Steep watersheds recently burned by fire often experience greater amounts of runoff and increased rates of sediment transport. Factors affecting rates of sediment transport, and also the initiation and growth of runoff-generated debris flows, include rainfall intensity and duration, vegetation cover, soil infiltration capacity, and sediment characteristics (e.g. grain size, erodibility). The model developed by represents rainfall, infiltration, fluid flow, and sediment entrainment and deposition processes, which makes it a useful framework for simulating runoff-generated debris flows in steep terrain . The model couples fluid flow, entrainment and deposition processes, and topographic change, such that the topography can evolve during a rainstorm in response to flow. We provide a brief overview of the governing equations, which we solve within the Titan2D framework .

The Titan2D code employs an adaptive mesh, finite volume scheme to solve hyperbolic PDEs describing shallow-water like mass flows over digital elevation models of real topography. Titan2D was originally developed to solve the depth averaged shallow-water mass flow equations by . Titan was modernized and restructured in 2019 to optimize storage and access for parallel adaptive mesh refinement, and to facilitate different representations of the physicals processes. Using Titan2D to solve the model equations proposed by therefore offers several advantages, especially for simulating debris flows over spatial scales of more than a few square kilometers. The computational efficiencies offered by a Titan2D implementation make it tractable to simulate end-to-end flows from initiation to inundation and/or run the ensembles needed to train GPs efficiently. Henceforth, we refer to Titan2D implementation of the model as T2D-M2017.

The equations representing the motion of fluid and sediment can be written as a set of depth averaged conservation laws, 1∂U∂t+∂F∂x+∂G∂y=S0+S1+S2, where U, F, and G denote the vector of conserved variables and their corresponding flux functions in the x- and y-direction, respectively. Specifically, 2U=huhvhc1h⋯ckhT,3F=huhu2+12gzh2huvhuc1⋯huckT,4G=hvhuvhv2+12gzh2hvc1⋯hvckT, where h, u, v, and ci are flow depth, velocity along x-axis, velocity along y- axis, and sediment concentration of particle size class i. Components of gravitational acceleration in the x, y, and z directions are given by gx, gy, and gz, respectively, and k denotes the number of particle size classes. S0, S1 and S2 are source terms. S0 denotes the contributions of mass sources and sinks associated with the effective rainfall rate, Peff, and the soil infiltration capacity, I, as well as momentum sources and sinks arising from variations in topographic elevation, and spatial variations in sediment concentration and debris flow resistance terms, Sx and Sy. Specifically, S0 is given as 5S0=Peff-I+∂z∂t-gxh+γx-ψSx-gyh+γy-ψSy0⋮0. The flow resistance terms are influenced by sediment concentration since flow behavior can change substantially as sediment concentration increases , though there is no universal approach for representing these changes in morphodynamic models where sediment concentration can change rapidly in space and time. Following , the debris flow resistance terms are scaled by ψ, which increases linearly from 0 to 1 as the volumetric sediment concentration increases from 0.2 to 0.4. This scaling factor gradually increases the importance of the debris flow resistance terms as volumetric sediment concentration approaches levels that are consistent with a transition from flood flow to debris flow. The terms γx and γy account for the effects of spatially variable sediment concentration and are given by 6γx=-(ρs-ρw)gzh22ρf∂c∂x, and 7γy=-(ρs-ρw)gzh22ρf∂c∂y. Here, c denotes volumetric sediment concentration, ρw=1000 kg m-3 the density of water, ρs=2600 kg m-3 the density of sediment, and ρf=cρf+(1-c)ρw the density of the flow. S1 accounts for flow resistance using a depth-dependent Manning’s formulation, and is given as 8S1=0gzη2huhu2+hv2/h7/3gzη2hvhu2+hv2/h7/30⋮0, where η is the Manning friction coefficient. The friction coefficient varies with flow depth according to 9η=η0h/hc-ϵh≤hcη0h>hc, where η0 is the hydraulic roughness coefficient, hc is a critical flow depth and ϵ is a phenomenological exponent. Soil infiltration capacity, I, is represented by the Green-Ampt model where 10I=ksZf+hf+hZf, with ks denoting saturated hydraulic conductivity, hf the wetting front potential, Zf=V/(θs-θi) the depth of the wetting front, V the cumulative infiltrated depth, θs the volumetric soil moisture content at saturation, and θi the initial volumetric soil moisture content. The source term S2 accounts for sediment entrainment and deposition processes, which are represented using the framework proposed . In particular, 11S2=000e1+er1+r1+rr1-d1⋮ek+erk+rk+rrk-dk, where ek and erk are sediment detachment and re-detachment rates due to raindrop impact for sediment particles in size class k, rk and rrk are rates of entrainment and re-entrainment due to runoff, and dk is the effective deposition rate. The model differentiates between original soil, which has not yet been entrained and transported during the modeled rainstorm, and deposited sediment, which has been detached and subsequently deposited. Detachment rates for entraining original sediment and re-entraining deposited sediment are computed differently. Sediment in the deposited layer can also fail en-masse . Rates of sediment entrainment and re-entrainment by runoff are given by 12rk=(1-H)pkF(Ω-Ωcr)J, and 13rrk=HmkmtF(Ω-Ωcr)ρs-ρfρsgh. Here, mk is the deposited sediment mass per unit area for sediment in size class k, mt is the total mass of deposited sediment per unit area, H=min⁡(mt/mt*,1) accounts for the degree to which deposited sediment shields the underlying bed from erosion, mt* is the mass of deposited sediment needed to completely shield original sediment from erosion, ρf is the density of the flow, ρs is the density of sediment, F denotes the fraction of stream power effective in sediment entrainment, Ω=ρfgSfuh2+vh2 is stream power, and Sf=η2(uh2+vh2)h-10/3 is the friction slope. In this work, we consider a single particle size class characterized by a representative particle diameter, δ. We do not attempt to link the value of δ to any particular metric (e.g., d50) related to the grain size distribution, but instead interpret it as a parameter that influences the rate and style of sediment transport through its control on the settling velocity.

The topographic surface evolves in response to sediment entertainment and deposition according to 14∂z∂t=1ρs(1-ϕ)∑k=1Kdk-ek-erk-rk-rrk. Here, ϕ=0.4 is the bed sediment porosity and ek and erk denote the sediment detachment and redatchment rates due to raindrop impact as defined by . Since the equations for fluid flow and evolution of the topographic surface are coupled, the model captures feedback between erosion and flow behavior even during individual rainstorms. For example, concentration of flow in one area can lead to erosion, which promotes increased flow concentration and enhanced sediment transport.

A digital elevation model (DEM) of the study area is input to the T2D-M2017 simulation. Here, we used a 1 m DEM derived from post-event airborne lidar . Elevations and slopes at locations required by the computational mesh were obtained using a 9 point (3×3) finite difference stencil to interpolate on the DEM grid reducing the effects of artifacts and noise in the data . Effects of uncertainty in the DEM could be propagated through the simulation and subsequent analysis , but we did not consider this in the present study.

Runoff and debris flows initiated in the study area in response to a short duration, high intensity burst of rainfall in the early morning hours of 9 January 2018 . All simulations used 1 min rainfall intensity data derived from the KTYD rain gauge for a 20 min time period that spans this short temporal window when rainfall intensity rapidly increased and debris flows initiated (Fig. S1 in the Supplement). The gauge was maintained by the Santa Barbara County Flood Control District and is located approximately 5 km west of the San Ysidro Creek watershed.

Simulations were designed to explore the extent to which inundated area and peak flow depths on the alluvial fan were influenced by rainfall intensity as well as several parameters that can play critical roles in debris-flow initiation and growth. Selection of these parameters was determined, in part, based on common effects of fire such as the tendency for moderate and high severity fire to reduce soil infiltration rates . The relative influence of these parameters on debris flow runout extent and peak flow depth, however, is less clear. We explored the effect of different rainfall intensities by multiplying the observed 1 min rainfall intensity time series by a rainfall intensity factor (RI_fac) that we varied from 0.4 to 1.5. We also varied the representative particle diameter, δ, from 0.05–0.3 mm, the fraction of stream power effective in entrainment, F, from 0.01–0.06, the hydraulic roughness coefficient, n0, from 0.03–0.2, and saturated hydraulic conductivity, ks, from 5–20 mm h-1. We further enforced a maximum soil thickness, rmax, that varied from 0.25–1.5 m to explore the role of sediment availability. All other parameters were fixed (Tables , ). We used a Latin hypercube sampling strategy to generate 64 different parameter sets from the ranges specified above . Each simulation took several hours to complete on an HPC cluster using up to 16 cores on an intel Xeon Gold 6226R processor.

We focused on exploring the effects of rainfall intensity, δ, F, n0, ks, and soil thickness (sediment availability) since they control different aspects of the debris flow initiation and growth process and, aside from rainfall intensity, they may all be strongly affected by fire in our study area . Peak rainfall intensity over sub-hourly durations, particularly the 15 min duration, is correlated well with runoff in recently burned watersheds in southern California . Peak 15 min rainfall intensity is also used in empirical models designed to predict postfire debris-flow likelihood and volume in the western USA . We therefore expect that variations in rainfall intensity during the relatively short (<0.5 h) portion of the rainstorm that we are modeling will influence debris flow processes.

We expect the representative grain size, δ, to be relatively small in areas of concentrated flow immediately following fire in our study area given the propensity for postfire dry ravel to transport hillslope sediment to channels and valley bottoms . Both δ and the amount of sediment available for transport, which we varied by enforcing a maximum soil thickness (rmax) throughout the model domain, may vary as a function of time since fire as sediment is exported from postfire rainstorms . Similarly, found that ks and the Manning coefficient were lowest during rainstorms in the first year following a high severity fire in the San Gabriel Mountains, southern California, and increased by factors of roughly 3–4 over the following 4 years (Fig. S2). Immediately after fire in southern California, values for the Manning coefficient and saturated hydraulic conductivity can be as low as 0.025–0.07 s m-1/3 and 1–6 mm h-1, respectively . used post-event, point scale measurements with a tension infiltrometer to estimate the geometric mean of saturated hydraulic conductivity at 20 mm h-1 in the days following the Montecito debris flows. The effective fraction of stream power, F, may be expected to increase immediately following fire due to reductions in roughness associated with ground cover and vegetation. Values of F≈0.005 have performed reasonably well at reproducing past events in steep, recently burned watersheds in southern California . Soil cohesion, particle size distribution, and ground cover, among other factors, are likely to influence F and could lead to considerable site-to-site variability.

We assessed the ability of the model to reproduce the observed inundation extent (Fig. ) by computing a similarity index . The similarity index, ω, is defined according to 15ω=α-β-Γ where α denotes the area of overlap between simulated and observed inundation, β denotes the area where the model underestimates inundation extent, and Γ the area where the model overestimates inundation extent. The similarity index varies between -1 and 1, with a greater value indicating a better fit between the model and observation.

Statistical emulators are effectively probabilistic models of computationally intensive physical model systems or simulators. Statistical emulators relate a set of user-defined inputs, often physical parameter specifications, to simulator output. Gaussian process emulators (GPs) are a popular class of surrogates for approximating and quantifying uncertainties in simulators as they (almost) interpolate model output . Further, the variance of the associated GP offers a quick mechanism to assess the uncertainty of using the emulator in place of the simulator for model prediction at untested inputs. Thus GP emulators offer a rapid and quantifiable mechanism to approximate output from physical process models that are computationally intensive to exercise. The parallel partial emulator (PPE) extends this surrogate model to field-valued output.

Inputs to GP emulators are user defined. They are typically influential parameters, which show up within the governing dynamics, the forcing terms, or boundary conditions, as opposed to independent variables in the physical model. For the model described in Sect. , we chose p=6 parameters to define our input vector, namely those described in Sect.  and given by q=[ks, rmax, η0, F, δ, RIfac]. We will discuss the relationship between GP emulators and sensitivity analysis further in Sect. .

The output under consideration, y, is the maximum (over time) flow depth at each of s=2.4M map points. The main objective of the emulator is to predict the output of the T2D-M2017 model at an untested scenario, q*, given a relatively modest set of N training or design runs qD and each of their corresponding inundation depth outputs, yjD, j=1,…,N. In this work, we took N=64 training runs and each output, yjD, was a 2.4M element vector recording the peak flow depth (inundation depth) at each map point. Collecting these outputs together resulted in YD, a 64×2.4M matrix of training run outputs. The 64 training run inputs were chosen by a Latin hypercube design covering the ranges of inputs listed in Table . Note that the training run design was intended to cover a broad range of possible scenarios resulting in flows that vary from relatively small inundation footprint areas to those with relatively large inundation footprint areas. All other parameters were fixed (Table ). To fit the emulator, these parameter ranges were normalized to a unit hypercube.

Given the training data, {qD,YD}, to approximate the inundation resulting from an untested scenario, q*, we used the predictive mean of the PPE given by 16ỹ(q*)=hT(q*)B+rT(q*)R-1(YD-HDB), where R is an N×N (64×64 in this work) matrix of correlations between pairs of design inputs, r(q*) is an N×1 vector of correlations between the untested input, q*, and each of the input scenarios in the design, qD. Further, h(q) is a l×1 vector of regression variables, often taken to be constant or linear in q (i.e., l=1 for the constant case used in this work and l=p+1 for the linear case), and HD is and N×l matrix where the jth row are the regression variables evaluated at the jth design point, hT(qjD). The matrix B is a l×s matrix of regression coefficients. Here, each of the s=2.4M outputs has its own set of regression coefficients, but a shared correlation structure. We used a Matérn 5/2 correlation function . For two scenarios, e.g., two input points qi=(xi1,…,xip)T and qj=(xj1,…,xjp)T, the standardized distance and correlation between these input scenarios are given by 17dm=(|xim-xjm|2θm2)1/2,m=1,…,pc(qi,qj)=∏m=1p(1+5dm+53dm2)exp⁡(-5dm), respectively. The predictive variance for each output dimension (pixel) of the PPE is given by 18vj(q*)=σj2(1-rT(q*)R-1r(q*)+(h(q*)-(HD)TR-1r(q*))T×((HD)TR-1HD)-1(h(q*)-(HD)TR-1r(q*))), where σj2, (j=1,…,s) is the scalar variance corresponding to each pixel's output. “Fitting” a PPE amounts to estimating the regression parameters in B, the scalar variances at of each output, σj2, and the range parameters {θm:m=1,…,p}. Note that range parameters, or correlation lengths, tell us how the function response changes as the associated input parameter changes. A small value of a range parameter indicates a strong change in response as that input parameter changes while a large value of a range parameter indicates little to no change in response as that input parameter changes.

In order to fit range parameters, we used the RobustGaSP package . Fitting a PPE to 2.4M pixels of output with N=64 training runs took roughly 10 min (on a single core of MacBook Pro with an M2 chip). Training the PPE scales as a cube of the number of training runs N, and linearly with the size of the simulator output, s.

GP emulators have been applied to Titan2D-based volcanic debris flows and recently to other Titan2D-based debris flows . In each of these studies, source terms (particularly debris mass or flux) were specified via ad-hoc parameterizations which are less appropriate for postfire, runoff-generated debris flows.

Evaluation of the GP emulator's mean quickly allows one to explore any output quantity of interest over the parameter space. Here we took the output quantity of interest, y, to be the maximum debris-flow depth at all locations. Additionally, the variance of the GP emulator accounts for the uncertainty introduced by evaluating the GP mean, ỹ, instead of the debris-flow process model. We have broken down our exploration of numerical experiments into four groups.

First, we performed leave-one-out experiments as a test of the PPE performance. This experiment amounted to excluding one simulation at a time, fitting a GP to the 63 remaining simulations, and then comparing the GP predicted inundation of the left-out scenario to actual simulated inundation for that scenario. This was repeated for each of the N=64 simulations. (Note, all other GP and PPE emulators were constructed with the full design of N=64 training simulations.)

Second, we explored the relative importance of different model input parameters using the GP's range parameters. Range parameters are positive numbers indicating the influence of each model parameter on the model response – the smaller the range parameter, the more influence the corresponding model parameter has on the debris flow model (i.e. maximum flow depth). As such, these range parameters act as an effective sensitivity analysis.

Third, we explored the effect of different rainfall scenarios, as quantified by varying I15, on debris flow inundation. Hazard assessments often involve quantifying the likelihood and magnitude of debris flow hazards in response to a forecast or design rainstorm. To do so here, we proposed an approach that uses two different emulators to generate a probabilistic inundation map for a given value of I15. One is the PPE emulator that has already been described, which predicts peak flow depth across the landscape. The other is a second emulator that we developed specifically to facilitate this set of numerical experiments. In particular, we fit a separate GP emulator to predict the volume of sediment eroded from the upper watersheds, which we view as a proxy for the amount of sediment mobilized by debris flows. Rather than a map, the output from this GP, which we refer to as the volume emulator, was a single scalar value (e.g., s=1) – the total volume of eroded sediment for a given simulation. Note that this volume emulator was trained on the same N=64 design simulations as the PPE inundation depth emulator and can likewise be evaluated at any untested combinations of parameters, q=[ks, rmax, η0, F, δ, RIfac]. We then used this volume GP emulator in a screening step to identify parameter sets that are consistent with the emergency assessment model.

Lastly, we applied the emulator to explore how inundation extent and peak flow depths are driven by temporal changes in saturated hydraulic conductivity and hydraulic roughness that occur as the soil and land surface evolve with time after fire. This analysis was designed to illustrate how emulators can be used to efficiently quantify the temporal persistence of debris flow hazards after fire. We focused, in particular, on exploring the effects of postfire changes in saturated hydraulic conductivity and hydraulic roughness since these two parameters are influenced by fire and provides guidance for parameterizing how they change with time since fire in the nearby San Gabriel Mountains (Fig. S2). Since the GP emulator enables rapid forward uncertainty quantification, we demonstrated how it can be used to accelerate a Monte Carlo probability of inundation calculation for two cases, namely when the observed storm occurs 2 months and 14 months postfire. These discrete times for evaluating the emulator were chosen since the parameterization from results in substantial changes in saturated hydraulic conductivity and hydraulic roughness over the first 14 postfire months and we therefore expected to see corresponding changes in the debris flow inundation footprint.

Emulators offer a significant advantage to probabilistic hazard analysis as compared to approaches that rely directly on debris flow simulations. If a GP is trained on a sufficiently broad set of design simulations covering the support of any reasonable prior distribution, then the probabilistic modeling of aleatory (scenario) uncertainty can be developed independently of the simulation set. Here we looked at two approaches, one where we screened a uniformly sampled parameter space for parameter combinations that matched the Gartner model (using the volume GP) and a second that relied on parameters samples from a validation study of postfire recovery from the work (using the inundation depth PPE).

Given the volume GP and inundation depth PPE emulators, the process for generating a probabilistic inundation map for a forecast or design rainstorm with a given I15 began by fixing the RI_fac input parameter (e.g., RIfac=0.5 for I15=38.5mmh-1) and sampling all other inputs randomly over the ranges given in Table . We then used the volume emulator to predict the total amount of sediment eroded from the upper watersheds for each sampled parameter set. We kept only those samples that lead to ±10% of the volume predicted by the Emergency Assessment Volume model developed by , which is used throughout the western USA to estimate postfire debris flow volumes for an individual rainstorm with a given I15. In general, there is roughly an order-of-magnitude of uncertainty associated with modeled debris-flow volumes using the Emergency Assessment Volume model . Here, however, we kept samples that lead to ±10% of the volume predicted since the Emergency Assessment volume model performed well in our study watersheds. The combined volume as estimated using the model for our two study watersheds was 320000 m3 compared with an observed volume of 309000 m3 . We then took these same sets of samples that resulted in ± 10 % of the target volume and evaluated the maximum flow depth GP at those parameter values and used them in a Monte Carlo (MC) simulation where the PPE inundation emulator was evaluated at each of these samples. This allowed us to estimate the probability of inundation, where a grid cell was considered to be inundated if the maximum flow depth was greater than or equal to 0.1 m. Involving the volume GP in this process provided a mechanism for focusing on the portion of the parameter space that led to realistic debris flow volumes (for a given I15) based on an empirical model developed for our study region. The relevant subset of the parameter space will vary with I15 since the Emergency Assessment Volume model predicts greater debris flow sediment volumes for greater values of I15. We considered three I15 values in our analyses – 38.5, 77, and 96mmh-1 – which yielded debris flow sediment volumes of 88 000, 230 000, and 358 000 m3, respectively. These values of I15 were chosen to be 50 %, 100 %, and 125 % of the observed I15 and correspond to RI_fac input parameter values of 0.5, 1.0, and 1.25, respectively.

The modeled area inundated, defined as the area below the watershed outlets where flow depth exceeded 0.1 m matches the observed area inundated well within a portion of the explored parameter space. The similarity index is generally greater when the hydraulic roughness coefficient, n0 is less than 0.1 (Fig. ). The amount of sediment eroded from the upper watersheds varies substantially from roughly 2000 m3 to nearly 700000 m3 (Figs. S3, ). The observed eroded volume of 307000 m3 falls near the upper range of most simulated erosion volumes (Fig. ). The similarity index generally increases with erosion volume and approaches its maximum value of roughly 0.23 where the simulated and observed erosion volumes are similar (Fig. ).

A good test of an emulator's performance is its ability predict left-out values using the rest of the design simulations, and so we examined leave-one-out predictions for our debris flow simulations. A GP emulator was re-fit 64 times, each with a set of N=63 design simulations and one left-out simulation; each GP was then used to predict its left-out simulation values. The range parameter estimates across these 64 emulators were very stable. More specifically, we found the coefficients of variation for each of the six range parameter values to be between 0.03–0.15 indicating that the relative influence of any input to the GP was not swayed strongly by any single flow simulation. For illustrative purposes, we focused on two points of interest – one located in a channel and one on the adjacent fan surface where flow is relatively unconfined. Note, the training simulations were designed to generate a large range of flow depth values. In Fig. , for both cases, we sorted the simulations by their left-out flow depths, y, and predicted each with a credible interval centered at the mean of the GP prediction, ỹ. We found root mean squared errors from these leave-one-out experiments of 0.16 and 0.36m for the locations on the fan and in the channel, respectively. Further, we observed that 89 % (fan location) and 94 % (channel location) of the simulated depths fall within their predictive credible intervals. These numbers are slightly below the anticipated 95 % coverage, but this is likely due to the relatively small training set. The rule-of-thumb for the size of a GP training set is at least 10× the number of input parameters varied . In the case of the fan location, roughly one third of the simulations resulted in no inundation. While this behavior was anticipated – some training scenarios do not lead to debris flow inundation on the fan – it also effectively reduced the size of the training set for the GP to learn how inundation depths change as scenarios change. Cases where a large fraction of training runs that result in zero (no inundation) outputs are challenging for GP emulation. Although introduced methodology to address this challenge in the scalar-output case, applying this methodology to the PPE is an on-going area of research.

A crucial step to fitting a GP is estimating the range parameters. Smaller range parameters indicate that the corresponding model parameter has more influence on the debris flow model output of interest (i.e. maximum flow depth). (See the leftmost column of Table ). Modeled peak flow depth was most sensitive to the hydraulic roughness coefficient, followed by effective grain size, rainfall intensity, fraction of stream power effective in sediment detachment, and saturated hydraulic conductivity; it was relatively insensitive to the maximum soil thickness.

One will often want to use GP surrogates in a predictive mode. We used surrogates to predict how the probability of debris flow inundation varies across three different rainfall scenarios as defined by rainstorms with a peak I15 of 38.5mmh-1 (RIfac=0.5), 77mmh-1 (RIfac=1), and 96mmh-1 (RIfac=1.25), respectively. Figure  shows scatter plots from (500 volume-targeted) samples of effective grain size (δ), effective sediment detachment fraction (F), and hydraulic roughness (η0) against each other along with histograms of samples in each parameter. As I15 increases, F goes from nearly uniform to skewed toward higher values while δ values cover the whole range of grain sizes for lower I15, but skew toward the smallest grain sizes for high values of I15. Greater values of I15 are associated with greater target debris flow volumes. The observed shifts in the relevant portion of the parameter space reflect this change. For example, greater values of F and smaller values of δ produce larger debris flows (Fig. S3).

Figure  shows a map of the probability of inundation for each rainfall scenario. The spatial footprint of area with a high probability of inundation increases substantially as I15 increases from 38.5 to 77mmh-1. Such a probabilistic analysis using only T2D-M2017 simulations would be computationally taxing.

developed parametric best-fit curves to model the change in saturated hydraulic conductivity and hydraulic roughness as a function of time following fire in the nearby San Gabriel Mountains. Using these relationships, and setting other parameters to the center of their respective ranges, we used the GP emulator to explore the effects of temporal changes in the hydraulic roughness coefficient and saturated hydraulic conductivity. Both peak flow depth and area inundated in response to the observed rainstorm would decrease substantially over the first six months following fire (Fig. ). For example, US Highway 101, which runs perpendicular to the direction of flow near the distal portion of the fan, would only be inundated when the rainstorm occurs within the first 3 months following fire. If the observed rainstorm were to have occurred 12 months following the fire, the simulated inundation area would be limited to channels near the fan apex.

We can also explore the effects of rainfall intensity and temporal changes in hydraulic roughness and saturated hydraulic conductivity following fire by examining flow depth at distinct points of interest. Again, we considered the same two points for illustrative purposes, one located in a channel and one on the adjacent fan surface where flow is relatively unconfined (Fig. ). For a given time since fire, peak flow depths are greater in the channel relative to the fan surface, as expected. Peak flow depth decreases gradually over the first several months at the point on the fan before dropping substantially after approximately nine months. Peak flow depths decrease over the first year following fire in the channel location from roughly 4 to 2 m. Visualizing peak flow depths as a function of time since fire and rainfall intensity can be helpful for assessing temporal shifts in the magnitude of rainfall associated with potential debris-flow impacts at different locations. For example, the area inundated by a flow in response to a rainstorm characterized by RIfac=1 would decrease substantially over the first 9 postfire months (Fig. ).

We further used the emulator to produce probabilistic maps of inundation at different times following fire (Fig. ). Differences in the spatial patterns of inundation likelihood are apparent between scenarios where the storm occurs 2 months following the fire versus 14 months. We identified a location as being inundated if peak flow depth exceeds 0.1 m. RI_fac was set to 1. All parameters were set to their central values except for saturated hydraulic conductivity and the hydraulic roughness coefficient. The latter parameter was sampled from the distribution suggested by while the former was set to the 2 and 14 month values, respectively, estimated from the same study. The probability MC calculation was carried out with 100 samples.

T2D-M2017 was able to provide reasonable estimates of inundation extent. Across the 64 T2D-M2017 simulations, which only sparsely cover the parameter space, the maximum similarity index of 0.23 is similar to what other debris flow runout models have achieved when calibrated to the study area. For example, a runout model based on an empirical flow routing algorithm achieved a similarity index of 0.25 when applied to the San Ysidro watershed and physics-based models (i.e. RAMMS, FLO-2D, D-CLAW) with various rheological assumptions had similarity indices between 0.23 and 0.26 . Here, the similarity index was generally greater when the volume of sediment eroded from the upper watersheds was closer to the observed volume eroded. This observation is consistent with flow volume being an influential factor in debris flow runout, including specifically in our study area .

Although the T2D-M2017 model was able to reproduce both the observed sediment volume eroded from the upper watershed and the inundation pattern on the fan, the resulting spatial patterns of erosion are not consistent with observations of widespread incision in higher order channels (Fig. S4). Modeled erosion patterns are characterized by incision on hillslopes in areas of concentrated flow and in low order channels, which was also observed after the debris flow event . We attribute the limited amount of modeled erosion in high order channels, in part, to model assumptions and limitations. For example, since there is no established and generally applicable methodology for quantifying sediment entrainment rates by debris flow, the model neglects sediment entrainment when the volumetric sediment concentration exceeds 0.4 . As a result, intense erosion that leads to high sediment concentrations and debris flow initiation in low order channels could result in higher order channels acting as transport zones for the debris flows that developed at lower drainage areas. We accepted this limitation since our main objective was to simulate the downstream effects of debris flows rather than their growth rates or erosion patterns. Improved representation of sediment entrainment processes, particularly in channels and in portions of flow with sediment concentrations approaching those associated with debris flows, would be beneficial for future applications.

Fire effects on soil and vegetation properties that affect the initiation and growth of runoff-generated debris flows are most extreme in the first few months following fire . Potentially rapid changes in hydrologic conditions following fire limit the time window for gathering data needed to constrain parameters for postfire runoff and erosion models, including the model used here. Aside from rainfall intensity, which will not be affected by the fire, we found that hydraulic roughness, the representative grain size, the fraction of stream power effective in sediment entrainment, and saturated hydraulic conductivity played the most important roles in controlling peak debris flow depth. Additional model testing across fire-prone regions in different geologic and climate settings is needed to assess model performance and determine the extent to which results related to parameter sensitivity are generalizable. Soil thickness, which provides a limit on the maximum depth of incision, could be more influential if the model better reproduced observed erosion depths throughout the channel network. In simulations, debris flow sediment was primarily sourced from more widespread shallow incision on hillslopes and low-order channels. Therefore, there were relatively few locations in the landscape where the maximum depth of incision was achieved. Nonetheless, these results provide observational targets that can help focus future efforts to collect perishable postfire data.

Peak flow depth was most sensitive to hydraulic roughness. We hypothesize that hydraulic roughness plays an important role in controlling inundated area and peak flow depths because of its influence on both modeled sediment detachment rates and flow resistance. Lower values of hydraulic roughness are associated with increased erosion from the upper watersheds (Fig. S3). Saturated hydraulic conductivity will influence the rate at which sediment is detached by overland flow since it plays a role in controlling runoff magnitude and stream power. Increased rates of sediment detachment lead to increases in flow volume, which in turn act to increase runout and inundation potential . Grain size,which is the second most influential parameter, similarly influences flow volume since a larger grain size will encourage more rapid deposition of sediment. The volume of sediment eroded from the upper watersheds decreases with increasing δ (Fig. S3).

Our evaluation of parameter sensitivity indicates that constraints on postfire values for hydraulic roughness, saturated hydraulic conductivity, fraction of stream power effective in sediment entrainment, and the grain size distribution of sediment entrained in debris flows would be beneficial for improving estimates of debris flow runout estimates. Burn severity is likely to play a substantial role in a fire's initial effect on these variables . In addition, attempts to capture changes in debris flow runout as a function of time since fire would benefit from methods to parameterize temporal changes in these influential parameters. Fire-driven reductions in hydraulic roughness are commonly cited as a cause for increased runoff and erosion as are increases in soil erodibility, which could be reflected in greater values of F. There are few constraints on the temporal changes in hydraulic roughness or F following fire, which may be facilitated by changes in vegetation cover, ground cover, and/or grain roughness. In addition to being parameterized based solely on time since fire, changes in hydrologic model parameters, such as hydraulic roughness or saturated hydraulic conductivity, could be tied to changes in remotely sensed vegetation indices .

Particularly in southern California and other tectonically active regions in the western USA , fire can promote substantial increases in dry ravel activity on hillslopes that can reduce hydraulic roughness by increasing the availability of fine sediment in channels. Hydraulic roughness may then increase over time as dry ravel deposits are progressively eroded during postfire rainstorms . Temporal changes in debris flow sediment source locations and coarsening of particle size distributions due to preferential erosion of fines would also influence the effective grain size in the model. In practice, it is not clear how to quantitatively connect this single grain size parameter to the particle size distribution of hillslope or channel sediment, especially when flows contain boulders. Postfire changes in saturated hydraulic conductivity can be inferred from calibration of hydrologic models , rainfall simulator experiments at the small plot scale , and point scale measurements . While some general patterns have been observed between time since fire and values of saturated hydraulic conductivity, there is substantial site-to-site variability . The level of uncertainty in influential model input parameters and how they change over time highlights the need for probabilistic assessments of debris flow runout, which emulators can help to achieve by facilitating rapid exploration of large parameter spaces.

Rainfall is a necessary driver for debris-flow initiation and the model was also sensitive to rainfall intensity, specifically a rainfall intensity factor which we used to scale an observed rainfall intensity time series. This finding is consistent with observations that postfire basin-scale sediment yields and debris flow volume increase with rainfall intensity averaged over durations of 60 min or less. Short duration (sub-hourly) bursts of high intensity rainfall are effective at generating infiltration-excess overland flow that can trigger debris flows in recently burned steeplands . Since the spatial and temporal distributions of rainfall can influence flood and debris-flow processes , future work could consider sensitivity of debris-flow runout to storm characteristics beyond peak intensity.

Emulators can be useful for generating probabilistic maps of debris flow inundation in response to design storms with different rainfall intensities or examining changes at particular points of interest. In cases where there are specific values at risk downstream of a burned area, rapid exploration of debris flow characteristics (i.e. peak flow depth) as a function of rainfall intensity could help define impact-based rainfall thresholds that could be used for planning and warning purposes. In other words, one could take advantage of the emulator's computational efficiency to determine not only the rainfall intensity required to initiate a debris flow, but also the rainfall intensity required to produce a debris flow that would impact a prescribed area of interest with some prescribed depth of flow. Such estimates could be particularly useful for assessing building damage .

The computational cost of many physically-based debris flow models is a limitation in applications that are time sensitive, such as rapid postfire hazard assessments. Postfire debris flows in the western USA, such as those that occurred near Montecito, may occur before the fire has been officially contained and within weeks or months of fire ignition. Approximately 23% of postfire debris flow events, for example, initiate in the first 60 d following fire . The emulator methodology presented here provides one avenue for minimizing computation times, since an initial suite of simulations can be used to train the emulator which can later be applied with substantially less computational effort to generate a probabilistic hazard map for a specific scenario. Further, as emulators do not depend on a prior distribution characterizing scenario randomness, modeling scenario randomness can be done in parallel or after emulator construction. Likewise, an emulator could even be trained prior to a fire. Analogous approaches have been employed in related applications . Within the context of postfire hazards, an emulator could be used to assess debris-flow runout and inundation downstream of a burned area in response to a design or forecast rainstorm. Atmospheric model ensembles, for example, can provide estimates of peak 15 min rainfall intensity over watersheds of interest that could be used to constrain a distribution of rainfall intensity factors .

We focused on applications of surrogate modeling for postfire runoff-generated debris flows since accelerating MC calculations could be particularly beneficial within the context of rapid hazard assessments after fire when there are time constraints. However, the methodology presented here could similarly be applied to debris flows in unburned settings as well as to a broader range of geophysical flows. Runoff-generated debris flows, for example, can initiate in unburned alpine environments through processes that are similar to those that operate in recently burned watersheds . There remain additional challenges to constructing robust and efficient emulators of geohpysical mass flow models that warrant future study: handling large numbers of zero-output pixels; and developing efficient adaptive design strategies that can reliably capture simulator behavior at every pixel of interest.