A Prefire Approach for Probabilistic Assessments of Postfire Debris‐Flow Inundation

Increases in wildfire activity and rainfall intensification are driving more postfire debris flows (PFDF) in many regions around the world. PFDFs are most common in the first postfire year and may even occur before a fire is fully controlled. This underscores the importance of assessing postfire hazards before a fire starts. Evaluation of PFDF hazards prior to fire can help strategize interventions lessening the negative effects of future fires. However, debris‐flow runout and inundation analyses are not routine in PFDF hazard assessments, partially due to time constraints and substantial uncertainties in boundary conditions. Here, we propose a prefire PFDF inundation assessment framework using a debris‐flow runout model based on the Herschel‐Bulkley (HB) rheology (HEC‐RAS v6.1). We constrain model inputs and parameters using Bayesian posterior analysis, rainfall‐runoff simulations, and a debris‐flow volume model. We use observations from recent PFDF incidents in northern Arizona, USA, to calibrate model components and then apply our prefire inundation assessment framework in a nearby unburned area. Specifically, we (a) identify yield stress as the most influential factor on inundation extent and arrival time in a HB model, (b) establish posterior distributions for model parameters suitable for forward modeling by leveraging uncertainties in field observations, and (c) implement a predictive forward analysis in an area that has not burned recently to evaluate PFDF inundation under several future fire scenarios. This study improves our ability to assess postfire debris‐flow hazards before a fire begins and provides guidance for future applications of single‐phase rheological models when assessing PFDF hazards.


Introduction
Postfire debris flows (PFDF) pose a threat to lives and infrastructure.PFDFs have been reported in a variety of geographic and climatic regions around the world, including Australia (Nyman et al., 2011), Canada (Jordan, 2016), China (Wang et al., 2022), Italy (Esposito et al., 2023), Greece (Diakakis et al., 2017), Portugal (Lourenço et al., 2012), Switzerland (Conedera et al., 2003), and the western USA.They initiate primarily from infiltration-excess overland flow, promoted by fire-driven reductions in ground cover and soil infiltration capacity, during short-duration high-intensity rainfall in headwater drainages (Kean et al., 2011(Kean et al., , 2012;;Nyman et al., 2011).PFDFs may flow for substantial distances down channels and unconfined alluvial fans, ultimately inundating and potentially damaging buildings, roads, and other infrastructure (Gorr et al., 2023;Kean et al., 2019).They are most common in the first year following fire (DeGraff et al., 2015;Hoch et al., 2021), and in some instances, PFDFs can initiate even before a fire has been contained or brought under control (Kean et al., 2019).Ongoing and anticipated increases in fire activity (Senande-Rivera et al., 2022) and rainfall intensification (Fowler et al., 2021) underscore the need to anticipate and prepare for postfire debris flow hazards even before fires ignite (Tillery et al., 2014;Wells et al., 2023).By proactively assessing PFDF hazards before a fire, authorities and communities can strategize and implement mitigation measures more effectively.
Assessing postfire debris-flow hazards involves identifying watersheds that are susceptible to debris flows, estimating the volume (magnitude) of a potential debris flow, and routing the debris flow across the landscape to determine its potential travel path.Previous work has established methods for estimating debris-flow likelihood (Staley et al., 2017) and volume (Gartner et al., 2014) at the watershed scale in the western USA based on terrain and soil characteristics, burn severity, and peak 15-min rainfall intensity.Models for postfire debris-flow likelihood and volume have also been developed and applied in other fire-prone regions, including the Mediterranean and Australia (Diakakis et al., 2023;Nyman et al., 2015).However, while a number of models, such as RAMMS (Christen et al., 2010), FLO-2D (O'Brien et al., 1993), D-Claw (George & Iverson, 2014;Iverson & George, 2014), and ProDF (Gorr et al., 2022), simulate PFDF inundation with comparable results (Barnhart et al., 2021), debris-flow inundation estimates are not currently standard practice in postfire hazard assessments (Barnhart, Miller, et al., 2023;Barnhart, Romero, & Clifford, 2023).This arises, in part, due to time constraints associated with completing hazard assessments following fire as well as uncertainties associated with representing the internal dynamics of debris flows.Additional time to perform debris-flow runout simulations in a prefire context, as opposed to a postfire context, makes it more feasible to augment existing hazard assessment frameworks with estimates of debris-flow inundation.Developing workflows that incorporate debris-flow inundation into hazard assessments would therefore benefit both prefire and postfire assessments, though additional modifications might be needed to streamline the process for postfire applications where time is more limited.
The Hydrologic Engineering Center's River Analysis System (HEC-RAS) is among the most widely used software in flood risk analysis (Gibson et al., 2022).Its single phase non-Newtonian module (Gibson et al., 2022) includes multiple rheology models, such as the Bingham model (Bingham, 1922), Quadratic model (O'Brien et al.,1985), Herschel-Bulkley model (Major & Pierson, 1992), and Voellmy model (Voellmy, 1955).These models can be used to estimate the movement of hyper-concentrated flows, mudflows and debris flows over complex topography to assess hazards.Although the verification and validation (Gibson et al., 2021) and application (Gibson et al., 2022) of non-Newtonian algorithms in HEC-RAS are available, systematic exploration of parameter sensitivity would assist with applying the model in practice to assess debris-flow inundation hazards.Determining appropriate parameter values, especially in postfire settings where there is limited data for constraining debris-flow runout and inundation models, remains a challenge.Therefore, a comprehensive global sensitivity analysis (GSA) to rank parameter importance is needed to better understand model behavior and to provide guidance on parameter optimization.
Efforts to quantify uncertainty associated with postfire debris-flow inundation modeling and to provide guidance for parameter choices are limited by a general paucity of postfire debris-flow inundation data.Detailed hydrogeomorphic field data, including the measurements of flow depth, extent of inundation, debris-flow timing and rainfall hyetographs for the triggering rainstorm, and soil-hydrologic properties may be required for model set up, calibration, and evaluation.Here, we take advantage of recently acquired field observations and hydrogeomorphic data from a runoff-generated debris flow following the 2022 Pipeline Fire in northern Arizona (Gorr et al., 2024).We propose a modeling framework for a probabilistic PFDF inundation assessment in a prefire context.The modeling framework is informed by a sensitivity analysis and model calibration using the Pipeline Fire observations.Specifically, we integrate the rain-on-grid and non-Newtonian algorithms in HEC-RAS with Bayesian posterior analysis (Aaron et al., 2019;Gregory, 2005).We apply the model framework to assess postfire debris-flow hazards in a nearby area that has not burned recently.More generally, this study introduces a prefire approach to PFDF inundation modeling that can be modified and generalized for other fire-prone regions.
which has not burned recently, as Fort Valley.We apply our prefire debris-flow inundation assessment framework at Fort Valley.
The elevation of the Copeland watershed ranges from 2077 to 3363 m above sea level with an average gradient of 22.6°(Figure 1c).This area consists of two headwater drainages (1.08 and 1.33 km 2 ) that converge near the apex of a Holocene alluvial fan.The average gradients of these two watersheds are 33.1°and28.3°, while the average gradient of the fan is 8.9°.Clastic materials are widely distributed on this rugged terrain underlain by a variety of Pleistocene-age dacites (Holm, 1988).The alluvial fan channels, with a length of approximately 7 km, coalesce near the fan apex and alternate downstream between expansion and contraction reaches due to variable confinement of the channels.
The Copeland headwater drainages were burned by the 2010 Schultz Fire and again by the 2022 Pipeline Fire.PFDFs were documented following both fires (Gorr et al., 2024;Youberg & McGuire, 2019).Data from the PFDF that occurred on 15 July 2022, shortly following the Pipeline Fire is used in this study.We refer to this particular flow as the Copeland debris flow.The Copeland debris flow traveled ∼7 km from the headwaters to the downstream neighborhood built on the unincised Holocene alluvial fan (Gorr et al., 2024).The Fort Valley site, which is located on the southwest slope of the San Francisco Peaks (Figure 1d), also consists of two watersheds (1.61 and 0.97 km 2 ) and an alluvial fan with elevations ranging from 2,355 to 3,766 m above sea level.These two watersheds, with average slopes of 30.1°and 29.4°, debouch onto an unconfined Holocene alluvial fan with an anastomosing channel network that has an average gradient of 6.0°.
The Fort Valley headwater drainages have not experienced significant fire in the last century (Heinlein et al., 2005); however, the lower slopes of the watersheds and the alluvial fan area were burned by the 2001 Leroux Fire (USDA Forest Service Southwestern Region GIS Data, https://www.fs.usda.gov/detailfull/r3/landmanagement/gis).
The mean monthly temperature of this area ranges between 2.8°C in January and 15.9°C in July with a mean annual temperature of 5.78°C.PRISM precipitation data shows the annual precipitation is 701 mm with 172 mm falling between July and August and 285 mm from December to March during the period of 1895-2022(PRISM Climate Group, 2023).The North American monsoon brings high-intensity rainfall to the area in summer months, which often leads to flash floods and debris flows in recently burned areas (Hoch et al., 2021;Raymond et al., 2020).During winter months, precipitation predominantly falls as snow (Sheppard et al., 2002).The vertical zonation of vegetation features ponderosa pine (Pinus ponderosa) forests at lower elevations, transitioning to Douglas-fir (Pseudotsuga menziesii) and Engelmann spruce forests (Picea engelmannii) before reaching alpine vegetation above tree line at 3,500 m (Pearson, 1931).

Data and Methods
We describe our data and methods in four subsections.In Section 3.1, we summarize field observations and data from the Copeland debris flow documented by Gorr et al. (2024).Section 3.2 presents the debris-flow routing model set-up, including inflow data, governing equations, and parameter sensitivity analysis (importance ranking).Section 3.3 describes how we determine the posterior probability of parameters for the debris-flow routing model.Section 3.4 demonstrates how to apply the debris-flow runout model (HEC-RAS) using the posterior probability distribution of parameters for prefire assessments of PFDF inundation.The prefire approach for probabilistic assessment of PFDF inundation is summarized in a flowchart (Figure S1 in Supporting Information S1).

Data
Five days after the Copeland debris flow, Gorr et al. (2024) mapped the extent of inundation.They then utilized these observations in combination with WorldView-3 satellite imagery taken on 22 July 2023 to estimate an inundation area of 478,267 m 2 .The volume of the debris-flow deposit was estimated at approximately 115,000 m 3 (Gorr et al., 2024).Three tipping bucket rain gauges (Figure 1) recorded rainfall during the event (Figure 2).A 1m bare earth DEM derived from airborne lidar is used to represent the pre-event surface topography in all simulations.The timing of the debris flow at the distal end of the fan is constrained by a video from a neighborhood resident (Movies S1).

Debris Flow Inundation Modeling
We simulated debris-flow inundation with the HEC-RAS single phase non-Newtonian module (Gibson et al., 2021).First, we determined the model inputs (Section 3.2.1),which required constraining inflow hydrographs from a rainfall-runoff analysis and estimates of sediment volume deposited by the debris flow.We provide a concise overview of the single phase non-Newtonian flow model in Section 3.2.2.Lastly, we introduce the model evaluation (Section 3.2.3)and the parameter sensitivity analysis methods (Section 3.2.4).

Rainfall-Runoff and Debris Flow Hydrographs
The inflow hydrograph, as the forcing of the debris-flow routing model, is a function of rainfall-produced runoff and the solid materials entrained in the flow.We estimated a runoff hydrograph using HEC-RAS rain-on-grid simulations (Costabile et al., 2021) which represent the rainfall-runoff processes over the headwater drainage areas (Figure 1a) where the debris flow initiated (Gorr et al., 2024).A 2D computational mesh was produced at 5 × 5 m spatial resolution over the overland flow area with a Manning's n value of 0.07 m s 1/3 , and 1 × 1 m mesh over the channelized area with a Manning's n value of 0.04 m s 1/3 .A perturbation (±0.01 m s 1/3 ) of the Manning's n values was used to test the variation of time-to-peak of the modeled hydrograph.In other words, a low Manning's n scenario (0.06 m s 1/3 for hillslopes and 0.03 m s 1/3 for channels) and a high Manning's n scenario (0.08 m s 1/3 for hillslopes and 0.05 m s 1/3 for channels) were also considered.A rain-on-grid precipitation timeseries was entered at a 1-min timestep by interpolating the rainfall records at the three rain gauges (Figure 1a).
The Green-Ampt method (Green & Ampt, 1911) was used to account for infiltration.Initial volumetric soil moisture was assigned as 0.29 which was inferred by Climate Forecast System Reanalysis soil moisture (0-5 cm and 0-25 cm) and Famine Early Warning Systems Network Land Data Assimilation System surface soil moisture (0-10 cm) (Figure S2 in Supporting Information S1).We estimated watershed-scale effective saturated hydraulic conductivity (Ks) and wetting front head (Hf) based on postfire flash flood and debris-flow simulations in similar burned areas (Hoch et al., 2021;McGuire & Youberg, 2020).In particular, McGuire and Youberg (2020) estimated values of Ks and Hf of roughly 4 mm hr 1 and 6 mm, respectively, following a moderate to high severity fire in a similarly forested setting in western New Mexico.We used three different values for Ks and Hf to test how variations influenced runoff timing and magnitude.We set Ks and Hf values as 1 mm hr 1 and 1 mm for a low infiltration case, 5 mm hr 1 and 5 mm for a moderate infiltration case, and 10 mm hr 1 and 10 mm for a high infiltration, respectively.We used the diffusion wave momentum equations (Gibson et al., 2022) to perform unsteady flow routing.
We then obtained a debris-flow hydrograph by bulking the runoff hydrograph with a volume of sediment consistent with the observed deposits.Assuming that the porosity of the debris-flow deposits is 0.4, the solid volume of the deposits is 69,000 m 3 (Gorr et al., 2024).Considering the uncertainty in field measurements, we varied the solid volume of the deposits by ±25%, and therefore the bulked solid volume ranged between 51,750 and 86,250 m 3 .We uniformly distributed the solid volume throughout the simulated water hydrograph to arrive at an inflow debris-flow hydrograph characterized by a volumetric concentration (VC) in the debris-flow routing model.

Debris-Flow Routing Modeling
To simulate debris-flow runout, we employed the Herschel-Bulkley (HB) rheological model within the HEC-RAS single phase non-Newtonian module (Gibson et al., 2021).The HB model fits the non-Newtonian terms (debris stress) into the depth-averaged Shallow Water Equations (SWE) model (Equations 1-3): where η is the flow surface elevation, t is time, h is the flow depth, υ h is the velocity vector and q is a source or sink term, to account for external and internal fluxes.The depth-averaged momentum conservation equation can be expressed as: where g is gravity, ν t is a turbulent eddy viscosity, C f is a bottom friction coefficient, τ is the non-Newtonian internal fluid stress, ρ m is non-Newtonian flow density.The bottom friction coefficient is computed as a function of Manning's roughness coefficient (n 0 ): where R is the hydraulic radius.
The non-Newtonian stress, τ, is then calculated from rheological models (stress-strain relationships).For the HB model (Equation 4), the debris stress, τ y can be expressed as: where τ y is the yield stress, γ is the shear rate or strain which is expressed as three times the ratio of velocity to depth (3|υ h |/h) in HEC-RAS (Gibson et al., 2021), K and c are the slope and exponent of the stress-strain curve, respectively.When c = 1, the stress-strain relationship is linear, so the HB model collapses to the Bingham rheological model.When c < 1, the flow can be described as having shear-thinning rheology and when c > 1, the flow presents a shear-thickening rheology.
A 2D computational mesh was produced at 3 × 3 m spatial resolution over the area inundated by the Copeland flow with a spatially uniform Manning's n value.We considered a mesh resolution of 3 m to adequately represent the topographic features within the simulation domain.Further refinement of the mesh size may not yield significant improvements in the accuracy of modeled inundation extent (Alipour et al., 2022).Given an inflow hydrograph (including runoff and solids), a set of HB model parameters (i.e., τ y , K, and c) and a Manning's n value Earth's Future 10.1029/2023EF004318 LIU ET AL.
(Table 1), HEC-RAS solves the governing equations using an Eulerian method with a varying timestep satisfying the Courant-Friedrichs-Lewy condition.The upstream boundary of the debris-flow simulation domain was selected at a location in the headwater drainage area where the flow was confined to a channel.Simulations were performed using HEC-RAS 6.1 (Linux version) on the University of Arizona's HPC, 94 cores 2.40 GHz, and 4 Gb memory per core.

Model Evaluation
We assessed the HB model's performance in terms of inundation and arrival time.We used a fitness metric to describe how well a model result, associated with a particular parameter set (i), matched the observed inundation area: where FP i denotes the number of false positives, namely the number of cells where a model simulated flow but there were no observed deposits and FN i denotes the number of false negatives, or the number of cells where debris-flow deposits were observed but not simulated by the model.A smaller fitness number represents a better agreement between the simulation and observation.A fitness number of zero occurs when the simulated inundation area perfectly matches the observation.
To evaluate the modeled arrival time, we used timeoff to describe how well the timing of a modeled debris flow, associated with a parameter set (i), coincides with the observed timing of the flow at the base of the fan: where SimTime i is the time when a simulated debris flow arrived at the base of the fan and ObvTime is the time when the debris flow was observed.We were able to determine the time that the debris flow arrived at the base of the fan because a local resident recorded a video of the event (Movies S1).

Sensitivity Analysis
We employed a Variogram Analysis of Response Surfaces (VARS) methodology (Razavi & Gupta, 2016a, 2016b), which provides a comprehensive characterization of the model response surface, to conduct the sensitivity analysis with the goal of ranking parameter importance.This method generalizes the variance-based Sobol approach (Sobol', 2001) and derivative-based Morris approach (Sobol', 2001) by varying the sample distances within the parameter space when quantifying the variogram variables (Razavi & Gupta, 2016a).We used the starbased sampling method (Razavi & Gupta, 2016b) to implement the VARS sensitivity analysis.
Specifically, we used fitness, timeoff, and the simulated inundation area (A sim ) to conduct global sensitivity analyses to rank the importance of each parameter.The relative importance of different parameters depends on the performance metric or model output that is used to assess sensitivity.The motivation for assessing sensitivity using fitness and A sim is that inundation extent is a basic model output that has relevance to hazard assessments, as it can be used to delineate areas downstream that may be directly impacted by a debris flow.The metric timeoff may be more relevant when using models to determine warning times, such as estimating the time between a triggering rainfall threshold being exceeded and the arrival of a debris flow at a downstream location of interest.
Both fitness and timeoff can be used to construct a metric response surface, which can be used to quantify how these two model performance metrics vary with the parameters.In contrast, A sim can be used to form a metric-free response surface because A sim is a model output variable rather than a performance metric.This metric-free response surface can also be used to quantify how A sim is controlled by different parameters.
Each of the five parameters, namely τ y , K, c, VC, and Manning's n, were assumed to follow a uniform distribution with lower and upper bounds (Table 1).The parameter ranges for τ y , K, and c were adopted from the comprehensive spans outlined in the studies by Gibson et al. (2022), Q. Liu et al. (2016), O'Brien and Julien (1988), and O'Brien et al. (1985, 1993).The VC range was established based on the uncertainty in the volume of debris flow deposits observed by Gorr et al. (2024) and the hydrograph simulations (Figure 2).Manning's n values were selected to cover a broad spectrum from 0.03 to 0.1 m s 1/3 , which align with the conventional values attributed to barren terrains and mixed forests, and are similar to those calibrated in recently burned watersheds in the Earth's Future 10.1029/2023EF004318 southwest US (Hoch et al., 2021;McGuire & Youberg, 2020).We sampled a total of 500 points within the 5parameter space and performed HEC-RAS simulations using these parameter sets.We then set up a random forest (RF) regressor as a surrogate model using the 500 HEC-RAS simulations and corresponding metrics (fitness and timeoff) and output variable (A sim ).Specifically, the inputs of the RF regressor included the 500 parameter sets while the outputs were corresponding to the fitness, timeoff, and A sim .The 500 simulations were partitioned into training (70%) and testing (30%) subsets.Constructing the surrogate model enabled us to explore the parameter space more thoroughly at much lower computational cost when compared to running a HEC-RAS simulation.
We sampled 10,000 parameters by using the same method as the original 500 points within the same parameter space and obtained corresponding fitness, timeoff, and A sim by using the random forest (RF) regressor.We then calculated a set of VARS-based sensitivity metrics by integrating the variogram across a range of scales (Table 2) using both the 500 simulations and the 10,000 RF-based predictions.To assess the uncertainty of the VARS products, we used a bootstrapping method with VARS-TOOL, to quantify the 90% confidence interval (CI) of each product.The ratio of parameter importance for a parameter is defined as a sensitivity metric value, such as IVARS 30 , of that parameter divided by the sum of that metric over all the parameters.Therefore, the ratio of parameter importance varies from 0 to 1, with a greater value indicating a greater parameter importance.

Bayesian Inference-Based Optimization
We optimized the debris-flow runout model in a Bayesian inference framework (Aaron et al., 2019) using the same 10,000 parameter sets that were used for sensitivity analysis.Aaron et al. (2019) detailed this posterior analysis.Here, we provide a brief overview of the approach.A uniform distribution is selected as the prior distribution for each parameter, π(p), where p denotes the input parameter vector.The model performance metric values of fitness (Equation 5) associated with each of the K parameter combinations can be vectorized as, Similarly, the timeoff (Equation 6) can be vectorized as, An overall evaluation metric can be written as,  (Saltelli et al., 2008).b VARS-ACE and VARS-ABE are equivalent to variations of the Morris approach, as explained in Razavi and Gupta (2015).
Due to uncertainty in both observed inundation area and arrival time, we assume each of the metrics follows a multivariate normal distribution with a mean of zero and an independent covariance matrix Σ, where σ f is the standard deviation of the fitness and σ t is the standard deviation of timeoff.Both σ f and σ t are determined by 95% CI of the errors.Specifically, considering the uncertainties from satellite imagery-including its 0.3 m resolution, unclear margins in some areas, and discrepancies between imagery-derived and fieldobserved boundaries-we adjusted the mapped inundation boundary within ±1 m, ±1.5 m, and ±3 m.This adjustment resulted in an inundation area change ranging from 5.6% to 5.6% (±1 m), 8.4%-8.3%(±1.5 m), and 16.8%-16.5%(±3 m) of the inundated area (see Figure S3 in Supporting Information S1).Then, σ f can be estimated by dividing their ranges by four under the normality assumption (Aaron et al., 2019).Since uncertainty in timeoff is introduced by imperfection of rainfall-runoff simulations, we estimated the value of σ t by examining the variation in the simulated timing of peak water discharge in the inflow hydrographs.We quantified the timing of peak discharge for a low and a high roughness case, took the difference in timing (minutes), and divided the difference by four to estimate σ t s.
In general, each of these three metrics ( f( p), t( p), and r( p)) can serve as the likelihood functions in Bayes law.
Each of these metrics provide a way of evaluating how well simulations with various parameter combinations ( p) perform relative to observed inundation area and arrival time.Here we used the overall evaluation metric (r( p)), and its likelihood function can be expressed as, where |Σ| is the determinant of the covariance matrix Σ (Equation 10).By maximizing the likelihood function, the most likely values of the parameter vector ( p) that minimize the evaluation metrics (r( p)) are then obtained and expressed as the posterior distribution (π( p|r)) of the calibrated parameters, where the denominator, equal to the sum of the numerator over the entire parameter space, serves as a normalizing constant.p ⋆ indicates that the integrand in the denominator is over the entire parameter space.
After obtaining the posterior distribution (π( p|r)) of the parameter sets, which is a list of probability values (π( p| r) i ), we calculate the highest posterior density region, also known as the credible region.This is achieved by sorting π i in decreasing order.The process begins with the highest π i value and sequentially incorporates smaller values.This inclusion continues until adding the next π i would exceed the predetermined threshold, C (e.g., C = 90%).Simultaneously, we track the associated parameter vectors, p i The credible region is thus defined as the range encompassing all p i values corresponding to the cumulative π( p|r) i values considered.The boundaries of the credible region are the minimum and maximum of these p i values.Parameter combinations that lead to optimized HEC-RAS simulation performance have greater probability and within the credible region.

Predictive Forward Analysis of Postfire Debris-Flow Inundation
We conducted a prefire analysis of postfire debris-flow hazards in the Fort Valley study area (Figure 1d).We explored four scenarios, designed to represent a range of rainfall and burn severity outcomes.Specifically, we considered cases where 50% and 100% of the watershed areas burned at moderate to high (Bmh) soil burn severity (SBS) for two design storms (2-year and 100-year) based on rainfall data from NOAA Atlas14 (NOAA, 2020 in the Fort Valley watersheds in response to a 2-year storm when 50% of the area burns at moderate to high SBS is approximately 99.9%, an assumed mean dNBR of approximately 470.We, therefore, assumed both 2-year and 100-year storms used in this study would trigger debris flows regardless of SBS. For each of the four scenarios, we obtained the input hydrographs with a method that is analogous to that described in Section 3.2.1.First, we determined the shape of runoff hydrographs in response to a rainstorm with the prescribed recurrence interval (Figure S4 in Supporting Information S1) by using a Ks of 5 mm hr 1 and Hf of 5 mm (i.e., the moderate infiltration case described in Section 3.2.1).Then, we bulked the hydrograph to account for a specific sediment volume.For the Copeland debris-flow analysis, the sediment volume was determined from estimates of the deposit volume.In our predictive model analysis at the Fort Valley study area, we employed a modified version of the emergency assessment volume model from Gartner et al. (2014).We modified the Gartner et al. (2014) volume model (Figure S5 in Supporting Information S1) by scaling the predicted volume by a factor (Rengers et al., 2023), which was determined by comparing modeled debris-flow volumes with observed debrisflow volumes following the 2010 Schultz Fire that burned the same area (Youberg, 2015).While the original Gartner et al. ( 2014) model overestimated debris-flow volume, dividing the predicted volume by a factor of 10 led to a reasonable agreement (Figure S5 in Supporting Information S1).After obtaining the deposit volume from the modified Gartner et al. ( 2014) model for each of the four scenarios, we adjusted the simulated runoff hydrograph for each watershed to ensure that the ratio of the modeled deposit volume to the combined volume of runoff and deposits is equal to the median value (57.05%) of the parameter VC range (Table 1).We then determined debrisflow hydrographs by further modifying the adjusted hydrographs so that they had a sediment concentration consistent with a sampled VC values.
We used a probability-proportional-to-size (PPS) sampling strategy (Skinner, 2016) to sample 100 members from the 10,000 parameter instances with known probabilities of selection, that is, their posterior probabilities (π( p| r) i ).π( p|r) i served as the weights of all members in the population.A random number between minimum and the maximum of the π( p|r) i was generated as a threshold to determine which instance of the 10,000 parameter instances was selected.We repeated this process until we obtained 100 members to represent 100 sets of HEC-RAS parameters that follow the posterior distribution.We then obtained 100 inundation maps using HEC-RAS (Section 3.2.2) for each of the four scenarios defined by different burn severities (50% or 100% burned at moderate-high severity) and design storms (2-and 100-year RI storms).The inundation probability of each cell was defined as the number of times (out of 100) that the cell was inundated divided by 100.As a comparison, we also ran simulations using 100 sets of parameters that follow a uniform distribution within the range specified in Table 1 for each of the four scenarios, as might be done in the absence of a local calibration site that could be used to determine the posterior distribution.

The Simulated Runoff and Inflow Hydrographs for the Copeland Watershed
We obtained two hydrographs for each of the three-infiltration scenarios (i.e., low, moderate, and high infiltration cases), one at the outlet of each of the Copeland watersheds (Figure 2).The peak flows produced by each of the three-infiltration scenarios ranged from 17.3 to 25.0 m 3 s 1 at the outlet of the northern watershed and 12.1-19.2m 3 s 1 at the outlet of the southern watershed.The total volume of runoff ranged from 39,975 to 57,515 m 3 with a runoff ratio of 0.58-0.84.
Time-to-peak is 42 min for low Manning's n scenario (0.06 m s 1/3 for hillslopes and 0.03 m s 1/3 for channels), while time-to-peak is 48 min for high Manning's n scenario (0.08 m s 1/3 for hillslopes and 0.05 m s 1/3 for channels).The difference in time-to-peak due to variations of Manning's n is 6 min, equivalent to σ t = 1.5 min.
To create the inflow hydrograph of a debris flow as described in Section 3.2.1,we added the solid volume of the observed debris-flow deposits with uncertainties into the simulated runoff hydrographs.The estimated VC is between 56.4% and 68.3% for the high infiltration case, 49.9%-62.4% for the median infiltration case, and 47.4%-60.0%for the low infiltration case.Given the substantial overlap in VC between the three infiltration levels, we selected the median infiltration-produced hydrograph to create the inflow hydrographs for the debris-flow routing simulation in this study.

GSA of HB Debris-Flow Routing Model
We estimated a set of VARS-based sensitivity metrics (Table 2) using 500 HB model simulations and the 10,000 RF-derived predictions (Figure 3).The out-of-bag (OOB) scores for all predictions (A sim , fitness, and timeoff) are larger than 0.96, indicating good performance of the RF regressor as a surrogate model.
The ratios of parameter importance for IVARS 30 , VARS-TO, and VARS-ABE show that the parameter rankings depend on model performance metrics (Figure 4).According to sensitivity metrics of VARS-ABE, IVARS30, and VARS-TO, the model is most sensitive to τ y , followed by n 0 and VC when A sim is used as the model response (Figures 4a and 4d).A sim is not sensitive to the other two parameters, which are the slope (K) and exponent (c) in the stress-strain curve.Earth's Future 10.1029/2023EF004318 When fitness, a performance metric describing the goodness of fit for the simulated inundation area relative to the observation, serves as the model response, τ y is the dominant factor followed by n 0 and VC showing moderate importance (Figures 4b and 4e).The ratio of parameter importance for τ y , is 0.51 for VARS-ABE, 0.82 for IVARS30 and 0.72 for VARS-TO.
When timeoff, a performance metric describing the goodness of timing, serves as the model response, n 0 and τ y are the two most important factors (Figures 4c and 4f) with the importance ratio of both parameters being larger than 0.4 for the three sensitivity metrics.Low to zero sensitivity is seen for parameters K and c, which is similar to what we found when A sim serves as the model response.
Parameter importance calculated from the 500 simulations and 10,000 RF predictions are consistent across the three metrics for both A sim -based analysis (Figures 4a and 4d) and metric-based analysis (Figures 4b, 4c, 4e, and  4f).The 90% confidence intervals for each sensitivity metric for each parameter are narrower when using the 10,000 RF predictions (Figures 4a-4c) than those when using the 500 simulations (Figures 4d-4f).This indicates that the parameter importance rankings are stable and reliable.Other sensitivity metrics (γ(h), IVARS10, IVARS50, VARS-ACE) also inform the same parameter ranking (Figures S6 and S17 in Supporting Information S1).
Overall, the results show that τ y exerts the strongest control on model behavior and is therefore considered the most important parameter.In contrast, n 0 and VC are moderately important, and the parameters K and c have little control on model performance using the metrics selected here.

Posterior Distribution of the Debris-Flow Model Parameters
Using the same 10,000 RF simulations (Figure 3) as those in Section 4.1, we obtained the posterior distribution of parameters using r( p), the metric that combine the fitness and timeoff, for assessing model performance.
Among 10,000 RF predictions, there are 175 parameter instances in the credible region of 90% (C = 90%; Figure S18), which are clustered in certain ranges that minimize both fitness and timeoff, as shown in the middle two panels of Figure 5. Within the credible region, n 0 ranges between 0.033 and 0.039 s/m 1/3 with a mean of 0.038 ± 0.003 s/m 1/3 , τ y ranges between 291 and 498 Pa with a mean of 397 ± 44 Pa, and VC ranges between 54.5% and 56.4% with a mean of 56.2% ± 0.55% when σ f = 4.2% (±1.5 m of the inundation boundary) and σ t = 1.5 min (±3 min of the timeoff).Within the 10,000 instances, the optimal parameter, labeled point a in Figure 5, has a posterior probability of 1.5% and it is associated with a fitness of 238,215 m 2 (Figure 6a) and timeoff of 1 min, while point b at the edge of the credible region has a posterior probability of 0.1% and is associated with a fitness of 239,700 m 2 (Figure 6b) and a timeoff of 4 min.In contrast, point c outside of the credible region has a posterior probability of 2.7 × 10 20 and is associated with a fitness of 247,666 m 2 (Figure 6c) and a timeoff of 11.5 min.
The three simulated inundation extents using parameter sets a, b, and c (Table 3) present a similar pattern where true positives (TP), which indicate overlap between the simulation and observation, are primarily located in areas where flow is confined.False negatives (FN, or underestimation) and false positives (FP, or overestimation) are more common in unconfined areas (Figure 6).The difference in simulated inundation between a and b is smaller than that between a and c.The major difference between a and c is that more FN (or underestimation) appears near the distal end of the fan for simulation a.

Application to PFDF Inundation Assessment in a Prefire Context
Sampled from the 10,000 parameters with the PPS strategy (Section 3.4), we obtained 100 parameter instances.
Only one of the 100 parameter instances is outside of the C = 90% range.We then obtained the posterior distribution-based inundation probability maps that represent the inundation likelihood at each cell for different fire (50% and 100% burned at moderate-high severity) and rainfall (2-year and 100-year RI storms) scenarios (Figures 7a-7d) by using the 100 optimized parameter instances.We also derived inundation probability maps using 100 parameter instances drawn from a uniform distribution for each scenario (Figures 7e-7h).
The high probability (>90%) inundation areas (i.e., light pink areas) include confined channels and portions of the alluvial fan (Figures 7a-7d) when using the posterior distribution.In contrast, the high probability inundation areas include primarily areas of confined flow and exhibit high spatial variability (Figures 7e-7h) when drawing Earth's Future 10.1029/2023EF004318 LIU ET AL.
parameters from a uniform distribution.Difference in the locations of high-probability inundation areas are most accentuated when comparing scenarios associated with 2-year and 100-year storm, such as those shown between Figures 7a and 7c, rather than scenarios with different amounts of area burned at moderate-to-high SBS (Figure 7).

Discussion
This study presents a prefire approach for PFDF inundation assessment by using a debris-flow model (i.e., the SWE model integrated with HB rheology model) that incorporates a Bayesian posterior analysis coupled with a rainfall-runoff model and a localized PFDF volume model.To better understand the debris-flow model's behavior, we conducted a sensitivity analysis to rank parameters' importance in both metric-free and metric-based approaches, which is discussed in Section 5.1.We then discuss the benefits and challenges of the proposed prefire PFDF inundation assessment framework in Section 5.2.

Parameter Importance
We assessed model sensitivity to rank the importance of five parameters (Table 3) by using a model output variable, A sim , and two model performance metrics, fitness and timeoff.Both A sim and fitness are related to inundation extent.When simulated inundation, A sim , is the response surface, the parameter importance represents how strongly each parameter influences A sim .We observed that A sim increases with n 0 and VC and decreases with τ y (Figure 5b), which agrees with our understanding of debris-flow routing.Specifically, greater roughness increases flow resistance and slows velocity.As the flow velocity decreases, the channel's ability to carry the same volume of flow diminishes.With a lower capacity to accommodate the inflow, flow starts to accumulate and spread out beyond the normal channel boundaries, resulting in increased inundation area (Z.Liu et al., 2018).Higher VC leads to increased momentum and increases in area inundated due to the positive relationship between VC and flow volume, discharge rate, and density (Iverson, 1997;Vallance & Scott, 1997).τ y , the intercept of the stressstrain curve of a rheology model, serves as a motion threshold.Higher τ y indicates more resistance and decreases momentum, tending to "stop" flow motion and produce deposition.Flow inundation is, therefore, negatively correlated with τ y .Furthermore, K and c, the slope and exponent of the stress-strain curve, respectively, have a weak effect on A sim .Therefore, the HB rheological model can be used with fixed K and c values or replaced with a Figure 6.A comparison between the mapped deposits and the simulated inundation using the three parameter instances (Table 3).Panel (a) is the results of the optimal parameter instance represented by point a in

Table 3
Comparisons of Model Performance Between the Optimal Parameter (Figure 6a), a Parameter at the Edge of the C = 90% (Figure 6b), and a Parameter Outside the C = 90% (Figure 6c) simpler rheological model, such as a Bingham model when the goal is to estimate the inundation extent.A simpler model with fewer parameters is usually more efficient and practical for practitioners in the context of decisionmaking for hazards mitigation or assessment.
When fitness is used to define the response surface, the analysis helps identify which parameters can be more easily determined.The information provided about model behavior and its relation to specific parameters is distorted when calculating the metric.For instance, A sim and τ y are negatively correlated, while fitness changes non-monotonically with τ y (Figure 5).The fitness-based sensitivity analysis, however, provides information that is helpful for model calibration and application.τ y is identified as the controlling parameter for inundation.The relative contributions of K and c are negligible so they can be removed from the model calibration in similar efforts in the future.VC can also be excluded from model calibration when debris-flow volume and runoff are well-constrained, although it may still be relatively influential in cases where uncertainties in sediment volume or runoff lead to a wide range of potential VC values.Therefore, the 5-parameter model we employ can be simplified into a single parameter (τ y ) model which is very friendly for optimization and application when computational resources are limited and/or the simulations need to be done in a timely manner in practice.
In summary, A sim -based sensitivity analysis effectively recognizes the importance of τ y , VC, and n 0 for controlling debris-flow runout extent and provides insights for how each parameter affects the runout extent.In contrast, the fitness-based sensitivity analysis effectively identifies that τ y is the dominant control on model performance as quantified by inundation area.Both n 0 and τ y are influential and contribute equivalently to timing of modeled debris flows.These findings also provide insights for debris-flow inundation assessment using the HB model in HEC-RAS.

Benefits and Challenges of the Prefire Approach
We used Bayesian calibration to update the initial uniform distribution of parameters in our debris-flow routing model.This method ensures statistical coherence between the model's output and the observed data, while also accounting for inherent uncertainties (Gregory, 2010), such as those in observed debris-flow volume (Gorr et al., 2024) and inundation boundaries (Figure S3 in Supporting Information S1).The resulting posterior distribution not only quantifies credible regions (Heckerman, 1998;Figure S18 in Supporting Information S1), but is also well-suited for probabilistic analyses and risk assessments (Figure 7).The posterior distribution determined here can be used as an initial guess for calibrating model parameters or as guidance for optimal parameters for predictive analyses when calibration data are not available.The direct transferability of the posterior distribution would be best where the physiographic properties are similar to our study area.
The range of optimized parameters in the 90% credible region (Figure 5) is consistent with the parameter ranking (Figure 4).For instance, τ y has the narrowest range among the five parameters across various uncertainty levels (Figure 5) and is ranked as the major control for inundation in terms of A sim (Figures 4a and 4d) and fitness (Figures 4b and 4e).Inundation probability maps based on the posterior distribution (Figures 7a-7d) exhibit fewer spatial variations compared to those generated from a prior uniform distribution (Figures 7e-7h).This is largely because the optimized parameter sets predominantly originate from a specific region within the parameter space, as opposed to the uniform distribution where parameters are scattered throughout the space.Only one out of the 100 parameter instances selected by using PPS method is outside of the credible region of 90% (C = 90%).The reduced spatial variability presents high consistency in simulated inundation extents, offering greater flexibility for parameter selection in practical applications.For instance, a reliable inundation simulation can be generated using a randomly selected parameter set from the posterior distribution within the C = 90% explored in this study (Figure 5 and Figure S18 in Supporting Information S1).This is illustrated by examining the variations in the simulated inundation patterns (Figure 6) using the three parameter sets in Figure 5.Moreover, maps based on the posterior distribution effectively focus attention on regions with consistently high probabilities (Figures 7a-7d), thereby eliminating the need for additional scrutiny of the large spatial variations seen in maps derived from the prior uniform distribution (Figures 7e-7h).
The overlap between the prior and posterior distribution-based inundation probability maps are confined to areas of concentrated flow for 2-year storm scenarios (Figures 7a,7b,7e,and 7f).For 100-year storms, more unconfined portions of the alluvial fan between those channels also experience inundation (Figures 7c,7d,7g,and 7h).This pattern is primarily due to the substantial difference in total volume of debris flows associated with a 2year storm and a 100-year storm.The former ranges between 13,073 and 22,362 m 3 and the later ranges from 95,999 to 164,209 m 3 when considering scenarios of 50% and 100% Bmh.This is consistent with the finding that model performance is more sensitive to flow volume and geomorphology than to flow properties (Barnhart et al., 2021).
The prefire hazard assessment framework that we propose here can be further improved and contextualized both for advancing scientific understanding and operational hazards assessment.The debris-flow routing model used in this framework, namely the SWE model integrated with HB rheology model in HEC-RAS's single phase non-Newtonian module (Gibson et al., 2021), could be replaced by alternative models.Other runout models include D-Claw (George & Iverson, 2014;Iverson & George, 2014), FLO-2D (O'Brien et al., 1993), RAMMS (Christen et al., 2010), and TITAN-2D (Patra et al., 2005) which represent different flow rheology and numerical schemes, as well as ProDF (Gorr et al., 2022) which employs an iterative variation of the multiple flow direction algorithm (Freeman, 1991;Pelletier, 2008) to route debris flows.
The Bayesian posterior analysis used here and described in detail by Aaron et al. (2019) can be employed to calibrate parameters of alternative runout models based on various objective functions.Although the objective function focuses on fitness for inundation extent, other model outputs such as flow depth and velocity could similarly be incorporated (Aaron et al., 2019).Moreover, this prefire approach has the potential for further development, specifically in linking model outputs to damage estimation for buildings and infrastructure within the inundation zones (Barnhart, Miller, et al., 2023;Barnhart, Romero, & Clifford, 2023).This would offer indepth insights into exposure and vulnerability, thereby serving as a strategic guide for prioritizing watersheds for intervention.
While the proposed approach offers benefits for PFDF hazards, it faces a few challenges.The general framework presented here provides a structure for conducting prefire assessments of postfire debris-flow runout in other settings where debris flows are initiated from runoff, but the choice of models within individual components of the framework and the parameterizations for those models would depend on site specific considerations.First, additional studies of PFDF inundation across a range of landscapes are needed to assess generalizability of the sensitivity analysis results presented here, which provide initial guidance on parameters that exert the greatest influence on inundation extent.This requires availability of observations, such as inundation boundaries, flow depth, sediment volume, and high-resolution rainfall data, that can be used for calibrating the debris-flow runout model.Collection of these data in recently burned areas is often hampered by logistical constraints, including minimal time between rainstorms that produce significant geomorphic responses and site access issues.The paucity of such data underscores the need for monitoring using a variety of techniques ranging from comprehensive in situ field investigations of PFDF events (Gorr et al., 2024), high-resolution remote sensing-based inundation mapping, pressure signal-based debris-flow timing measurement (Kean et al., 2012), and LiDARbased terrain analysis to quantify sediment depth (Rengers et al., 2021).
Second, prefire assessments of debris-flow runout in other regions should take into account the intensity of rainfall needed to generate debris flows.We simulated debris-flow runout in response to 2-year and 100-year RI rainstorms, which are generally of sufficient intensity to initiate runoff-generated debris flows in the first year following fire in the southwest US (Staley et al., 2020).The design storm(s) selected for modeling in other regions could be based on the intensity and duration of rainfall associated with past storms that have triggered debris flows in the region.Alternatively, models designed to assess debris-flow likelihood based on terrain, burn severity, and rainfall characteristics (Nyman et al., 2015;Staley et al., 2017) could be used to provide quantitative estimates of debris-flow occurrence for different burn severity and rainfall scenarios.In addition to exploring a range of generic burn severity scenarios, modeling approaches can be used to generate synthetic burn severity data that can be used as input into debris-flow likelihood and volume models (Staley et al., 2018;Wells et al., 2023).
Third, it is challenging to objectively define upper and lower limits for sediment concentration.The range of debris-flow sediment concentrations (49.9%-62.4%)at our site was constrained by a combination of fieldobserved sediment volumes at the Copeland study area and data that allowed us to estimate runoff.In particular, we relied on data from several nearby rain gauges and estimates of hydrologic model parameters, namely saturated hydraulic conductivity, wetting front potential, and a hydraulic roughness coefficient, from past studies in the area to estimate runoff (Hoch et al., 2021;McGuire & Youberg, 2020).Additionally, our general approach for rainfall-runoff modeling is based on infiltration-excess overland flow being the dominant runoff-generation mechanism in the first year following fire.This is a reasonable assumption for our study region (Gorr et al., 2024;Schmidt et al., 2011), but would be expected to vary from place to place depending on rainfall intensity, soil properties, and burn severity.In the absence of data or observations that can be used to constrain runoff and debris flow volume, as may be the case in other regions, the range for debris-flow sediment concentration would be greater.A large range in sediment concentration may change the relative importance of the parameter VC and the posterior distribution (e.g., Barnhart et al., 2021).
A related challenge for prefire assessments involves estimates of debris-flow volume, which are essential for defining inflow hydrographs for debris-flow runout modeling.In our prefire assessment, we estimated the debris-Earth's Future 10.1029/2023EF004318 flow sediment volume using a modified version of the model presented by Gartner et al. (2014) that benefitted from observed debris flow volume data collected nearby (Youberg, 2015).Postfire debris flow volume models have been developed in the western USA and Australia (Gartner et al., 2014;Nyman et al., 2015;Santi & Morandi, 2013;Wall et al., 2023) while PFDFs have been reported much more broadly (e.g., Conedera et al., 2003;Esposito et al., 2023;Wang et al., 2022).Employing PFDF volume models outside regions where they were developed has been met with mixed results (Gorr et al., 2024;Nyman et al., 2015) and is likely to increase uncertainty associated with flow volume, a critical input for inundation models (Barnhart et al., 2021).An alternative approach to estimating inflow hydrographs for debris-flow runout modeling is the use of a bulking factor (e.g., Gusman et al., 2009).Debris flow hydrographs could be generated by applying a range of bulking factors to increase the discharge associated with a clear water hydrograph.This approach would be most applicable in situations where there is no reliable PFDF volume model and there are constraints on fire effects to hydrologic model parameters (e.g., Basso et al., 2020;Ebel & Moody, 2020) that would be needed to simulate clear-water hydrographs for different design storms.Regardless of the method used to estimate debris-flow volume, errors approaching an order of magnitude are possible (e.g., Gartner et al., 2014).In our prefire assessment approach, we did not explore the uncertainty introduced from using a model to determine a debrisflow volume for a given rainfall and burn severity scenario, but it highlights future research needs and a challenge for generalizing the framework presented here to other settings.In addition, most debris-flow runout models, including the HB rheology model in HEC-RAS's single phase non-Newtonian module, do not account for changes in flow volume due to sediment entrainment along the flow path which can alter inundation patterns (e.g., Frank et al., 2015).
Lastly, we found that the single-phase HB model with fixed parameters was effective for simulating the timing and inundation limits of the Copeland debris flow.However, our approach does not explicitly consider interactions between different parameters, such as yield stress and sediment concentration.We did not prescribe correlations between parameters of the HB model in the prior distribution and did not observe any substantial correlations in the posterior distributions.In contrast, experiments of mudflow and fine-grained slurry properties indicate that yield stress and viscosity generally increase with sediment concentration (e.g., Major & Pierson, 1992;O'Brien & Julien, 1988).Further understanding of these relationships in large-scale, natural debris flows would benefit from a combination of field measurements, back-analysis of past events, and physical experiments, considering the intrinsic variability and complexity of debris flows.

Conclusion
PFDFs triggered by short-duration, intense rainstorms over recently burned watersheds endanger downstream areas and can occur even before a fire is fully contained.This underscores the urgent need for preemptive postfire PFDF hazard assessment.We present a prefire approach to PFDF inundation assessment and apply it in a region where fire size and severity are increasing.The modeling framework integrates detailed hydro-geomorphic field observations with a debris-flow runout model based on the HB rheology, Bayesian posterior analysis, rainfallrunoff simulation, and a debris-flow volume model.
We employed a GSA to rank the HB rheology model parameters in two ways, a metric free approach and a metricbased approach, offering insights into the model's behavior.The metric-free approach effectively recognizes the importance of τ y , VC and n 0 in governing the extent of debris-flow runout.In contrast, the metric-based analysis reveals that τ y predominantly influences model performance in terms of inundation area, while both n 0 and VC are influential and contribute equivalently to debris-flow timing.This finding suggests that future studies applying a single phase non-Newtonian rheology to simulate debris-flow runout should prioritize placing constraints on τ y .Application of the model framework in a prefire context highlights the importance of flow volume on inundation extent and demonstrates the benefits of having historical events that can be used to determine posterior distributions for model parameters.

Figure 1 .
Figure 1.(a) Location of study areas in Arizona; (b) Copeland watersheds and Fort Valley watersheds on the slope of the San Francisco Peaks, in northern Arizona; (c) Map of the Copeland area including two headwater drainage areas and the Copeland debris-flow inundation zone; (d) Map of Fort Valley watersheds and the alluvial fan.

Figure 2 .
Figure 2. (a) The simulated hydrographs of three infiltration levels at two outlets.Q1 denotes the discharge rate at the outlet of the northern watershed and Q2 the southern watershed; (b) The total flow volume of debris flow; (c) The volumetric concentration of debris flow with an uncertainty of ±25% of the observed debris flow volume.

Figure 3 .
Figure 3. Random forest regressor derived versus HB model simulated A sim , fitness, and arrival time based on 500 simulations.

Figure 4 .
Figure 4. VARS parameter importance ranking.Parameters are Manning's n (n 0 ), yield strength (τ y ), the slope (K) and exponent (c) of the stress-strain curve, and volumetric concentration (VC) of the debris flow.The sensitivity metrics including ratio of VARA-ABE (mean absolute elementary effect across scales, Δh = 5% of the parameter range), IVAR30 (Integrated Variogram Across a Range of Scales, Δh ranges from 0% to 50% of the parameter range), VARS-TO (Variance-based Total-Order effect), are calculated based on (a) simulated inundation area, (b) fitness, and (c) arrival time using 500 simulations.Subplots (d-f) showing the same sensitivity metrics with a random forest surrogate model derived 10,000 simulations.

Figure 5 .
Figure 5. Scatterplots in top panel showing the model output, simulated inundation area (A sim ), and two middle panels showing the model performance metrics (fitness and timeoff) varying with parameters at uncertainty level of σ f = 4.2% (±1.5 m of the inundation boundary) and arrival time uncertainty of σ t = 1.5 min (±3 min of the timeoff).Scatterplots in the bottom pannel showing posterior probability of parameters (π( p|r)) and 90% credible region (C).Point a is the optimal parameter instance, point b is at the edge of the C = 90%, and point c is outside of the C = 90%.
Figure 6.A comparison between the mapped deposits and the simulated inundation using the three parameter instances (Table 3).Panel (a) is the results of the optimal parameter instance represented by point a in Figure 5. Panels (b) and (c) are the results of the parameter instances represented by points b and c in Figure 5. Areas where the model overlaps with the observations are shown as true positive (TP).False negatives (FN) represent instances where the model underestimated flow, despite sediment being observed in the field.Overestimation or false positives (FP) occur when the model predicts flow, but no deposits are mapped.

Figure 7 .
Figure 7.A comparison of the simulated inundaton probabilities for four scenarios designed to represent a range of rainfall and burn severity scenarios: 50% and 100% of the watershed areas burned at moderate to high soil burn severity (Bmh) for two design storms (2-year and 100-year) based on rainfall data from NOAA Atlas14 (NOAA, 2020) in the Fort Valley.(a-d) Inundation probability based on parameters following the optimized posterior distribution; (e-h) Inundation probability based on parameters following prior uniform distribution.Debris-flow inundation zones abruptly end along straight lines, which correspond to the downstream boundaries of the model.

Table 1
Parameters of the Herschel-Bulkley (HB) Rheological ModelThe range of VC is determined by introducing uncertainty of observed sediments as described in Section 3.1 and 4.1. a

Table 2
The VARS Products for Global Sensitivity Analysis Used in This Study a VARS-TO is effectively the same quantity as Sobol's variance-based Total-Order effect