherently sensitive to pixel size, but this provides an opportunity for multi-scaled variety analysis.The relationship between GFV and pixel resolution provides an opportunity for geomorphic form analysis because different resolutions may reveal different aspects of variety, and multiple pixel sizes may be deliberately chosen to investigate variety at different scales.As a proposed multi-scaled metric approach, the pixel resolution of GFV variety investigation may be adjusted and support specified morphological variability research questions including reference to bankfull widths (Woodworth & Pasternack, 2022), for example, and river restoration objectives more generally.
Our study included high-resolution (pixel size equivalent to median bed particle diameter) flume DEMs for evaluation of GFV component metric responses to fine-scaled topographic surface changes including bed roughness related to spatial sorting of particle size.These DEMs were provided as GeoTIFF files and will be described in Section 2.4.Coarser pixel resolutions may not respond to surface roughness, compared with DEMs used in our study, suggesting that pixel size should be selected based on the interpretation of the meaning of GFV maps for morphology assessment or design decision-making and metrics should be made in relation to the pixel resolution and spatial extent (Hengl, 2006).

| GFV variety maps for comparative analysis
Component metric variety rasters may be useful for mapping local morphological variability and spatial patterns to identify features and forms that influence variety values related to aspects of channel morphology.Figure 1 shows the GFV workflow, starting with component metrics flow direction, aspect and planform curvature applied to a source DEM example (Figure 1a) and variety computation of input metric rasters to derive metric variety class maps (Figure 1b).We merged component metrics by averaging output metric variety rasters with Map Algebra in ArcGIS Pro (Figure 1c), and the resulting GFV raster provides a statistical variety value.Finally, hotspot analysis using the Getis-Ord Gi* statistic (Esri, 2018) was applied to component variety metrics and GFV variety class rasters, resulting in component metric and GFV hotspot cluster maps for comparative analysis (Figure 1d).
Several methods for merging different metric types exist, including for habitat suitability criteria (Moniz et al., 2019) and ecohydraulics (Kammel et al., 2016), for example, and weighted averaging regression can calibrate a metric order most relevant to specific research objectives (Ban et al., 2022;Mokarram & Hojati, 2017;ter Braak & Juggins, 1993).The aim of our study was to evaluate GFV functionality, so the weights of different component metrics were not determined but offered opportunity for further investigation.We chose an arithmetic approach because our analysis was not driven by absolute statistical variety values, which would require more robust and diverse datasets, but rather by understanding how variety computation responds to morphological feature changes.with confidence levels of 99%, 95% and 90% (Esri, 2018).For a hotspot to be significant, a feature would have a moderate-to-high value, extracted from variety class maps in ArcGIS Pro (e.g., value of >3), and be within close proximity to raster cells of correspondingly similar variety value (Melelli et al., 2017).The Getis-Ord Gi* statistic formulae and descriptions are provided in Getis and Ord (1992), with z-scores resulting from Gi* returns.Figure 2b  By providing multiple stages of potential analysis (e.g., Figure 2a, variety class maps, and/or Figure 2b, hotspot cluster maps), the GFV may be capable of multi-scaled assessment of topographic variability and surface roughness, or topographic surface variability, given the selected pixel size and analytical objectives.Applying hotspot maps may provide further information on fluvial processes and geomorphic form configurations that contribute to variety cluster adjustments and help to understand the relationship between GFV and differences in morphology.

| Synthetic and flume derived channel dataset descriptions
GFV performance was tested on 20 channel surfaces to analyse the response to controlled and quasi-controlled adjustments of bedform topography and different planform patterns.These channels included two sets of data types, (1) River Builder-derived synthetic digital surfaces and (2) two high-resolution physical model flume experiments completed in previous studies to which we had access to GeoTIFF file DEM data (Papangelakis et al., 2021;Peirce et al., 2021).Synthetic channels were selected for GFV testing because River Builder provides a unique opportunity to systematically adjust specific topographic variables (Brown et al., 2014).Unlike DEMs derived from real-world river surveys where unknown variables may contribute to variety changes, River Builder channels can be readily controlled to test GFV responses to specific geometric changes.Synthetic channels were designed to specifically evaluate GFV responses to geometric aspects of pool frequency and pool amplitude adjustments.
Flume channel DEMs included two planform datasets, each containing four bed channel surfaces of identical, fixed boundaries and planform geometry but different extent and amplitude of bar development from deposition within the fixed channels.One dataset was of regular meander channel pattern, and the second was of more irregular planform shape with width variation and provided an opportunity to evaluate GFV responses to quasi-controlled morphology adjustments and known differences in morphological complexity within a fixed planform.Channel lengths were all similar, with DEMs generated from a 10-m-long tiltable flume surveyed by Papangelakis et al. (2021) and Peirce et al. (2021), and pixel resolution was standardized to control the effect of river-bed topography development.
A resolution of 0.1 m was chosen for comparative channel analysis because a 0.1-to 0.15-m resolution range was defined as most suitable for potential morphological feature mapping applications (Belletti et al., 2017;Wheaton et al., 2015;Wyrick et al., 2014).Flume experiments were scaled at a ratio of 1:40 relative to field surveys (Peirce et al., 2021) at a pixel size of 0.0025 m for these DEMs.This resolution was resampled with Data Management Tools in ArcGIS Pro from the original DEM resolution of 0.001 m, and pixel size was comparatively similar to coarser bed material particle size (Papangelakis et al., 2021;Peirce et al., 2021) to analyse the GFV response to microtopographic changes of surface roughness.

| Generation of river synthetic surfaces and description of flume channel DEMs used for GFV approach
River Builder is a software application developed by Brown et al. (2014).The software derives digital topographic representations of synthetic river channels and valleys with a theoretical framework that incorporates principles of fluvial geomorphology towards 3D modelling (Brown et al., 2014).Synthetic digital channels have been used to study topographic design of form-process linkages for river restoration (Brown et al., 2015) and compared with alternative methods of digital river synthesis (Brown & Pasternack, 2019).River Builder relies on cumulative 2D surface planes that are integrated by mathematical functions known as geometric element equations.These planes are used to build 3D representation of topographic surface designs with computer-aided design (CAD) software applying explicit or implicit mathematical equations drawn from research completed by Mortenson (1997) and Ju et al. (2005).This allows river designs to go beyond existing empirical methods of channel design by using implicit process-based theories applied through geometric element equations and established spatial correlations among channel properties (Pasternack & Brown, 2013).
Ten synthetic channels were designed with River Builder to be tested as a preliminary approach to understand how the GFV responds to controlled topographic geometry differences.The domain parameters included a straight planform pattern, 1000-m length and critical Shields stress of 0.047.The channel extent was selected to represent a mid-scale river restoration project (Belletti et al., 2017), and critical Shields stress value was suggested in the River Builder Manual (Pasternack & Zhang, 2021).A channel width-depth ratio was selected based on typical ranges for small, stable channels and width was standardized to 10 m for each synthetic channel.TIN surfaces were created in ArcGIS Pro from point files generated in River Builder and then converted to raster surfaces.Channel parameters began with plane beds and progressively developed pool forms increasing the thalweg input frequency by a doubling series 1, 2, 4 and 8.These channels were labelled as 1F, 2F, 3F and 4F, respectively (Figure 3).
Although pool shapes and spacing may not mimic real channel forms, River Builder channels were used simply to evaluate GFV sensitivity to differences of this type and understand variety responses to controlled geometric changes.
River Builder channels were then used to investigate GFV responses to elevation changes of existing pools.Channel 4F (Figure 3) was used to investigate the influence of increasing pool amplitudes, beginning with an amplitude of 0.1 m that was increased to 0.2, 0.4 and 0.8 m, also comparing the plane bed channel in the dataset.These channels were labelled as 1A, 2A, 3A and 4A, respectively.Rivers are most often designed with pool spacing of five to seven times the channel width (Hudson, 2002)  These experiments were done previously as part of another research project related to formation of alluvial cover, bed topography and bar formation in channels with differing bed material supply rates (Papangelakis et al., 2021;Peirce et al., 2021) but provided a useful set of data on which to run GFV analysis.The applied feed rates and fixed planform patterns were controlled variables and contributed to systematic morphological differences between channels because of differences in the bar dimensions, amplitude and position (Papangelakis et al., 2021;Peirce et al., 2021).These flume channels were selected for GFV investigation because they provided opportunities for quasi-controlled comparisons of bed topography development and morphology to evaluate the relationship to variety metrics in real physical channels.Flume channels were constructed as fixed beds with trapezoidal cross-sections, whereas channel dimensions and discharge were modelled from a field prototype (Papangelakis et al., 2021;Peirce et al., 2021).Different bar extents and amplitudes were produced from differences in bed material feed rates producing different bed topography, and these bar form characteristics were the basis for GFV investigation.
DEMs were derived from Structure-from-Motion (SfM) photogrammetry using the software package Agisoft Photoscan 1.4, and each dataset contained four high-resolution channels of varying in bed topography, including a plane bed channel as a baseline reference (Papangelakis et al., 2021;Peirce et al., 2021;Welber et al., 2020).
Figure 4a shows DEMs for the regular channels representing bed topography differences and bar development between channels (also see figure 8a,c,e,g in Papangelakis et al., 2021).Figure 4b shows the irregular channel planform pattern and DEMs with different bar morphology used for analysis of GFV response (also see figure 10 in Peirce et al., 2021).
Bars developed on the inside of bends, whereas outside bends remained generally bare for all regular channel surfaces tested (Figure 4a).Comparatively, the irregular channels initially developed bars downstream from the first bend apex.Bars developed near and along the toe-of-banks where the bend radii were lower, whereas higher radii bends caused bars to develop away from bank slopes (Figure 4b).

| GFV response to changes in channel bed shape
River Builder channels showed progressively increasing variety values with greater pool frequency (Channels 1F to 4F).Increased pool amplitudes (Channels 1A to 4A) also showed progressive increases in GFV metrics, showing that component GFV metrics were responsive to channel vertical elevation oscillation frequency and amplitude.Results showed also that component GFV metrics responded independently to specific geometric surface changes.
Aspect was most sensitive to pool frequency increases (Figure 5a), whereas planform curvature was most sensitive to pool amplitude (Figure 5b).
The relatively low average metric variety values (<2.4), compared with maximum possible class value of 9, resulted from the absolute number of processing cells showing a variety class of 1 (60%-84% of proportional raster cells).River Builder channels showed two main geometric changes resulting from increased pool frequency.Firstly, concave pools decreased in relative size that caused planform curves to be more oval and pool lengths reduced as a result (Figure 3).
Considering that aspect is measured relative to compass direction, a F I G U R E 3 River Builder-derived DEM channels with contour lines showing increased pool frequency from 0 (bare-bed) to frequency 8 at a spatial extent of 10 000 m 2 .more circular planform shape would increase directional slope variety.
Secondly, as pool lengths reduced, the gradient of pool slopes increased because the elevation range was constant and the pool length decreased.
Figure 6a shows aspect variety hotspot maps of Channels 1F to 4F, illustrating progressively higher variety confidence level clusters forming along pool margins.As clusters became denser with increased pool frequency, variety clusters moved from the centre of the channel towards the banks, following steeper oval-shaped contours.Although clusters developed along convergent pool forms, the steepest pool slopes along bank intersections showed no statistical variety for each channel.A similar spatial relationship was found in planform curvature hotspot maps, where converging pool slopes were of greatest statistical variety, whereas the steepest planar slopes showed no clustering (Figure 6b).Flow direction showed the lowest proportional variety value change and comparatively minimal cluster changes between channel surfaces.These findings showed that aspect and planform curvature were not sensitive to more planar surface steepness but rather responsive to the curvature of increasingly concave slope forms.A relationship between pool shape and slope may not be assumed, but the frequency of pool forms showed a direct relationship to increased GFV values (see Figure 5a), showing denser clusters of high confidence aspect and planform curvature variety values.The combination of both planform curvature and aspect is responding to most aspects of topographic feature geometry.River Builder results showed that GFV component metrics are each responding differently to topographic variable changes, illustrated by their sensitivity difference to pool frequency and amplitude changes.
Curved feature slopes influence aspect and planform curvature, but the interior surface of concaved pools showed no significant variety, suggesting that feature margins are geometrically more complex in relation to variety values.Although the regular channels developed bars of similar shape, the irregular planform dataset had variation in channel width and resulting bar shapes.Figure 9a shows that wider bends contained wider bar features (bend 6) and narrower bends produced correspondingly narrower bar shapes (bend 7) resulting from bend curvature and width variability in the irregular Channel 4I compared with more uniform bars shown in Channel 4R.Although higher variety cells were expected to cluster along smaller and larger bars, increasing bar lengths with more sinuous edges contribute to variety cluster locations resulting from planform bar shape, which is illustrated by hotspot maps in Figure 9a.The orientation of bar development resulting from planform geometries also contributed to differences in GFV clusters for irregular channels.Although bars were fixed to streambanks in the regular meander channels, bars that developed away from streambanks were more exposed in irregular channels (Figure 9a).This provided more marginal bar surface area compared with fixed bars that were partially supported by constructed banks in the regular channels.Figure 9c demonstrates topographic bed variability at bend apex 3 in Channel 4I (Figure 9a) showing GFV clusters along feature margins developed

| GFV response to bar amplitude changes
Bar amplitudes increased through DEMs of both regular and irregular planform channels.As bars expanded laterally, the flat bar tops covered larger spatial areas and GFV findings showed these areas to be of low-to-no variety (Figure 9b). Figure 10a

| GFV response to bed surface roughness
Connecting riffle features that developed in the regular meanders reduced upstream bar margin slopes, but riffles were of coarser sediment deposits (see figure 11 in Papangelakis et al., 2021).The variety statistic was sensitive to grain-scale roughness, or micro-topographic variability, as shown in GFV variety class maps.Figure 11 shows GFV value class cell distributions, where mid-to-high class values developed on locations of coarse deposits in Channels 2R and 4R.Although GFV hotspot clusters formed along steeper bar margins, these locations had coarse deposits resulting from mechanics of particle size sorting.Variety class maps show that the GFV also responded to riffle feature roughness of more dispersed coarse sediment (Figure 11) although these locations did not show hotspot clusters (Figure 10d) and response depends on relative pixel size.

| GFV response to morphological feature variables tested
The GFV approach was developed to quantify bed surface form for river channels, based on the Focal Statistics-type Variety trialling a larger set of potential metrics and evaluating the selected metrics relative to known experimental differences and changes in channel morphology to understand the response of GFV metrics to these differences and the potential use and response of GFV and to provide informative maps and statistics of these differences.At this stage, GFV investigation is not intended to define a proportional variety increase as good (or not) or identify the ideal variety value for river restoration design because such an evaluation would require robust GFV analysis of extensive channel types and feature morphologies to build a sufficient dataset of statistically significant variety values.Raster-based measures of landscape pattern, such as contagion, show similar correlation to pixel data because the quantity of processed cells increases with resolution (Ricotta et al., 2003) and statistical values will adjust according to changes in class cell counts.However, with further research and GFV applications, statistical variety values may be proposed as quantitative river design objectives with reference to pixel resolution.exploring an approach of multi-scaled channel characterization that is based on quantifiable and repeatable variety metrics using continuous, high-resolution topographic data.The multi-stage investigation of component metrics and combined GFV variety class maps were demonstrated, including how variety processing cells respond to river-bed surface roughness and how hotspot analysis is particularly valuable for spatially mapping complex geometry aspects of topographic surface forms.Results also demonstrated how the GFV mapping applications of hotspot variety clusters are able to illustrate geomorphic surface forms of statistically significant variety clusters.All of this contributes to understanding the connection between specific aspects of channel morphology and GFV metric responses to potential geomorphic process linkages and therefore the utility of GFV as a tool for measuring topographic complexity.
To summarize the above findings, GFV responds to changes in channel bed shape and picks up differences in morphological details.
Doubling the frequency of pools in River Builder was intended to show how the GFV responds to increasing the number of features per unit length of channel and amplitude remained constant.Because spatial extent must be a constant for comparison between channels, pool spacing reduced as frequency increased.The size, or spatial area, of pools also decreased with spatial increased frequency (Figure 3).Increased frequency and amplitude of vertical oscillations from pool forms show a distinct GFV response in River Builder channels that also identifies differences in, for example, pool shape and contour curvature related to the frequency and amplitude of morphological features (Figures 6 and 7).GFV identifies differences in bar form shape and orientation related to the differing morphology in the regular and irregular channel datasets (Figure 9a,c).Results clearly showed that bar margins had the highest confidence hotspot clusters (Figure 9b) and suggest these locations as major variety components.Bar shape and orientation resulting from irregular channel geometry increased the length of bar margins within the channel, therefore increasing surface area for GFV hotspot clusters.Bar amplitudes increased areas of greater curved slopes in irregular channels (Figure 10a,b) but decreased upstream slopes as riffles connected to adjacent depositional features in regular channels (Figure 10c,d), suggesting that overall curved slopes were reduced as cover increased and flatter connecting riffle features developed (also see figure 8 in Papangelakis et al., 2021).Although the concentration of variety values decreased as a result, surface roughness of coarse-grained riffles was contributing to GFV class values resulting from pixel size relative to grain scale (Figure 11), illustrating how subtle the GFV may be in responding to morphological change related to specific processes at a given spatial extent and pixel resolution.

| Channel planform geometries and fluvial processes
Channel planform geometry exerts a strong control on the morphological features within sinuous channels with partial alluvial cover (Peirce et al., 2021).GFV cluster locations suggest that the shape and orientation of bars were a consequence of different bend curvatures and width variability of more irregular planform geometry (Figure 8b).This supports geometric River Builder results of variety responding to increased morphological feature frequencies by increasing the total surface area of exposed curved feature margins of variety cluster locations.One example from the sample channels is the development and location of steeply curved features along some bar margins, with clusters forming along curved surface forms and the flat top interior bar surfaces had low variety (Figure 9b).Outer bends where the initial plane bed was exposed also showed low variety values, illustrating the influence of planform geometry and fluvial processes on responding GFV variety patterns.
Planform patterns and bar development of semi-and alluvial channels are influenced by fluvial processes controlling erosion and deposition of sediment with reciprocal process-form response relationships (Finotello et al., 2020).Generally, flow velocity is greater along outside meander bends where erosional processes create pool features, whereas inner bends are of lower velocity flows, resulting in depositional bar features and local bed material particle size sorting patterns associated with bar development (Mishra et al., 2018;Nelson et al., 2014;Papangelakis et al., 2021;Turowski, 2018).Fundamentals of particle distribution suggest that coarser sediment will travel outward because of momentum and gravity, whereas finer particles travel towards lower shear stress areas (Lanzoni, 2000;Seminara et al., 1997).The experimental channel DEMs respond to these process controls, as do the resulting GFV variety class metric maps (Figure 11).GFV then helps to identify the process-form linkages related to, for example, bend geometry and bar development and topography.The multi-stage analysis potential and flexibility of the GFV may provide opportunities for multi-scaled morphological evaluation from reach-scale geometric forms to feature grain size roughness.

| Complementary approaches for geomorphic river assessment and GFV applications
Results suggest how the GFV may be applied to geomorphic river assessment and reveal features of morphology and complexity that are not revealed in other approaches, such as geomorphic unit (GU) mapping (Wheaton et al., 2015), especially because of the GFV focus on continuous spatial variation in form and spatial gradients of form change.However, combined GFV and GU analysis may allow correspondence to, for example, geomorphic covariance structure analysis (Pasternack et al., 2021), and GU maps may be particularly valuable for further analysis of GFV responses to bar characteristics and spatial feature roughness, so extending spatial and process linkages between statistically significant GFV variety clusters and GU configuration.
Figure 12 illustrates this with a case example of how hotspot variety clusters may be overlayed on GUT-derived maps.Hotspots show cluster locations along mound forms and follow polygon edges as mounds expanded laterally and also show that GFV clusters revealed the topographic development before it was apparent in GUT mapping.This example highlights the potential value of the GFV as a complementary approach to GUT mapping applications by, for example, showing the GUs associated with statistically significant variety hotspot cluster locations.
Process-based software able to design synthetic rivers can also benefit the use of GFV as a practical metric for evaluating, F I G U R E 1 1 GFV value class maps showing variety cells developing in Channels 2R and 4R at areas of coarse sediment deposits (also see figure 11 in Papangelakis et al., 2021).[Color figure can be viewed at wileyonlinelibrary.com] comparing and designing rivers to meet interdisciplinary objectives for river restoration (Hobbs et al., 2014).River Builder design variables can be systematically adjusted to derive site-specific planform channel patterns in the planning phase where synthetic river design scenarios may be presented, tested and compared with variety class values or hotspot cluster maps.River hydraulic assessment methods using the Surface Water Modelling System (Brown et al., 2015) or HEC-RAS software (Bilali et al., 2021) can also complement the GFV for site-specific flood risk and channel design planning, referencing additional analytical approaches such as DEMs of Difference (DoD) to link GFV variety to sediment storage, transport or budget adjustments (Capito et al., 2023;Merz et al., 2006;Wheaton et al., 2009).
Multidisciplinary decision-making may then consider objectives relating to public aesthetic preferences, ecological habitat enhancements and fundamental process-form relationships in fluvial geomorphology (Silva et al., 2013).With ecological restoration objectives often calling for enhanced habitat heterogeneity (e.g., Laub et al., 2012;Palmer et al., 2010;Xue et al., 2018;Yarnell et al., 2006), for example, the GFV can map spatial variety clusters of topographically complex conditions.These complementary methods can provide a quantifiable representation of conditions contributing to geomorphic complexity for comparative analysis and river restoration objectives more generally and expand assessment methods beyond traditional procedures of channel form evaluation (Papangelakis et al., 2023a(Papangelakis et al., , 2023b)).
The continuous mapping of GFV values at pixel level reveals important information on channel form that differs between morphology types and changes, and this is the important outcome rather than the overall GFV as a score for morphological quality.With improvements in data resolution, extent and frequency, stream conditions may become more quantifiable and temporal changes can readily be compared with more confidence that will increase opportunity for development and applications of variety metrics and maps.When considering topographic data as a continuous spatial variable, finer scales are involved in a hierarchical assemblage (Pasternack et al., 2018a;Pasternack & Wyrick, 2017) and the GFV can also be seen as a scalable approach to morphology and morphological change analysis applicable to rivers of a range of sizes.
Future testing of variety computation on additional properties such as channel width, substrate and especially flow linkages may allow further understanding of GFV relevance to potential morphodynamic evaluation (Papangelakis et al., 2023a(Papangelakis et al., , 2023b)).Applying the GFV with 2D hydrodynamic models could support the assessment of ecosystem processes and interactions with spatial patterns of surface-form variety and habitat structure contributing to the diversity of analytical metrics used in river management (Kammel et al., 2016;Pasternack et al., 2018b;Strom et al., 2016).Beyond DEM data, the GFV may be applied directly to output 2D flow rasters to map flow variety and connect eco-geo-hydro variety maps with spatial hotspot clusters to quantify aspects of morphodynamics and statistical correlation to process-form linkages.
GFV applications may also consider greater spatial extents to determine if the approach remains functional beyond in-channel morphological features.This could include floodplain and valley topographies and reference hydrodynamic-ecological estuarine modelling, for example (Ganju et al., 2016).By evaluating continuous river surfaces at the reach scale and greater, relationships between river restoration decision-making, resulting morphology change and catchment scale impacts may then be further identified (Gurnell et al., 2016).Existing geomorphic river design strives for dynamic system resilience (Tullos et al., 2021) and highlights important characteristics of natural geomorphic process-form linkages required for enhanced ecological habitat conditions (Johnson et al., 2019) and consideration of catchment-scale causes of degradation (Vietz et al., 2016) in stream restoration.Various complexity metrics exist (Tranmer et al., 2022), but increasing availability of high-resolution topographic data allows topographic variety to be measured, mapped and compared to provide an additional approach for linking geomorphic forms and morphological processes, beyond traditional surveying techniques.If scalable, the GFV may be further proposed as a metric for quantifying channel, reach and floodplain geomorphic form variety as a holistic, repeatable representation of complexity for channel morphology assessment and geomorphic process restoration design and planning.
As a proposed multi-scaled metric approach, the pixel resolution and neighbourhood window size of GFV variety investigation may be adjusted, and support specified morphological complexity research questions.However, because the variety statistic is sensitive to raster pixel resolution, GFV application for real-world data must consider data quality and equivalent spatial raster extents.Real-world topographic data may also contain unknown variables not included in our controlled study and will require further investigation to propose GFV as an analytical component for more reliable river design in practice.

| CONCLUSIONS
Increasing availability of high-resolution topographic data is providing advantages for morphological analysis and geomorphic process linkages.The proposed GFV is a scalable metric derived from spatially continuous variety class maps and is an approach to quantifying topographic variability of morphological feature surface form and pixel-level surface roughness resulting from channel planform geometry differences.The GFV was tested on controlled synthetic channels derived from River Builder software and quasi-controlled flume experiment channels to analyse metric responses to morphological feature aspect changes resulting from regular and irregular planform geometries.
Component GFV metrics, including flow direction, aspect and planform curvature, responded independently to specific geometric surface changes and were sensitive to morphology changes including pool frequency and amplitude.Channel geometry showed a strong control on feature shape, orientation and amplitude, which contributed to the development of steeply curved surface slopes, whereas planar surface slopes showed no variety.Hotspot clusters were particularly valuable for spatially mapping metric variety class cells, finding major components of topographic variety existing along the steepest curved surfaces of bar margins.Variety class maps supported the multi-analytical capability and scalability of the GFV, where variety class cells responded to coarser-grained sediment distributions of pixel-level surface roughness.
GFV is an element of resilient river systems, and the GFV can provide a multi-scaled and multi-analytical approach to comparing channel forms of all types and may support the decision-making process for river restoration planning and design, particularly when geomorphic complexity enhancement is a design objective.Analysing geomorphic form variety with statistical hotspot cluster maps allows GFV correspondence to systematic feature changes assessed, providing an analytical approach for river morphology change comparison, channel design and geomorphic process restoration.Future GFV applications with complementary feature mapping methods, hydraulic condition analysis and habitat heterogenicity metrics may also support our understanding of fluvial processes contributing to resulting variety value cluster distributions and provide a greater range of tools for more-resilient river design.
Figure 2 is an illustration of the outputs from GFV and hotspot analysis showing the culmination of the process and a demonstration of the spatially continuous form information that comes from it.Figure 2a illustrates the raster averaging process, where component metrics are applied to an experimental flume channel DEM provided by other authors, which was a F I G U R E 1 Workflow diagram illustrating (a) the process of applying component metrics to a DEM, (b) deriving component metric variety rasters using the Focal Statistic-type 'Variety', (c) deriving GFV variety class map from averaging component metric maps and (d) applying hotspot analysis for generating component metric and GFV variety cluster maps.[Color figure can be viewed at wileyonlinelibrary.com] surface used for testing and GFV analysis in this study.The approach produces raster variety class maps for each metric and a single map for the combined GFV metric raster.Analysing metric variety class values with complementary statistical hotspot cluster maps allows assessment of GFV correspondence to systematic feature changes, providing a potential application for river morphology comparison, channel design and geomorphic process restoration.Hotspots can indicate variety cluster locations that are statistically significant according to z-score and p-value calculations, Figures illustrating all flume channel DEMs and corresponding hotspot variety cluster maps used for GFV investigation are provided in the Supporting Information (see File 2).Hotspots indicate statistically significant variety cluster locations of mapped confidence levels (Esri, 2018) allowing a clearer visual representation of the spatial configuration of variety value classes for comparative channel analysis.
and Channel 4F pool spacing was within that fundamental range and therefore used for amplitude adjustments.Variety computation for these systematic differences in channel form can identify which GFV component metrics are most sensitive to pool frequency compared with amplitude increases.More detailed investigation of GFV responses to morphological differences was then completed on quasi-controlled flume channel DEM surfaces.Two flume datasets of different planform geometries were evaluated.The symmetrically meandered, or regular, channel DEMs were taken from experiments reported by Papangelakis et al. (2021), who analysed alluvial cover formation, morphology and bedload transport resulting from controlled supply feed rates into the fixed-geometry channels.Experiments reported in Peirce et al. (2021) provided the second flume dataset of irregularly meandered planform channel geometry with varying channel widths, where the impact of supply rate on sediment cover dynamics and bar formation and bed morphology were also studied.

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I G U R E 4 Illustration showing (a) symmetrical DEM channels of bar development differences between channels, also see figures 8, 10 and 11 in Papangelakis et al. (2021) for differences in bed elevation, cumulative distribution of alluvial cover and surface texture, and (b) irregular DEM channels illustrating bends and bar development differences between flume channels derived by Peirce et al. (2021).[Color figure can be viewed at wileyonlinelibrary.com]Although aspect was most sensitive to pool frequency increases, planform curvature was most sensitive to amplitude changes and showed the clearest relationship between hotspot cluster differences compared with aspect and flow direction.Although flow direction was more sensitive to pool amplitude increases compared with pool frequency (see Figure 5a,b), hotspot analysis showed very few variety clusters compared with planform curvature.Figure 7 shows planform curvature hotspot maps illustrating variety responses to increasing pool amplitudes of Channels 1A to 4A.As pools deepened and bed slopes increased, variety cells of high confidence clustered on concave slope margins.Variety responded most to high radius planform curves along the centre of Channel 4A, illustrating the influence of geometric shape convergence of pool shapes as amplitudes increase.
To further analyse GFV response to more-realistic channel geometry and morphological feature changes, experimental flume channels were tested.Variety comparisons were completed on flume channel DEMs to evaluate the GFV response to planform geometry (regular vs irregular) relating to changes of bar shape and bar orientation.GFV testing then demonstrates how these topographic, spatial and surface-form conditions influence variety values and spatial patterns by referencing fundamental fluvial processes that contribute to the development of morphological features in these physical model experiments.As in the case of the River Builder channels, component GFV metrics responded differently to topographic DEM variables, but these channels were of more complex topography with regular and irregular planform geometries showing different morphologies.Regular channels developed bars of similar shape simultaneously in fixed locations around each bend and bars expanded laterally through analysed DEMs (Figure4a), whereas irregular channels developed bars downstream from the first bend apex and existing bars expanded and new bar shapes developed at downstream apexes (Figure4b).

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I G U R E 5 Bar graph showing (a) progressive component GFV metric variety value growth from pool frequency increases showing aspect variety was most sensitive and (b) progressive component GFV metric variety value growth from pool amplitude increases showing planform curvature was most sensitive.[Color figure can be viewed at wileyonlinelibrary.com]Hotspot maps are the basis for showing how overall variety values, and specifically, high confidence level clusters are apparent along bar edges and can be further explored with the multi-stage GFV approach.Figure 8 shows this phenomenon, illustrating how GFV hotspot clusters form along developing bars in both regular (Figure 8a) and irregular (Figure 8b) channel datasets.The regular Channel 1R showed fixed bars developed around each meander bend and corresponding GFV clusters forming along bar edges.Comparatively, GFV clusters formed along the early-developed bars in the irregular Channel 1I, whereas the downstream channel bed showed no bar development or GFV clustering.As new bars developed and existing bars expanded in both channel DEMs, GFV clusters followed the evolving bar edges in most channel DEMs.The component metric variety hotpot maps that contribute to these GFV hotspot maps are provided in the Supporting Information (see File 2) and support this GFV-bar edge relationship.

Figure
Figure 9b illustrates a developed bar at bend apex 4 in Channel 4I (Figure 9a) and Channel 1R (Figure 8a), showing high confidence GFV clustering along steep bar margins and no variety on flat bar tops for both regular and irregular channels.This relationship shows that geometric shape of developed bars influences GFV values, where steeply curved feature margins were major components of topographic variety and further supported River Builder channel findings where variety clusters formed along steeply curved pool margins (Figure 6a,b).

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I G U R E 6 Illustration showing (a) aspect variety hotspot maps of progressive high confidence clusters developing along pool margins as frequency increased from 1 to 8 pools and (b) planform curvature variety hotspot maps of progressive high confidence clusters developing along pool margins as frequency increased.[Color figure can be viewed at wileyonlinelibrary.com] away from streambanks, increasing variety values through the width of the channel bend.
illustrates a developed bar at bend apex 5 in Channel 2I (Figure 8b) showing hotspot GFV clusters along the steepest downstream bar margin.As bar amplitude increased in Channel 4I (Figure 10b), high-confidence GFV clusters spread laterally towards the outer bank.The increased bar amplitude also developed a steep upstream bar face along the inner bend, where high confidence clusters also formed.

Figure
Figure10cillustrates a developed bar at bend apex 4 in the regular Channel 1R (Figure8a) and increased bar amplitude in Channel 4R (Figure10d).A similar relationship was found between hotspot GFV clusters forming along steep downstream bar margins.However, the upstream bar margin showed an inverse relationship as amplitude increased, forming a connecting riffle feature.Channel 1R shows GFV variety of 95% confidence clusters along the initial steep bar face (Figure10c).As bar amplitude increased in Channel 4R (Figure10d), this location showed no variety, illustrating an opposing GFV response to feature morphology resulting from planform channel geometries (Figure10a,b).The above relationships further demonstrate that the GFV is responding to feature changes and is sensitive to morphology differences resulting from regular and irregular planform geometries.In the

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I G U R E 7 Planform curvature hotspot maps showing progressively high confidence variety clusters forming along pool margins as amplitude increased, illustrated with contour lines.[Color figure can be viewed at wileyonlinelibrary.com] computation of multiple input surface-form metric rasters.Component metrics suited for continuous surface-form analysis were selected by

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I G U R E 8 GFV hotspot maps of both (a) regular and (b) irregular channel dataset DEMs showing high confidence GFV hotspot clusters forming along bar edges in both channel beds.[Color figure can be viewed at wileyonlinelibrary.com]Different component metrics picked up different aspects of the morphological differences (Figure 5a,b), and there may be scope for adding metrics to those selected to identify other morphological characteristics of interest.Results demonstrated how GFV variety values respond to systematic morphology and spatial patterns by

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I G U R E 9 Illustrations showing (a) a comparison of developed flat-topped bar shapes of irregular Channel 4I and more uniform fixed bar shapes of regular Channel 4R of high confidence GFV hotspot clusters along bar edges, (b) GFV hotspot maps of high confidence clusters along bar margins of irregular and regular bend apex 4 and (c) GFV hotspot map showing variety clusters along depositional feature margins through the width of Channel 4I bend apex 3. [Color figure can be viewed at wileyonlinelibrary.com]

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I G U R E 1 0 GFV variety clusters spreading laterally as bar amplitude increased from (a) Channel 2I to (b) Channel 4I, and (c) GFV clusters on the upstream bar face of Channel 1R and (d) low-to-no clustering on the developed connecting riffle feature of lower slope in Channel 4R.[Color figure can be viewed at wileyonlinelibrary.com]

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I G U R E 1 2 Example diagram of GUT map overlays of regular and irregular channels showing densely concentrated hotspot clusters near each mound.Variety clusters also formed on a plane form at bend apex 4 in Channel 1I, where a low elevation depositional feature formed, and hotspot clusters spread in Channels 2I and 3I.[Color figure can be viewed at wileyonlinelibrary.com]