Objective definition of discharge thresholds for post-fire debris flows

. Runoff-generated debris flows are a common post-fire hazard in the western United States and a growing number of regions around the world. As wildfire continues to emerge across a broader range of geographic regions and plant communities, there is an increasing need for generalizable methods to predict post-fire debris-flow initiation. The prediction of post-fire debris flow during intense rainstorms has traditionally relied upon empirical rainfall thresholds. Rainfall intensity-duration thresholds are often developed based on rainfall data and the hydrologic response to those rainstorms. They are most applicable to the specific regions where data are collected. Here, we present a new predictive approach that utilises processes-based models with fundamental physics and machine learning methods to estimate discharge thresholds for runoff-generated debris-flow initiation in four recently burned areas in the western United States. We assess the performance of the objectively defined discharge threshold-based predictions for post-fire debris-flow initiation from our hybrid framework, which utilises debris-flow timing within rainstorms, physically based numerical simulations of runoff, and the support vector machines method. The proposed thresholds have a good balance between true and false predictions for debris flow and floods. Importantly, our method permits the direct estimation of rainfall intensity-duration thresholds for areas where post-fire debris flow observations are limited.


Introduction
Wildfire significantly changes vegetation and soil hydrologic properties, making burned areas more likely to produce increased runoff and debris flows than unburned areas [1,2]. Specifically, wildfires usually decrease canopy interception and reduce surface water storage, allowing more rain to reach the ground and generate runoff [3]. Flow may be further enhanced by reduced infiltration due to soil sealing, waterrepellency, and hyper-dry soil conditions induced by wildfire [4]. The increased surface water flow following fires can initiate post-wildfire debris flows in many burned drainage basins in the Western United States [e.g., 4,5].
Post-fire debris-flow hazard assessments often employ regionally specific rainfall thresholds. However, many different site-or region-specific variables control rainfall thresholds, such as antecedent conditions, topography, lithology, soil properties, initiation mechanism, sediment source, and land cover [6]. Rainfall intensity-duration (ID) thresholds are the most common method for predicting debris flow and landslide occurrence [e.g., [5][6][7]. Most studies have traditionally defined thresholds using a subjective approach. Some attempts have been made to define rainfall ID thresholds objectively using statistical methods [e.g., 6], machine learning models [e.g., 7], or theoretically process-based models [e.g., 8]. However, Corresponding author: Hui Tang, htang@gfz-potsdam.de defining thresholds with these methods is still challenging when direct observations and data constraining the precise timing of debris-flow activity are unavailable. Another general challenge for rainfall thresholds derived from statistic or machine learning methods is that they usually can not reveal the physical mechanisms that control the initiation of debris flow and the hydrologic response of the drainage basin after a fire.
Past studies have suggested that debris-flow initiation is related to discharge in the channel, which is slope-dependent [10][11][12]. Runoff-generated debris flows initiate once discharges exceed slope-dependent dimensionless discharge thresholds during debris-flowproducing storms on a catchment scale [11][12]. [12] tested the applicability of dimensionless discharge for predicting debris-flow activity at the basin scale using a post-fire debris flow dataset in the San Gabriel Mountains of southern California. They also showed that these rainfall intensity-duration thresholds derived from discharge thresholds are comparable to empirical rainfall thresholds in different catchments with less than a 20% difference. Processes-based models offer an opportunity to explore the links between rainfall intensity, discharge, and debris-flow initiation [13]. However, these physics-based models usually require meteorological, hydrological, and topographic data for model calibration. Consequently, a combination of a machine learning model and a process-based model for predicting the temporal occurrence of debris flows could overcome the drawbacks of these two types of models and potentially improve post-fire debris-flow predictions.
This study proposes and tests a new hybrid framework combining process-based models with machine learning methods to objectively determine discharge thresholds that can be applied to areas with no established empirical thresholds. Specifically, our new framework derives discharge thresholds for post-fire debris flows from two process-based models based on the Kinematic wave equations (KWE) and shallow water equations (SWEHR) with the support vector machines method. We use numerical models and monitoring data that constrains the timing of 135 postfire debris flow or flood events from 28 monitored basins to estimate the discharges associated with debrisflow initiation and flood-dominated events. The debris flow and flood are identified based on monitoring data and field surveys. We then apply the support vector machine method to derive dimensionless discharge thresholds for debris-flow initiation. We also use receiver operating characteristics analysis to evaluate the performance of our derived thresholds.

Monitoring and Field Measurements
Our study sites include 28 watersheds in the western USA, burned by four different wildfires (Station Fire, California, 2009; Fish Fire, California, 2016; Pinal Fire, Arizona, 2017; Buzzard Fire, New Mexico, 2018; see Figure 1). Study basins from the 2009 Fish Fire and the 2016 Station Fire have similar climate, vegetation, burn severity, and topographic characteristics [19,20]. Table  1 summarise some basic features of wildfires and basins in this study. For details, we refer to the respective papers provided in the notes to Table 1.

Process-based Models
This study uses two process-based models with basic physics to estimate discharges. For study sites at Las Lomas (burned by the Fish Fire) and Arroyo Seco (burned by the Station Fire), where data exist to calibrate a sediment transport model, we applied a model based on the shallow water equations with advection equations for sediment transport and the Hairsine and Rose soil erosion model (SWEHR). The details of SWEHR can be found in [19] and [20], which include components representing the fluid flow, size-selective sediment transport, infiltration, rainfall interception, and mass failure of bed material. Runoff and sediment transport are simulated for all runoff-generating rainstorms at Las Lomas and Arroyo Seco to compute the unit discharge for every channel location in the basin by where ℎ is the water depth, and and are the flow velocity in the x and y-direction, respectively. For the other study sites affected by the Station Fire, Pinal Fire, and Buzzard Fire, we employed the kinematic wave (KW) model described in [22], which approximates the full Shallow Water equations for simulated discharge. The water flux was determined by the Manning equations: where 0 and 0 are the ground surface slope in the x and y-direction, respectively. is the Manning coefficient. Then, the discharge is: To estimate dimensionless discharge thresholds, we quantify dimensionless discharge when debris flows are initiated or when there is a flood response that does not include any debris-flow activity. We calculate the dimensionless discharge following [11] and [23] where is the discharge from the SWEHR model ( ) or KW model ( ). 50 is the median grain size. and are the density of water and sediment, respectively. is the gravity acceleration. Here, we constrain 50 based on the grain size distribution from debris-flow deposits at each study site. We follow the approach from [12] and compute time-averaged values of * immediately prior to the time a debris flow was observed at the basin outlet if it is known. In cases where debris-flow timing within a debris-flow-producing storm is unknown, we use the peak discharge. In the case of a flood response to a rainstorm, we also use the modelled peak discharge.

Receiver Operating Characteristics analysis (ROC)
We used a receiver operating characteristic (ROC) analysis to objectively analyse discharge thresholds. ROC is an optimisation method that assesses classification model performance and has been widely applied for rainfall intensity-duration thresholds [6,17]. This method generally maximises the number of correct predictions while minimising the amount of falsepositive and false-negative predictions [24][25]. Each discharge for a given bed angle was classified based on https://doi.org/10.1051/e3sconf/202341503029 , 03029 (2023) E3S Web of Conferences 415 DFHM8 whether a debris flow was recorded from monitoring data and field observations. The boundary between debris-flow events and water-dominated flow events is the optimal discharge threshold to distinguish debrisflow-producing events from non-debris-flow-producing events. Here, we use the ROC analysis to objectively define a slope-dependent discharge threshold based on numerical simulations and compare it with discharge thresholds for the San Gabriel Mountains from [12]. We further evaluate the performance of the discharge thresholds from support vector machines. The ROC method can determine the best threshold for a given bed angle by using the true-positive rate ( + , where TP is the number of true-positive cases and FN is the number of false-negative cases) and false-positive rate ( + , where FP is the number of false-positive cases, and TN is the number of true-negative cases). The AUC value is defined as the area below the ROC curve. The perfect prediction model will yield an AUC value of one. The discharge threshold derived from the ROC method will subsequently be referred to as the ROC threshold.

Support Vector Machines (SVM)
The Support Vector (SV) algorithm is a class of nonlinear search algorithms based on a statistical learning theory developed by [26] and [27]. The SV algorithm has been successfully developed as a classification tool, such as the Support Vector Machines (SVM) based on a supervised learning system. Sample classification is determined by finding a hyperplane that produces the optimal separation between classes. The SVM differs from other approaches because the hyperplane is constructed from the training dataset points for linearly separable binary problems. If two categories are not linearly separable in a dataset, kernel functions are used to map the data into a higherdimensional space to separate the classes with a linear hyperplane. The best hyperplane for an SVM is the one with the most significant margin between the two categories. Margin means the maximum slab width parallel to the hyperplane with no interior data points. This study applies the support vector machine with a kernel function to define the boundary to separate the dimensionless discharges that produce debris flows and water-dominated floods as two categories. The defined threshold model from this method will subsequently be referred to as the SVM threshold. Figure 2 shows the preliminary results from both the ROC and SVM methods. For comparison, we added the original discharge thresholds derived from processesbased model simulations in the San Gabriel Mountains [12]. There are two types of thresholds in [12]: upper limit thresholds (the averaged value of discharge during debris flows, UL, red dash line) and lower limit thresholds (the lowest value of discharge during debris flows, LL, blue dash line). See [12] for details. The ROC thresholds are very close to these LL thresholds from numerical model simulation. In contrast, the SVM thresholds are generally higher than ROC thresholds but smaller than the UL thresholds from [12]. The discrepancy between SVM thresholds and those previously derived from the processes-based modelling may arise because the previous study only focused on one study site (Fish Fire). Meanwhile, the significant difference in the grain size of debris-flow deposits from different sites (1 to 15 mm) increases variability in the dimensionless discharges. The AUC value for the SVM threshold is about 0.870-0.940. The AUC value for the ROC threshold is slightly smaller than the best SVM threshold with a radial basis kernel, which is about 0.904. Since only limited data points are available, we cannot directly compare the AUC value for the UL and LL thresholds from [12]. The UL usually has lower true-positive rates, but LL usually has higher false-positive rates. More generally, the SVM threshold makes a good balance between the true predictions and false predictions for all channel bed angles. This is because the SVMs need to simultaneously consider all data from different channel bed angle. However, ROC, LL, and UL get the best performance for each channel bed angles individually.

Results and Discussion
Numerical simulations of debris flow for rainstorms with different durations and intensities from monitored basins demonstrate that dimensionless discharges needed to initiate debris flows are similar across four recently burned areas. Therefore, an advantage of establishing dimensionless discharge thresholds for debris-flow initiation is that they are not strongly based on the hydrologic properties of the catchment. Thus, generating rainfall ID thresholds based on dimensionless discharge thresholds with an approach similar to that in [12] can provide a promising alternative to empirical methods for assessing the potential of runoff-generated debris flows in areas with limited or no historical data on debris flows after a wildfire. In summary, our results indicate that slopedependent dimensionless discharge thresholds can predict runoff-generated debris flow following fire, in agreement with prior work that utilised a more limited dataset. This study proposes a new physics-informed machine learning approach to discriminate debris-flow events from water-dominated flood events. Thresholds derived from support vector machines (SVMs) fall between the UL and LL thresholds from [12] and show good performance relative to those derived with receiver operating characteristics analysis.