Figure 3: Examples of varying landscape subdivision (A and B; Deer Creek, OR), channel edge density (B and C; North Fork Teanaway River, WA), and class level subdivision of wood (D and E; Snoqualmie River, WA). Multiple metrics show differences in spatial configuration between contrasting reaches, all with slightly differing utility and interpretation.
Landscape level spatial configuration metrics (i.e., those computed across multiple geomorphic unit classes) can be a proxy for characteristics that can be difficult to directly measure. For instance, if lacking quantitative bathymetric data, one could use aerial imagery to map the channel using a geomorphic unit schema defined by units with varying relative elevations (e.g., pools, chutes, riffles, and bars, in order of increasing relative elevation). The subdivision or edge density of those units could indicate variability in channel bed elevation and resulting hydraulic roughness.

4          Temporal Geomorphic Heterogeneity

Geomorphic heterogeneity includes both spatial and temporal components. While spatial heterogeneity describes variability in geomorphic units from one place to another, temporal heterogeneity describes variability in geomorphic units through time in a single place. Temporal heterogeneity is simply the rate at which geomorphic units change, or turnover, from one unit to another.
Some degree of disturbance and turnover of geomorphic units is usually a prerequisite for sustained spatial heterogeneity, as disturbances rearrange geomorphic units (Rice et al., 2012; Townsend, 1989) and the riverine habitat mosaic (Arscott et al., 2002; Stanford et al., 2005; Ward et al., 2002; Willig & Presley, 2018). Rivers that lack either driving forces (e.g., geomorphically effective flows) or forms (e.g., wood that increases roughness) that generate disturbance and turnover of geomorphic units tend towards a more homogenous state (flow regulation is a good example; Gendaszek et al., 2012; Graf, 2006). However, constant, high-magnitude disturbance is not necessary to sustain a heterogeneous character — different river systems (e.g., monsoon-dominated versus snowmelt-dominated) and different portions within a river system (e.g., channel versus floodplain) will require different turnover rates, or different disturbance regimes, to sustain different aspects of spatial heterogeneity.
Although one-time alterations, perhaps due to river restoration (e.g., Stoffers et al., 2020) or unusually high-magnitude flows (e.g., Gendaszek et al., 2012), can generate high spatial heterogeneity, estimating turnover rate can help determine whether those short-term gains are likely to be sustained, or whether an alternative state has been reached (Livers et al., 2018; Phillips & Van Dyke, 2016). For example, comparing post-restoration to pre-restoration turnover rate could indicate restoration effects on the overall erodibility of the valley bottom. Turnover rate might indicate whether that restoration simply made the landscape more heterogeneous or has reactivated the processes needed to sustain that heterogeneity.

4.1      Measuring Temporal Heterogeneity

Temporal heterogeneity can be expressed as a turnover rate (change per unit time) or its reciprocal, turnover time (time required for the entire landscape or portion of the landscape to change), both typically derived from many observations of change. Temporal heterogeneity can be calculated at the level of individual landforms, analogous to class level spatial heterogeneity metrics (e.g., floodplain turnover; Beechie et al., 2006; O’Connor et al., 2003) or for entire areas, analogous to landscape level spatial heterogeneity metrics (e.g., the turnover rate of all in-channel landforms). It is important to note that different portions of the river corridor may be expected to change at different rates, so measuring temporal heterogeneity over the entire river corridor may be misleading, whereas measuring it separately, for instance, for active channels versus floodplain surfaces versus terraces, may better represent the real propensity of the landscape to change.
Regardless of spatial scale, the interpretation of temporal heterogeneity metrics depend strongly on the definition of geomorphic units. In a meandering river, for instance, a geomorphic unit schema defined only by channel and floodplain units will have a longer average turnover time than one defined by low flow wetted channel, bars, early successional floodplain, and late successional floodplain, as the more detailed geomorphic unit schema will be more sensitive to frequent changes, such as vegetation succession.
Two pieces of contextual information are required to unbiasedly assess temporal heterogeneity: observation frequency and disturbance frequency. Observation frequency dictates the maximum detectable turnover rate, as geomorphic units cannot be observed to change more times than there are observations. Observations should be timed appropriately to the frequency with which geomorphic units are expected to change. For example, observations every 5 years will only provide a minimum estimate — likely a dramatic underestimate — of the turnover rate of fast-changing geomorphic units, like grain size patches in a gravel-bed river. Disturbance frequency, or the frequency of events that can change geomorphic units, sets the expectation for maximum potential temporal heterogeneity. A system with very low in-channel geomorphic unit turnover rate might be behaving just as expected if there have been no geomorphically effective flows in the period of measurement, but that same turnover rate over a period of multiple major floods would likely indicate a channel boundary that is very resistant to change, assuming observations were timed appropriately. Dating of floodplain strata or the use of historical imagery can be effective ways of extending the period over which temporal heterogeneity is measured, which can be key to measuring turnover rates for slow-changing geomorphic units. It stands to reason that normalizing turnover rate by dividing it by disturbance frequency can be a useful way of comparing across sites with similar geomorphic processes but differing rates of those processes (e.g., different flow regimes).

5          Contextualizing Geomorphic Heterogeneity

Geomorphic heterogeneity metrics pair well with descriptors of process space, utilization of process space, and geomorphic trajectory. Confinement, or process space (sensu Ciotti et al., 2021) describes the proportion of the valley bottom over which fluvial processes are active (e.g., the proportion of the valley bottom occupied by channels and floodplains). Process space utilization is the degree to which the river is actively reshaping the space available to it (e.g., the proportion of the channel and floodplain area occupied by channels either at a given time or cumulatively over a period of time). Rivers with greater process space can exhibit higher lateral connectivity, lower longitudinal connectivity, and higher spatial heterogeneity (Choné & Biron, 2016; Fotherby, 2009; Williams et al., 2020; Wohl et al., 2018; Wohl & Iskin, 2019). Because process space can regulate spatial heterogeneity, it can help to measure process space to aid in interpreting either spatial or temporal variability in heterogeneity metrics.
Although measurements of process space can identify constraints on the river corridor, measurements of process space utilization, or the degree to which the river is actively reshaping its fluvial process space, can identify deficiencies in the ingredients necessary to reshape the available space (e.g., flow, wood, sediment, etc.). Process space utilization can be measured by the proportion of the channel and floodplain area occupied by channels either at a given time or cumulatively over a period of time. Temporally, process space utilization can be measured as the turnover rate of the fluvial process space. Process space utilization can indicate the effects of different magnitudes of forcings (e.g., various flood magnitudes or durations), provide key context for observed temporal heterogeneity, or highlight hotspots of change that have caused changes in landscape scale spatial heterogeneity.
Finally, geomorphic trajectory (Fryirs et al., 2012; Mould & Fryirs, 2018; Surian et al., 2009), especially the trajectory of spatial heterogeneity metrics, can indicate whether the geomorphic processes that sustain heterogeneity are indeed active. If geomorphic heterogeneity is being sustained or increasing through time and the metrics used to infer geomorphic heterogeneity adequately reflect active geomorphic processes, then it stands to reason that geomorphic processes are active or even increasing in magnitude or rate.

6          Summary Recommendations for Applying Geomorphic Heterogeneity

Here, we have discussed how metrics of spatial and temporal heterogeneity can represent geomorphic processes and characteristics. Table 1 summarizes the aspects of heterogeneity discussed above and their meaning.
 
Table 1: Aspects of heterogeneity and their meaning in a geomorphic context.
Heterogeneity Aspect
Typical Application Level
Meaning
Spatial Heterogeneity
Diversity (richness and/or evenness)
Landscape
Richness describes how many unique geomorphic unit types exist in an area.
Evenness describes the relative abundance of each unit type, with low evenness indicating that some units take up most of the landscape, and high evenness indicating that most units take up the same proportion of the landscape.
Spatial Configuration
Class or Landscape
Describes the geometry and arrangement of individual or groups of geomorphic unit types, including, but not limited to, their subdivision, edge density, and aggregation.
Temporal Heterogeneity
Class or Landscape
Describes the rate at which geomorphic units change, or turnover rate.
 
In conclusion, we suggest eight considerations when evaluating geomorphic heterogeneity in river corridors:
There is no single degree of geomorphic heterogeneity that defines a well-functioning river corridor. Geomorphic heterogeneity expectations (or, for restoration, goals) will differ depending on active geomorphic processes, process space, disturbance regime, the metrics and geomorphic unit schema being applied, and what people value about a river corridor.
Carefully design geomorphic unit schemas to address specific objectives. To evaluate a specific process, consider what components of the river corridor that process tends to alter, and how to define those components in a way that is relevant for that process.
Select heterogeneity metrics that will describe the characteristics of geomorphic unit assemblages that relate to specific objectives. Different metrics are sensitive to different characteristic, such as spatial configuration, evenness, or turnover. Choose metrics that conceptually represent characteristics of interest, then check that they match qualitative observations across a range of possible conditions.
Interpret spatial heterogeneity metrics in the context of potential heterogeneity, fluvial process space, and the processes that drive geomorphic change. Measurements of spatial heterogeneity should usually be contextualized by expectations about the maximum level of heterogeneity a river corridor might be expected to achieve, based on available fluvial process space, flows of water, wood, and sediment, and ecological function.
Use multiple metrics to achieve more holistic descriptions of heterogeneity. For instance, evenness and subdivision metrics can together describe overall valley bottom heterogeneity better than either metric alone.
Set expectations for heterogeneity that are specific to the system in question. Applying the same geomorphic unit schema across rivers with very different characteristics and active processes may yield misleading comparisons unless the schemas and metrics applied are comparable. It may be more appropriate to compare heterogeneity metrics based on unique geomorphic unit schemas specific to each river system, but that reflect analogous processes.
Frame expectations of heterogeneity based on scale. Landform spacing is scale-dependent (e.g., pool-riffle spacing depends on channel width; Gregory et al., 1994), meaning that so too are heterogeneity metrics that describe them (e.g., compare the narrow versus wide channels in Figure 1). Similarly, measurements of turnover rate depend on the observation interval relative to disturbance frequency. Varying spatial or temporal scales will produce different and potentially incomparable heterogeneity metrics.
Provide context for spatial heterogeneity using temporal heterogeneity. Although spatial heterogeneity alone can be useful, it may produce misleading conclusions without the context that comes from evaluating turnover rate. This is especially important when the sustainability of a heterogeneous state is in question.
Geomorphic heterogeneity, or the spatial and temporal variability in geomorphic units, is a useful tool that allows investigators to infer geomorphic processes and quantify characteristics hypothesized to regulate those processes. By applying the concepts discussed here, we hope that investigators can continue to develop novel and effective applications of geomorphic heterogeneity to improve our ability to describe river forms and processes.

7          Acknowledgements and Data Availability

Funding for the development of methods discussed here was provided by the United States Forest Service National Stream Aquatic Ecology Center and Pacific Northwest Research Station, as well as the McKenzie Watershed Council. Thanks to Brian Collins, Todd Hurley, Kate Meyer, Jared Weybright, Johan Hogervorst, Matt Helstab, and Ryan DeKnikker for discussions that helped refine the ideas presented here. Thanks to two anonymous reviewers for comments that helped improve the manuscript.
Data shown in this manuscript is solely to provide illustrative examples of various heterogeneity metrics, not as accurate representations of the landscapes shown.

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