IntroductionIt is understood widely that key components of the natural flow regime and associated physical conditions and processes, such as water temperature and sediment regime, are critical for sustaining native aquatic species (Poff et al. 1997; Yarnell et al. 2020; Gasith and Resh 1999). Streamflow alterations for water diversions, hydropower, and flood control have contributed to rapid loss of aquatic biodiversity and ecosystem services (Häder and Barnes 2019; Tickner et al. 2020; Tonkin et al. 2019), prompting widespread development of environmental flows. Environmental flows are flow regimes designed to achieve a set of desired ecological outcomes—defined as riverine species or processes of management interest that can be maintained through flow management—while also sustaining human needs (Arthington et al. 2018).Researchers and natural resources agencies have used numerous approaches to develop environmental flows which all rely on different methods, assumptions, and data requirements (Tharme 2003). For example, the Tessman (1979) method is based on average annual and monthly natural flows, whereas the functional flows approach (Yarnell et al. 2020) focuses on maintaining key natural flow aspects that are understood to support a suite of critical river processes (e.g., peak flows and spring recession flows). Although methods vary widely, all approaches are similar in that ecological outcomes are characterized in part by flow–ecology relationships—models linking ecological characteristics to the flow regime, either directly or through mediating physical habitat conditions (e.g., sediment composition, water temperature, and hydraulics) or biological factors (e.g., food web dynamics) (Horne et al. 2019; Wheeler et al. 2018). Flow–ecology relatsionships thus are fundamental for developing and adaptively managing environmental flows (Horne et al. 2018).Researchers have used Bayesian network (BN) models to portray inferred causal links between flow and ecological outcomes—described using conditional probabilities—to evaluate alternative flow management decisions with respect to these outcomes (Horne et al. 2018). For example, Stewart-Koster et al. (2010) used a BN model to estimate the likelihood of low dissolved oxygen events based on expert understanding of water velocity and riparian cover. Shenton et al. (2011) depicted the spawning and recruitment potential of native fish under different frequencies and magnitudes of seasonal flow events to inform development of environmental flows under different climate conditions. BN models can incorporate multiple information types (Castelletti and Soncini-Sessa 2007), and inherently represent uncertainty through probability distributions (Chen and Pollino 2012; Uusitalo 2007).Because river ecosystems are inherently complex and composed of multiple interacting and uncertain flow–ecology relationships (Acreman et al. 2014; Colloff et al. 2018; Poff 2018; Williams et al. 2019), opportunities remain to improve how BN models account for uncertainty. Uncertainty—defined as any departure from a complete understanding of a system—can result from inherent variability, incomplete knowledge, or both (Horne et al. 2017; Walker et al. 2003). Uncertainty can be categorized into four levels that extend from a known range of values (Level 1) to deep uncertainty (Level 4) (Courtney 2003; Marchau et al. 2019; Wang et al. 2020a). Traditional BN models represent uncertainty through a single probabilistic relationship (Level 2). However, this may underrepresent uncertainty for complex flow–ecology relationships that cannot be understood as a single set of conditional probabilities. In these instances, BN models may produce inconclusive results (e.g., Shenton et al. 2011) or fail to fully communicate uncertainty. Thus, an approach is needed that allows multiple possible scenarios with no known likelihood function (Level 3) to be incorporated within BN modeling of natural systems.Uncertainty in flow–ecology relationships can derive from (1) using limited data to develop or test relationships; and/or (2) an incomplete understanding of the critical attributes—or inherent characteristics—of each relationship, such as the geomorphic setting, climate, or antecedent conditions (e.g., Lynch et al. 2018; Walters 2016). This information is fundamental for informing relationships and model boundaries (Walker et al. 2003). Although some studies have used literature reviews to improve the understanding of flow–ecology relationships (e.g., Greet et al. 2011; Miller et al. 2013; Poff and Zimmerman 2010) and inform BN modeling efforts, there is an additional need to consider the attributes that underpin these relationships, given the mounting interest in developing and implementing environmental flows at watershed or larger scales with limited data (Arthington et al. 2018).This study presents a novel quantitative approach for representing higher (Level 3) uncertainty and data gaps within flow–ecology BN models to facilitate development of effective catchment-scale environmental flows using common and accessible tools (Fig. 1). First, flow–ecology relationships for the study watershed were identified and characterized through a rigorous literature review to extract key categorial, temporal, and spatial attributes and identify research gaps. These findings informed an exploratory BN model and sensitivity analysis that accounts for the different levels of uncertainty and data gaps revealed by the literature review. Following a description of the case study, we present the methods and results first for the literature review and then for the BN model and sensitivity analysis. Although this paper focuses on a single case study watershed due to data and resource limitations, the proposed framework for representing data gaps and uncertainty in flow–ecology relationships is readily applicable to the many other watersheds and regions seeking to develop effective catchment-scale environmental flows with limited data and understanding.Literature ReviewMethodsPeer-reviewed studies that relate directly to the study watershed and pertain to flow, in-stream physical habitat, or desired ecological outcomes for the watershed were compiled systematically, and key categorical, temporal, and spatial attributes were recorded. The set of studies considered initially was identified through a search of peer-reviewed journal articles published on or before May 26, 2020 in the database Scopus using the keyword–abstract–title search criteria “South Fork of the Eel River” OR “South Fork Eel River” OR “Eel River Basin.” Twenty-five of the 91 studies that were returned described SFER riverine relationships in the following categories: flow–species, flow–habitat, species–species, habitat–species, habitat–habitats, or species–habitat (Table S1). In this case, flow describes streamflow characteristics (e.g., duration, timing, and magnitude) or seasonal components (e.g., summer baseflow and peak flows), and habitat describes flow-influenced variables or other physical habitat conditions (e.g., light and contributing area). Forward and backward citation chaining (e.g., Jalali and Wohlin 2012) then was used to identify any additional articles cited by or within these 25 studies that met the aforementioned criteria, resulting in 109 studies, which were reduced to 66 based on reapplication of the initial criteria. Most studies did not reference the SFER watershed in their abstract, title, or keywords, which means that they were found only through several citation chaining iterations. Theses, grey literature, and review articles were not considered. Flume and laboratory experiments were included only if they used species sourced directly from the SFER watershed.The selected studies were read to extract key categorial, temporal, and spatial attributes expected to improve the conceptual understanding of flow–ecology relationships and identify research gaps. Relationship attributes were recorded and coded in the qualitative software ATLAS.ti version 8.4 according to the protocols in Tables S1 and S2. Data were analyzed and visualized using R version 3.5.1 and ArcGIS Pro version 2.5.1. Categorical attributes recorded included relationship type (e.g., flow–species or species–species) and variables related to flow, habitat (e.g., hydraulics and stream temperature), and species within each relationship. Flow characteristics (e.g., duration and magnitude), life stages (e.g., seedling, juvenile, and adult), and interactions (e.g., breeding and predation) were specified when possible. Summary statistics portray the distribution of relationship types and variables across relationships. Categorical attributes were used to create a conceptual network diagram of relationships, in which degree centrality (number of links from variables) determined the node size and link weight (summed across relationships) determined the line thickness (Csardi and Nepusz 2006). Multiple attribute relationships (e.g., light to algae and macroinvertebrates) were split into multiple links (e.g., light to algae and light to macroinvertebrates).Additional categorical attributes extracted from the studies to inform the BN model sensitivity analysis included a short description of methods and key findings for each relationship, as well as any specified units (e.g., cubic meters per second), thresholds (e.g., bankfull flow), or states (e.g., greater than bankfull flow). Uncertainty in relationships was ranked from Level 1 to 4 (low to high) based on author judgement (Marchau et al. 2019; Wang et al. 2020a). Uncertainty can be based on a specific range of values (Level 1), a known probability distribution (Level 2), or several potential scenarios with no known likelihood (Level 3). For example, a Level 2 uncertainty is the likelihood of winter flows greater than bankfull, and a Level 3 uncertainty is the composition of invasive fish to native fish within the SFER in 50 years. No relationships were classified as deep uncertainty (Level 4), which pertains to events we have not experienced and of which we have no understanding.Interannual and seasonal hydrologic variability influence ecological outcomes in Mediterranean rivers (Gasith and Resh 1999), and therefore are expected to play a pivotal role in SFER flow–ecology relationships. The dates of data collection were used to assess seasonality and representation of water year types (WYTs) within and across relationships. To be consistent with past studies (e.g., Kelson and Carlson 2019) and the California Department of Fish and Wildlife, five WYTs (i.e., very dry, dry, moderate, wet, and very wet) were defined based on annual flow quintiles over the period of record (1967–2019) at the Elder Creek USGS gage (USGS 11475560). Elder Creek is an undammed tributary to the SFER which flows through the Angelo Coast Range Research Reserve—a pristine environment with cool, groundwater-fed tributaries and high-quality habitat in which local researchers have focused significant data collection efforts for decades (e.g., CDFW 2014; Greer et al. 2019; Wang et al. 2020b). WYTs were assigned to each study year within relationships. Binary presence-and-absence counts of WYTs were used to determine the number of unique WYTs encompassed within a relationship and the total composition across relationships.Finally, data collection locations were spatially referenced to reach segments to characterize the spatial coverage and resolution of relationships across channel types (Guillon et al. 2020) using coordinates, maps, or study area descriptions provided within the studies. Spatial referencing indicates that data were collected in the vicinity, but not the density or method of data collection (i.e., points versus transects), which often were not provided. Similar to the WYT analysis, binary presence and absence counts of channel types were used to determine the number of unique channel types encompassed within a relationship and the total composition of channel types across relationships. The location and relative density (kilometer per square kilometer) of relationships across the watershed were visualized in ArcGIS Pro using the line density tool, which sums the length of studied segments and divides by a search area (radius=1,240  m2).ResultsCategorical AttributesThe literature review resulted in 88 unique flow–ecology relationships for the SFER; 26% of relationships were described using a threshold (e.g., greater than bankfull), and only 12% were represented probabilistically (i.e., proportions, probabilities, or Type 2 uncertainty). Only six relationships were portrayed both probabilistically and with thresholds. Of the total relationships, 49% were habitat–species, 33% were species–species, and only 15% were flow–species. Although several relationships discussed how flow affects species through mediating physical conditions, no specific relationships between flow and habitat were identified. Studies were unbalanced across flow, species, and habitat. Algae (17.5%), aquatic macroinvertebrates (17.5%), foothill yellow-legged frog (FYLF, 16%), and steelhead (16%) made up 67% of all 11 species groups identified from the studies, and several species accounted for only 1%–4% of relationships. Across all species, 56% of specified life stages were juvenile, and 44% were adult. In terms of biological factors, 47% related to feeding (e.g., predation and food webs), and 4% discussed predation by an invasive species. Rearing (26%) relationships were more frequent than breeding (19%) or migration (4%). No relationships related to coho and chinook salmon despite their federally threatened status in the watershed.Water temperature was the most common physical habitat condition (27%), followed by general habitat (15%), which commonly was used for multispecies relationships such as a habitat assessment for a native fish assemblage. Velocity (12%) and nutrients (10%) were the next most common, followed by light, depth, sediment, width, and shear stress. Although many relationships considered flow as an inherent site condition, only 13 included direct links to the flow regime. Peak flow (40%) and the spring recession (40%) were the most represented flow components, followed by dry-season baseflow (13%) and wet-season initiation flows (7%). Flow often was described by magnitude (65%), and was described explicitly in terms of WYT 23% of the time. Timing was used twice to describe flow, and no relationships were described explicitly in terms of duration or frequency.The conceptual network diagram highlights the disproportionate amount of information present within SFER peer-reviewed studies of aquatic species and physical habitat compared with flow (Fig. 2). The most well-studied relationship (n=10) was between water temperature and FYLF. Others included species–species relationships between algae and macroinvertebrates, and the relationship between physical habitat and algae. The most well studied flow–species relationships were those of spring recession–FYLF and peak flow–algae.Temporal AttributesMost flow–ecology relationships used data collected only during summer months [Fig. 3(b)]. In fact, 57% used data collected only from June through September, which coincides with low-flow conditions [Fig. 3(a)]. An additional 23% of studies (80% total) used data spanning May–October. Within any given year, few relationships included data over periods longer than 180 days (10%), and only 8% used data collected in November, December, and January.A histogram of the number of unique WYTs used to develop relationships has a right-skewed distribution [Fig. 4(a)]. A total of 56 relationships (∼65%) used data spanning only 1 of 5 possible WYTs, 13 (15%) spanned 2 WYTs, and few included more than 2. Fig. 4(a) also illustrates the specific WYTs used to develop each relationship, organized by the number of unique WYTs represented. The total composition of WYTs across all relationships was more equally distributed, with very dry and dry WYTs slightly more common than wet or very wet WYTs [Fig. 4(c)].Spatial AttributesThe distribution of unique channel types across relationships also was right skewed, with 64 relationships (74%) based on only 1 channel type, 18 relationships (21%) based on 2 channel types, and no relationships spanning all 7 channel types [Fig. 4(b)]. Across all relationships, channel types SFE04 (confined, high width-to-depth ratio, gravel–boulder, uniform) and SFE05 (confined, low width-to-depth ratio, gravel–cobble, uniform) were most common, composing 58% and 30% of the channel types represented across relationships, respectively [Fig. 4(d)]. Monitoring sites were particularly underrepresented relative to their occurrence in the watershed in confined high-gradient cobble–boulder step-pool streams (SFE07) and partly confined gravel–cobble uniform streams (SFE05) [Fig. 4(d)].Data collection was clustered spatially in the watershed. High-density data collection occurs in a public state park in the northern watershed with a range of channel types, including mainstem (SFE01 and SFE04) and tributary (SFE05 and SFE07) settings, and in a research reserve to the south that has been the focus of significant data collection for decades (e.g., CDFW 2014; Greer et al. 2019; Wang et al. 2020b) (Fig. 5). The reserve contains channel types SFE04 and SFE05, contributing to their high representation across relationships relative to their prevalence in the watershed [Fig. 4(d)]. Data collection also occurs along the mainstem SFER, which parallels a highway and intersects several small towns.DiscussionCritical Data Gaps and ImplicationsCritical data gaps identified in the flow–ecology literature for the study watershed highlight the challenges of relying on data collected for other purposes to inform broadly applicable flow–ecology relationships. Just as few of the 66 articles we reviewed provided explicit and quantifiable ecological responses to flow, a review of 359 data sets for the Murray–Darling watershed in Australia found that only 9% were useful for developing or testing flow–ecology relationships (Colloff et al. 2018). Of those studies that did directly link biological or physical habitat data with streamflow attributes, categorical, spatial, and temporal data gaps limited the ability to confidently extrapolate these relationships across and between watersheds. It is well established that ecological outcomes for rivers can vary by season (Gasith and Resh 1999; Tattam et al. 2017), WYT (Null and Viers 2013; Rheinheimer et al. 2016) and environmental gradients (Acreman et al. 2014). However, echoing past studies (e.g., Walters 2016), data available to inform flow–ecology relationships were skewed to a small subset of species, functional flow components (i.e., peak flows and spring recession flows), flow characteristics (i.e., magnitude), WYTs, and geomorphic settings.Although various watershed characteristics (e.g., channel type and geology) across the SFER create a gradient of environmental conditions and ecohydraulic responses to flow (Dralle et al. 2018; Guillon et al. 2020; Hahm et al. 2019), most identified relationships relied on data collected in relatively pristine environments and a limited number of geomorphic settings (Figs. 4 and 5). The temporal bias toward low-flow periods also has major management implications for the study watershed, in which wet season diversion limits are needed for cannabis growers and to size off-stream storage tanks appropriately (SWRCB 2019). These examples underline mounting concerns of ecological data being collected from a small subset of streams (George et al. 2021; Poff and Zimmerman 2010) and spatial autocorrelation in data (Bruckerhoff et al. 2019) that then is used to inform environmental water management over much larger spatial scales. Study findings highlight the challenge of using empirical flow–ecology relationships developed for a subset of species and physical conditions to develop comprehensive ecosystem-scale environmental flows, which increasingly is required by holistic approaches (Horne et al. 2017; Mierau et al. 2018; Yarnell et al. 2020). Ultimately, these data gaps represent a critical limitation for water managers, who often require specific flow thresholds and defensible evidence linking streamflow to desired ecological outcomes for decision-making (Acreman 2005; Colloff et al. 2018; Miller et al. 2018).Study results point to a larger need for rigorous approaches to (1) extrapolate available information or relationships within and outside a study watershed, and (2) quantitatively account for uncertainty in flow–ecology relationships in subsequent modeling and management. River ecosystems are inherently complex and are composed of multiple interacting relationships (Acreman et al. 2014; Colloff et al. 2018; Poff 2018; Williams et al. 2019). Expanding the watershed literature review to include other information sources such as state agency data sets, grey literature (e.g., Asarian et al. 2016; Higgins 2013), laboratory and flume studies, and studies from outside the study area inevitably would reduce some data gaps, and is recommended for future work.Although the “more data needed” mantra applies to this problem, researchers will never have sufficient data to describe flow–ecology relationships fully. Given this reality, there are inevitable trade-offs between incorporating more data within BN models and increased uncertainty in the modeled ecological outcomes. There also are trade-offs between reducing uncertainty in a relationship or ecological outcome through space (i.e., more monitoring sites) versus time (i.e., continued monitoring at established sites). For example, a relationship developed across consecutive dry years may be stronger than one developed across different WYTs; however, it would not account for the role of interannual variability in ecological response (Lynch et al. 2018). Some of these trade-offs can be explored quantitatively using the sensitivity analyses described herein. However, the appropriate balance ultimately will depend on the identified ecological management objectives for a watershed and available resources. Regardless, particularly in regions with high hydrologic and/or geomorphic variability, systematically documenting and characterizing the critical spatial and temporal attributes inherent in specific relationships as demonstrated here for the SFER could facilitate more-meaningful interpretation and application of flow–ecology relationships (Bruckerhoff et al. 2019).Representing Uncertainty through Sensitivity AnalysisA major challenge for water managers is contextualizing the impacts of flow with other limiting variables such as physical habitat or food web dynamics, which may or may not be impacted by flow (Poff 2018). A benefit of BN models is their ability to highlight additional relevant variables to an ecological outcome. However, due to data limitations, elicitation of relationships and conditional probabilities often are subject to expert opinion—which inevitably is uncertain (Cook 1991). Our approach for combining Level 2 and Level 3 uncertainty within BN models removes the need to specify a single accurate conditional probability for relationships. Because only 6 of 88 identified relationships for the SFER were presented both probabilistically and using thresholds, elicitation of relationships and conditional probabilities still was subject to our own opinions. However, it allowed us to evaluate BN model sensitivity in light of the biases and uncertainties observed across several dimensions of the flow–ecology literature (i.e., what, when, and where).Performing sensitivity analyses within a BN modeling framework enabled more extensive consideration of the inherent complexity in flow–ecology relationships than in past studies. Despite uncertain conditions and incomplete knowledge, natural resource managers are tasked with making decisions to support aquatic ecosystems, and they require tools to do so (Acreman 2005; Pullin et al. 2004). Previous BN studies primarily focused on Level 2 uncertainty, using a combination of literature and personal judgement to assign node states and conditional probabilities (e.g., Chan et al. 2012; Stewart-Koster et al. 2010). To the authors’ knowledge, this is the first study to apply sensitivity analysis to explore several sets of possible conditional probabilities for flow–ecology relationships to account for Level 3 uncertainty. Such an approach removes the need to specify a single accurate conditional probability for relationships. Key outcomes of these analyses are summarized in the rest of this section.The inconsistency of BN model outcomes under moderate conditions compared to dry or wet conditions across Scenario A runs suggests that uncertainties in flow–ecology relationships are more significant in certain water years than others. Depending on the management objectives, watershed managers could use these findings to support additional monitoring or to implement more ecologically conservative diversion policies during hydrologic conditions exhibiting higher uncertainty in ecological outcomes. Observed sensitivity of juvenile salmonids to Level 3 uncertainties further indicates that BN models that do not consider these sources of uncertainty may lead to false confidence in a particular outcome. Although it is unlikely that near-perfect certainty in flow–ecology relationships would ever be achieved, it often is assumed for modeling purposes. For example, Shenton et al. (2011) specified conditional probabilities for triggering spawning as 0% (triggered) or 100% (not triggered) for different combinations of environmental conditions. Horne et al. (2018) characterized the state of two nodes (macroinvertebrate biomass and diversity and existing overall condition) as 100% good condition. Scenario A results [Fig. 8(a)] demonstrate that even small uncertainties in the BN model base conditional probabilities may substantially alter the expected ecological outcome. Alternatively, near-perfect certainty in middle-node relationships (e.g., algae to food supply) significantly increased the likelihood of a certain outcome under wet and dry hydrologic conditions [Fig. 8(b)]. Taken together, these results emphasize that assuming near-perfect certainty in a conditional relationship may inflate confidence in a certain ecological outcome, and that this bias may be magnified under certain hydrologic conditions.Uncertainty in our understanding of limiting variables in river ecosystems was also shown to have a large impact on the expected ecological outcome [Fig. 8(b)]. Under the same hydrologic conditions, the likelihood of a good outcome for juvenile salmonids varied across Scenario B runs depending on whether food supply and fish growth or longitudinal connectivity was considered most important. Although it was outside the scope of this study, the importance of these variables also may change through time or by location. For example, disease may become more prevalent across a watershed as streams warm with summer air temperatures (Schaaf et al. 2017) and additional environmental stressors, such as nonnative predation, may become more important as invasive species expand throughout the Eel River basin (Moyle et al. 2017). Given challenges in isolating limiting variables that impact aquatic species (Holmes et al. 2018) and knowing how relationships will hold through time (Horne et al. 2019), traditional BN modeling using one set of assumptions risks making incorrect assumptions and drawing inaccurate conclusions.Limitations and Future ResearchThe main purpose of this study was to exemplify how information extracted through a rigorous review of the peer-reviewed literature can be compiled into a BN model that explicitly represents various levels and types of uncertainty. As a result, the model does not consider the full range of conditions that are important to juvenile steelhead or other ecological outcomes, including other steelhead life stages. Similarly, BN models have a limited ability to represent cyclical loops or dynamic systems (Hart and Pollino 2009; Uusitalo 2007), so there are inherent limitations in using them to model systems that change from year to year, such as ecosystems or Mediterranean climates in general. The BN model structure and conditional probabilities also reflect the personal judgment of the authors, and do not include insights from other experts or relationships derived from other information sources. Although including additional information sources likely would improve representation of ecological outcomes, this sensitivity analysis can be applied to existing or future BN models to provide insights under multiple levels of uncertainties and in light of additional information across watersheds. For example, researchers could apply different uncertainty ranges in base conditional probabilities to existing BN models or develop new scenarios to explore other Level 3 uncertainties, such as those related to model structure or the effects of different management actions such as flow diversion limits, forest management, or habitat improvement projects.Simplifications and assumptions were needed to handle the complexity, variability, and number of flow–ecology studies reviewed. For example, WYTs calculated using streamflow data from a single gauge with a long record were applied over the entire watershed. However, effects of this decision on study outcomes are expected to be minimal because climate is relatively uniform across the watershed. We also considered only whether a given WYT was represented in a flow–ecology relationship, and not the number (e.g., 3 dry years) or sequence (e.g., dry–wet–dry) of WYTs. 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