Investigation of Post-Fire Debris Flows in Montecito

Debris flows in a burned area, post-fire debris flows, are considered as one of the most dangerous geo-hazards due to their high velocity, long run-out distance, and huge destruction to infrastructures. The rainfall threshold to trigger such hazards is often reduced compared with normal debris flow because ashes generated by mountain fires reduce the permeability of the top soil layer, thus increasing surface runoff. At the same time, burnt material and residual debris have very poor geo-mechanical characteristics, e.g., their internal friction angle and cohesion are typically low, and thus an intense rainfall can easily trigger some debris flows. Studying post-fire debris flow enables us to get a deeper understanding of disaster management. In this paper, the debris flow that occurred in Montecito, California, USA, and was affected by the Thomas Fire was used as a case study. Five major watersheds were extracted based on the digital elevation model (DEM). Remote sensing images were used to analyze the wildfire process, the extent of the burned areas, and the burn severity. The hypsometric integral (HI) and short-duration rainfall records of the watersheds around Montecito when the post-fire debris flows occurred were analyzed. Steep terrain, loose and abundant deposits, and sufficient water supply are the important conditions affecting the formation of debris flows. Taking watersheds as the research objects, HI was used to describe the geomorphic and topographic features, open-access rainfall data was used to represent the water supply, and burn severity represented the abundance of material sources. An occurrence probability model of post-fire debris flow based on HI, short-duration heavy rainfall, and burn severity was developed by using a logistic regression model in post-fire areas. By using this model, the occurrence probability of the post-fire debris flow in different watersheds around Montecito was analyzed based on the precipitation with time. Especially, the change characteristics of occurrence probability of debris flows over time based on the model bring a new perspective to observe the obvious change of the danger of post-fire debris flows and it is very useful for early warning of post-fire debris flows.


Introduction
Vegetation in the forest has the function of conserving water and preserving soil erosion.The branches and leaves of forest trees can greatly reduce the impact of rainwater on the surface of the earth.The spongy vegetation layer on the ground surface of wooded areas not only can dampen the impact of rainwater, but it can also absorb a large amount of water, thus reducing water infiltration into the ground.At the same time, a large number of interlaced forest roots can help to stabilize soil.Forest fire is a common disaster in wooded areas, which poses a serious threat to the environment, life, and property [1,2].Forest fire burns vegetation on hillsides, which enhances water infiltration into the ground and decreases the stability of slopes [3].On the other hand, because there is no vegetation protection, the rate of water absorption by the soil may not exceed the rate of rainfall, and it could cause surface runoff.Cannon et al. [4] found that most post-fire debris flows are generated through the process of progressive entrainment of material eroded from hillslopes and channels.Post-fire debris flows are initiated by runoff and erosion, rather than shallow landslides [5], and they can grow in size through erosion and scour during the movement [6].When a large amount of ash layer and loose sediment appear as rich material sources after mountain fires, the process of progressive entrainment may become easier.As a result, the threshold of rainfall to induce debris flows is reduced obviously.Short et al. [7] conducted research about a post-fire debris flow in western Montana, and they found that the probability of debris flows increased by 50% after forest fires.Debris flow often damages highway, railway, and other transportation facilities, even villages and towns, causing huge losses [8][9][10][11][12][13].
Compared with traditional debris flow, some important parameters related to post-fire debris flow should be taken into consideration when modifications of the conventional debris flow prediction method are made.Currently, the prediction of post-fire debris flow mainly focuses on occurrence probability, frequency, and scale.Cannon et al. [14,15] studied the peak flow prediction of post-fire debris flows using the basin gradient, burn extent, and storm rainfall.Gartner et al. [16] also used variables, including basin areas burned at different severities, measures of basin morphology, rock type, soil material properties, rainfall amounts, and intensities for storms, to construct empirical models to predict the volumes of debris flows.Cannon et al. [17] used areal burned extent, soil properties, basin morphology, and rainfall from short-duration and low-recurrence-interval convective rainstorms to construct models to describe post-fire debris-flow probability.Gartner et al. [18] proposed a prediction function of erosion probability for post-fire debris flows based on channel slope, plane curvature, and channel length by conducting a comparison among factors, including contributing drainage basin area, channel slope, planform curvature, burn severity, and the upstream channel length, resulting in good accuracy and precision.It could be stated that different researchers use various evaluation factors.Based on the data-driven theory and genetic programming algorithm, Friedel [19] considered that the nonlinear equation established by three indexes, the average slope of the basin, the total burned area, and the total rainfall, has the least prediction error for the debris-flow scale.If only rainfall factors are used, it is also a common method to judge the occurrence probability of post-fire debris flows by rainfall intensity-duration thresholds [4,20].
The above research shows that it is feasible to construct forecast models to study the occurrence probability of post-fire debris flow.Complicated parameters in models would increase the uncertainty [21] and difficulty of the model application.Therefore, using a reduced number of key factors, like Friedel [19], may be easier to establish forecast models of watersheds affected by wide fire.Steep terrain, loose and abundant deposits, and sufficient water supply are the important conditions affecting the formation of debris flows.Currently, the geomorphic and topographic characteristics of debris flow can be better reflected by hypsometric integral (HI) [22] based on watershed units.Burn severity calculated from remote sensing images could represent the abundance of material sources.An important effect of mountain fires is the change of the rainfall threshold for hazards occurring due to the amount of ash produced by mountain fires.Different from the effect of long-time antecedent rainfall for ordinary debris flow, short-duration rainfall plays an important role in the occurrence of post-fire debris flows [23].In this paper, HI, burn severity, and short-duration rainfall characteristics of the watersheds around Montecito were analyzed firstly.After that, these three factors were selected to conduct the logistic regression model of occurrence probability of the debris flows in post-fire areas.The change of post-fire debris flow occurrence probability around Montecito was then analyzed from the newly proposed model based on different rainfall in different periods.

Background and Study Area
Montecito is an unincorporated community and census-designated place in Santa Barbara County, California, located east of the City of Santa Barbara.Located at the foot of the Santa Ynez Mountains, which are young and mostly of sedimentary origin, the peak elevation around Montecito has an altitude of 981 m.Montecito experiences a cool Mediterranean climate characteristic with warmer winters and cooler summers compared with places further inland because of Montecito's proximity to the ocean.
Early on the morning of 9 January 2018, about one month since the start of Thomas Fire, a heavy rainfall caused mud and boulders from the Santa Ynez Mountains to flow down creeks and valleys into Montecito [24] (Figure 1).This post-fire debris flows consisting of mud, boulders, and tree branches were up to 15 feet (5 m) in height, moving at estimated speeds of up to 20 miles per hour (30 km/h) into the lower areas of Montecito [25].The cascading processes resulted in 21 deaths and two missing persons [26].About 163 people were hospitalized for various injuries [27].The disaster caused at least $177 millions of property damage, at least $7 millions in emergency responses, and another $43 millions in cleaning costs [28,29].

Background and Study Area
Montecito is an unincorporated community and census-designated place in Santa Barbara County, California, located east of the City of Santa Barbara.Located at the foot of the Santa Ynez Mountains, which are young and mostly of sedimentary origin, the peak elevation around Montecito has an altitude of 981 m.Montecito experiences a cool Mediterranean climate characteristic with warmer winters and cooler summers compared with places further inland because of Montecito's proximity to the ocean.
Early on the morning of 9 January 2018, about one month since the start of Thomas Fire, a heavy rainfall caused mud and boulders from the Santa Ynez Mountains to flow down creeks and valleys into Montecito [24] (Figure 1).This post-fire debris flows consisting of mud, boulders, and tree branches were up to 15 feet (5 m) in height, moving at estimated speeds of up to 20 miles per hour (30 km/h) into the lower areas of Montecito [25].The cascading processes resulted in 21 deaths and two missing persons [26].About 163 people were hospitalized for various injuries [27].The disaster caused at least $177 millions of property damage, at least $7 millions in emergency responses, and another $43 millions in cleaning costs [28,29].The post-disaster high-resolution remote sensing image of WorldView-2 on 13 January 2018 was downloaded from Google Earth.By using image interpretation of the post-disaster remote sensing image, it was found that debris flows affecting Montecito mainly broke out in five watersheds (Figure 2).The impact areas of these post-fire debris flows are different, among which the debris flows coming from the first and third basins caused huge damage to downstream residential areas (Table 1).Mass mud came out of the mountain passes and rushed into the city to form a number of debris-flow accumulation areas with different thicknesses of mud.The post-disaster high-resolution remote sensing image of WorldView-2 on 13 January 2018 was downloaded from Google Earth.By using image interpretation of the post-disaster remote sensing image, it was found that debris flows affecting Montecito mainly broke out in five watersheds (Figure 2).The impact areas of these post-fire debris flows are different, among which the debris flows coming from the first and third basins caused huge damage to downstream residential areas (Table 1).Mass mud came out of the mountain passes and rushed into the city to form a number of debris-flow accumulation areas with different thicknesses of mud.

Materials
Free remote sensing data are easy to obtain through the internet, such as Landsat satellite images [32,33] of National Aeronautics and Space Administration (NASA) which can be downloaded from the United States Geological Survey (USGS) Global Visualization Viewer (GloVis) website (https://glovis.usgs.gov/),and Sentinel satellite images [34,35] of European Space Agency (ESA), which can be downloaded from the Copernicus Open Access Hub website (https://scihub.copernicus.eu/dhus/#/home).These free satellite data can be used to analyze nature hazards -mountain fires [34], landslides [36], debris flow [37]

Fire Analysis Using the Shortwave Infrared (SWIR) and Normalized Burn Ratio (NBR)
Shortwave infrared (SWIR) can penetrate haze and smoke, so SWIR images can be used to identify and analyze active forest fire [39,40].Taking Landsat 8 data as an example, the fire smoke can be seen in the true color image (Figure 3a).However, it is difficult to identify the burned areas.In the false-color composite image of shortwave infrared data (R = SWIR2, G = SWIR1, and B = SWIR2), the burned areas are revealed as pink color (Figure 3b).

Materials
Free remote sensing data are easy to obtain through the internet, such as Landsat satellite images [32,33] of National Aeronautics and Space Administration (NASA) which can be downloaded from the United States Geological Survey (USGS) Global Visualization Viewer (GloVis) website (https://glovis.usgs.gov/),and Sentinel satellite images [34,35] of European Space Agency (ESA), which can be downloaded from the Copernicus Open Access Hub website (https://scihub.copernicus.eu/dhus/#/home).These free satellite data can be used to analyze nature hazards -mountain fires [34], landslides [36], debris flow [37]

Fire Analysis Using the Shortwave Infrared (SWIR) and Normalized Burn Ratio (NBR)
Shortwave infrared (SWIR) can penetrate haze and smoke, so SWIR images can be used to identify and analyze active forest fire [39,40].Taking Landsat 8 data as an example, the fire smoke can be seen in the true color image (Figure 3a).However, it is difficult to identify the burned areas.In the false-color composite image of shortwave infrared data (R = SWIR2, G = SWIR1, and B = SWIR2), the burned areas are revealed as pink color (Figure 3b).A normalized burn ratio (NBR) is defined to index the severity of a burn using multispectral remote sensing images.The formula for NBR uses the near-infrared band and the short-wave infrared band [41,42]: where NIR is the near-infrared band and SWIR is the short-wave infrared band For a burned area, NBRb is calculated before the burn and NBRp is calculated from an image immediately following the burn.Burn severity is calculated by taking the difference between these two index layers [43]: Table 2 from the USGS FireMon program can be used as a first approximate for interpretation of NBR differences corresponding to different burn severity [44].A watershed is a hydrological unit commonly used in natural resource management and planning.It is more common to study debris flow hazards based on watershed units [45][46][47].In this paper, the extraction of watersheds by creating a watershed boundary was achieved using 30-m resolution Shuttle Radar Topography Mission (SRTM) Digital Elevation Model (DEM) data in ArcGIS software.A sink is a cell or set of spatially connected cells, which are surrounded by higher value grids.Sink filling is the first step for the extraction of the basin boundary.It can help to A normalized burn ratio (NBR) is defined to index the severity of a burn using multispectral remote sensing images.The formula for NBR uses the near-infrared band and the short-wave infrared band [41,42]: where NIR is the near-infrared band and SWIR is the short-wave infrared band For a burned area, NBR b is calculated before the burn and NBR p is calculated from an image immediately following the burn.Burn severity is calculated by taking the difference between these two index layers [43]: Table 2 from the USGS FireMon program can be used as a first approximate for interpretation of NBR differences corresponding to different burn severity [44].A watershed is a hydrological unit commonly used in natural resource management and planning.It is more common to study debris flow hazards based on watershed units [45][46][47].In this paper, the extraction of watersheds by creating a watershed boundary was achieved using 30-m resolution Shuttle Radar Topography Mission (SRTM) Digital Elevation Model (DEM) data in ArcGIS software.A sink is a cell or set of spatially connected cells, which are surrounded by higher value grids.Sink filling is the first step for the extraction of the basin boundary.It can help to remove small imperfections in the data by adding the pixel value to the lowest pixel value around it [48].Grids whose sinks are filled can be used to obtain the drainage direction of each grid.The widely used method of determining the flow direction is the eight-direction method, which makes the direction of a pixel to point to a lower pixel, which has the maximum gradient in the surrounding eight pixels [49].Based on the flow-direction grids, the flow-accumulation grids list the number of pixels flowing through it for each pixel.Using the conditional function to set a threshold of the flow-accumulation grids, drainage networks could be obtained.Pour points at the outlet of each watershed were drawn.Using the ArcGIS "watershed" tool, in which the flow-direction raster and pour-point data were used as input data, the contributing areas above different pour points in a raster data were determined as ranges of watersheds.

Hypsometric Integral
The hypsometric integral (HI) is a quantitative index to study the relationship between the horizontal section area and its elevation in a watershed.HI reflects the landform erosion stage and evolution process.Strahler [50] divided geomorphic development into three stages-young stage (HI ≥ 0.60), mature stage (0.35 < HI < 0.60), and old stage (HI ≤ 0.35).When calculating an HI value, contour i is chosen as a reference, the whole watershed area is A, and the area above the contour of the basin is a i .The height difference between the contour and the lowest point of the basin is h i , and the height difference between the highest and lowest points is H.The variables are set as: where x i is the area percentage based on contour i, and y i is the percentage of relative height difference.
The points (x i , y i ) in the rectangular coordinate system are spread to form a curve.It can be defined as: On this basis, the HI value is calculated by a definite integral value in the range of 0 to 1 in Equation ( 6) by using integral curve method: Formula ( 6) belongs to the integral curve method.Pike and Wilson [51] proposed the elevation-relief ratio method by deducing a mathematical formula to estimate the HI value.It can be defined as: where H ave is the average elevation value of a watershed, H min is the minimum elevation value of a watershed, and H max is the maximal elevation value of a watershed.By comparing the above methods, Chang et al. [52] found that the HI values calculated by the methods are almost the same.However, the method of the elevation-relief ratio is more efficient and simpler to calculate the HI value.Therefore, the elevation-relief ratio method used in Equation ( 7) was chosen in this study.

Logistic Regression Model
Although there are many advanced techniques and methods, such as the neural network method [53], conditional probability, and support vector machine [54,55], used in developing a susceptibility mapping of natural hazards, the logistic regression model, as a widespread methodology with the advantage of being simple and easy to implement [55][56][57][58][59], was used in this study to develop the occurrence probability model of post-fire debris flows.Logistic regression (LR) is a regression model in which the dependent variable (DV) is categorical.The output can take only two values, "0" and "1", which represent outcomes, such as win/lose, alive/dead, or healthy/sick [60].Logistic regression is an example of a qualitative response model.In this study, the relationship between the probability of the occurrence of post-fire debris flow and the influencing factors can be expressed as [61]: In this study on the occurrence probability of post-fire debris flow, P represents the probability, and the output result ranges from 0 to 1. Z is the sum of linear weights after superposition of variables, and Bi is a regression coefficient.

Technical Route
The methods used in this paper are described above.In order to make the structure of the article clearer, the technical route is made (Figure 4).In this study, we attempted to take the post-fire debris flow that occurred in Montecito, California, USA as a case study to develop the occurrence probability of post-fire debris flows using several important influencing factors.Firstly, three factors affecting post-fire debris flow in five basins around Montecito were analyzed.We then selected a wider range of data in a larger area covering Montecito to build the logistic regression model in order to enhance the robustness of the model.The model was applied to the five watersheds around Montecito to find the change of occurrence possibility of post-fire debris flow along with rainfalls at different times.
ISPRS Int.J. Geo-Inf.2019, 8 FOR PEER REVIEW 7 two values, "0" and "1", which represent outcomes, such as win/lose, alive/dead, or healthy/sick [60].Logistic regression is an example of a qualitative response model.In this study, the relationship between the probability of the occurrence of post-fire debris flow and the influencing factors can be expressed as [61]: In this study on the occurrence probability of post-fire debris flow, P represents the probability, and the output result ranges from 0 to 1. Z is the sum of linear weights after superposition of variables, and Bi is a regression coefficient.

Technical Route
The methods used in this paper are described above.In order to make the structure of the article clearer, the technical route is made (Figure 4).In this study, we attempted to take the postfire debris flow that occurred in Montecito, California, USA as a case study to develop the occurrence probability of post-fire debris flows using several important influencing factors.Firstly, three factors affecting post-fire debris flow in five basins around Montecito were analyzed.We then selected a wider range of data in a larger area covering Montecito to build the logistic regression model in order to enhance the robustness of the model.The model was applied to the five watersheds around Montecito to find the change of occurrence possibility of post-fire debris flow along with rainfalls at different times.5a), Sentinel 2A data on 13 December 2017 (Figure 5b), Landsat 7 data on 17 December 2017 (Figure 5c), and Landsat 8 data on 25 December 2017 (Figure 5d) were selected to get the burned area using SWIR false color composite images in which burned areas are displayed with pink color.According to the contrast of multi-temporal images, the Thomas Fire, which began on 4 December, had not affected Montecito until 9 December 2018.While on 13 December 2017, mountain fires were burning fiercely, and the range of fires in this area no   5a), Sentinel 2A data on 13 December 2017 (Figure 5b), Landsat 7 data on 17 December 2017 (Figure 5c), and Landsat 8 data on 25 December 2017 (Figure 5d) were selected to get the burned area using SWIR false color composite images in which burned areas are displayed with pink color.According to the contrast of multi-temporal images, the Thomas Fire, which began on 4 December, had not affected Montecito until 9 December 2018.While on 13 December 2017, mountain fires were burning fiercely, and the range of fires in this area no longer expanded on 17 December 2017, which is more than 20 days from 9 January 2018 when the disaster occurred.
ISPRS Int.J. Geo-Inf.2019, 8 FOR PEER REVIEW 8 longer expanded on 17 December 2017, which is more than 20 days from 9 January 2018 when the disaster occurred.The Landsat 8 images on 23 November 2017 and on 26 January 2018 were used to calculate the normalized burn ratio (NBR) separately as the pre-fire NBR and post-fire NBR.Burn severity was calculated by taking the difference between the above two indexes (Figure 6).It can be found that most of the watersheds were moderate burn that could provide a great deal of loose source for debris flow.The Landsat 8 images on 23 November 2017 and on 26 January 2018 were used to calculate the normalized burn ratio (NBR) separately as the pre-fire NBR and post-fire NBR.Burn severity was calculated by taking the difference between the above two indexes (Figure 6).It can be found that most of the watersheds were moderate burn that could provide a great deal of loose source for debris flow.

Geomorphic Development Stage
Using DEM data downloaded from the USGS website [62] and the HI calculation method, the hypsometric integral curves of the different watersheds around Montecito were drawn, as shown in Figure 7, and HI values of them were calculated.According to the HI-based geomorphic development stages proposed by Strahler, and the summary of watershed hypsometric integral curves [22,63], the geomorphic development in Figure 7 is divided into three stages.The area above the convex curve is considered as the young stage; the area below the concave curve is treated as the old stage; the area between the two curves is known as the mature stage [8].According to Figure 7, the five watersheds around Montecito belong to the mature stage.In the infancy and mature periods, the terrain was steep and the solid content of the gully was rich.Therefore, mature stage watersheds have a susceptible geomorphic environment for debris flows.

Geomorphic Development Stage
Using DEM data downloaded from the USGS website [62] and the HI calculation method, the hypsometric integral curves of the different watersheds around Montecito were drawn, as shown in Figure 7, and HI values of them were calculated.According to the HI-based geomorphic development stages proposed by Strahler, and the summary of watershed hypsometric integral curves [22,63], the geomorphic development in Figure 7 is divided into three stages.The area above the convex curve is considered as the young stage; the area below the concave curve is treated as the old stage; the area between the two curves is known as the mature stage [8].According to Figure 7, the five watersheds around Montecito belong to the mature stage.In the infancy and mature periods, the terrain was steep and the solid content of the gully was rich.Therefore, mature stage watersheds have a susceptible geomorphic environment for debris flows.

Geomorphic Development Stage
Using DEM data downloaded from the USGS website [62] and the HI calculation method, the hypsometric integral curves of the different watersheds around Montecito were drawn, as shown in Figure 7, and HI values of them were calculated.According to the HI-based geomorphic development stages proposed by Strahler, and the summary of watershed hypsometric integral curves [22,63], the geomorphic development in Figure 7 is divided into three stages.The area above the convex curve is considered as the young stage; the area below the concave curve is treated as the old stage; the area between the two curves is known as the mature stage [8].According to Figure 7, the five watersheds around Montecito belong to the mature stage.In the infancy and mature periods, the terrain was steep and the solid content of the gully was rich.Therefore, mature stage watersheds have a susceptible geomorphic environment for debris flows.

Rainfall
Post-fire debris flows generally are triggered by one of two processes: Surface erosion caused by rainfall-induced runoff, and slope failure caused by rainfall infiltration.Runoff-dominated processes are by far the most prevalent because fires commonly reduce the infiltration capacity of soils, which increases runoff and erosion [64].Rainfall conditions are often regarded as the

Rainfall
Post-fire debris flows generally are triggered by one of two processes: Surface erosion caused by rainfall-induced runoff, and slope failure caused by rainfall infiltration.Runoff-dominated processes are by far the most prevalent because fires commonly reduce the infiltration capacity of soils, which increases runoff and erosion [64].Rainfall conditions are often regarded as the triggering factor of post-fire debris flows.The influence of mountain fire makes the threshold of rainfall decrease obviously compared with ordinary debris flow.The debris can begin to slide after a rain of 8 mm with a time interval of only 30 min [23].In the current study, Montecito's local rainfall data of a three-hour interval were obtained through the World Weather Online website [38].The first measurable rain for the area around Montecito fell on 8 January 2018.Ash and loose sediment generated by forest fires formed the source of debris flow development.A heavy rainfall process occurred at about 3:00 on the morning of 9 January 2018 (Figure 8a), causing mud and boulders from the gullies to flow along several drainage basins into Montecito.The comparison of rainfall data during the same period in 2018 with 2017 is shown in Figure 8.It is observed that no debris flow occurred on 9 January 2017, with a cumulative precipitation of 16.5 mm (Figure 8b), while the debris flow occurred on 9 January 2018, with a cumulative precipitation of less 13.7 mm (Figure 8a).It can be seen that the rainfall threshold of debris flows in the burned area decreased obviously after the Thomas Fire.
triggering factor of post-fire debris flows.The influence of mountain fire makes the threshold of rainfall decrease obviously compared with ordinary debris flow.The debris can begin to slide after a rain of 8 mm with a time interval of only 30 min [23].In the current study, Montecito's local rainfall data of a three-hour interval were obtained through the World Weather Online website [38].The first measurable rain for the area around Montecito fell on 8 January 2018.Ash and loose sediment generated by forest fires formed the source of debris flow development.A heavy rainfall process occurred at about 3:00 on the morning of 9 January 2018 (Figure 8a), causing mud and boulders from the gullies to flow along several drainage basins into Montecito.The comparison of rainfall data during the same period in 2018 with 2017 is shown in Figure 8.It is observed that no debris flow occurred on 9 January 2017, with a cumulative precipitation of 16.5 mm (Figure 8b), while the debris flow occurred on 9 January 2018, with a cumulative precipitation of less 13.7 mm (Figure 8a).It can be seen that the rainfall threshold of debris flows in the burned area decreased obviously after the Thomas Fire.

Model Construction of Post-Fire Debris Flow Occurrence Probability
In the current study, burn severity, HI, and short-duration rainfall in the Montecito post-fire debris flow disaster were discussed to illustrate their relationships with post-fire debris flows.To build a good model between them, we needed more data to support it.The watersheds affected by the Thomas Fire were detected using an SWIR false-color image of Sentinel-2A data on 12 January 2018 (Figure 9).Firstly, 380 rainfall records of 63 watersheds at different times were obtained.Remote sensing images of Google Earth at different times after the Thomas Fire were used to

Model Construction of Post-Fire Debris Flow Occurrence Probability
In the current study, burn severity, HI, and short-duration rainfall in the Montecito post-fire debris flow disaster were discussed to illustrate their relationships with post-fire debris flows.To build a good model between them, we needed more data to support it.The watersheds affected by the Thomas Fire were detected using an SWIR false-color image of Sentinel-2A data on 12 January 2018 (Figure 9).Firstly, 380 rainfall records of 63 watersheds at different times were obtained.Remote sensing images of Google Earth at different times after the Thomas Fire were used to determine whether or not the post-fire debris flows occurred.We took the scouring and accumulation characteristic of debris flows at the outlet of the main river course as the criterion for the occurrence of debris flow.Debris flows that did not produce significant deposits at the basin outlet were not included.1 and 0 were used to express the occurrence and non-occurrence and were input into the logistic regression model.As shown in the figure below, 30 of the watersheds used to construct the model can be identified traces of debris flows in the main channel.determine whether or not the post-fire debris flows occurred.We took the scouring and accumulation characteristic of debris flows at the outlet of the main river course as the criterion for the occurrence of debris flow.Debris flows that did not produce significant deposits at the basin outlet were not included.1 and 0 were used to express the occurrence and non-occurrence and were input into the logistic regression model.As shown in the figure below, 30 of the watersheds used to construct the model can be identified traces of debris flows in the main channel.To construct a model between the debris flow occurring and the factors, including burn severity, HI, and short-duration (3-h interval) rainfall data, rainfall data were obtained through the World Weather Online website [38].The burn severity image was calculated using the Landsat 8 data of 23 November 2017 and 26 January 2018 (Figure 10).The percentages of burn severity, including low-severity burn and high and moderate severity burn, of each watershed were calculated.The HI factors could also be calculated using DEM.Taking the burn severity, HI, and rainfall as the independent variable, and the debris flow occurring as a dependent variable, by using the logistic regression model, the prediction model of the post-fire debris flow occurrence probability was constructed according to Equation ( 9), as To construct a model between the debris flow occurring and the factors, including burn severity, HI, and short-duration (3-h interval) rainfall data, rainfall data were obtained through the World Weather Online website [38].The burn severity image was calculated using the Landsat 8 data of 23 November 2017 and 26 January 2018 (Figure 10).The percentages of burn severity, including low-severity burn and high and moderate severity burn, of each watershed were calculated.The HI factors could also be calculated using DEM.
determine whether or not the post-fire debris flows occurred.We took the scouring and accumulation characteristic of debris flows at the outlet of the main river course as the criterion for the occurrence of debris flow.Debris flows that did not produce significant deposits at the basin outlet were not included.1 and 0 were used to express the occurrence and non-occurrence and were input into the logistic regression model.As shown in the figure below, 30 of the watersheds used to construct the model can be identified traces of debris flows in the main channel.To construct a model between the debris flow occurring and the factors, including burn severity, HI, and short-duration (3-h interval) rainfall data, rainfall data were obtained through the World Weather Online website [38].The burn severity image was calculated using the Landsat 8 data of 23 November 2017 and 26 January 2018 (Figure 10).The percentages of burn severity, including low-severity burn and high and moderate severity burn, of each watershed were calculated.The HI factors could also be calculated using DEM.Taking the burn severity, HI, and rainfall as the independent variable, and the debris flow occurring as a dependent variable, by using the logistic regression model, the prediction model of the post-fire debris flow occurrence probability was constructed according to Equation ( 9), as Taking the burn severity, HI, and rainfall as the independent variable, and the debris flow occurring as a dependent variable, by using the logistic regression model, the prediction model of the post-fire debris flow occurrence probability was constructed according to Equation ( 9), as shown in Equation (11).The fitting parameter, R 2 , of the model was 0.887, which indicates a good fitting degree.Z = −22.563+ 20.263(HI) + 0.554(PR) + 8.6(%Low-severity burn) + 7.404(%High and moderate severity burn), (10) In the above equations, P represents the probability of the occurrence of post-fire debris flows.Z is the sum of linear weights after superposition of variables.The probability of debris-flow occurrence (P) of an individual basin can be estimated as a function of: -Hypsometric integral value of each basin (HI); -total three-hour precipitation (PR, mm); -percent of the low-severity burned area in each basin (%Low-severity burn); and percent of the high and moderate severity burned area in each basin (%Low-severity burn).
A receiver operating characteristic (ROC) curveis a graphical plot that illustrates the diagnostic ability of a binary classification system as its discrimination threshold is varied.The area under the curve is defined as the AUC (area under the curve, with a numeric range between 0.5 and 1).The classifier with a larger AUC value would be much better [65].In this research, 22 new watersheds (Figure 11a), which were different from the ones for conducting the model, were used to generate the ROC curve.Using the "ROC curve" tool in the SPSS software, taking the P value calculated using the above model as the "Test Variable", the actual occurrence of the debris flow as the "State Variable" ("Value of State Variable" is 1), the ROC curve could be drawn, and the AUC value was calculated automatically (Figure 11b).Fressard et al. defined five classes of accuracy using the parameter, AUC: Excellent accuracy for AUC values from 0.9 to 1.0; good accuracy for AUC values from 0.8 to 0.9; fair accuracy for AUC values from 0.7 to 0.8; poor accuracy, for AUC values from 0.6 to 0.7; and fail accuracy, for AUC values from 0.5 to 0.6 [66].In this paper, the AUC value was 0.876, which indicates a good accuracy of the prediction ability of the logistic regression model.We used 0.5 as the threshold of probability of the logistic regression model to judge the occurrence of debris flows.For the prediction of debris flow occurrence probability in the 22 watersheds, only one basin showed a negative result.The classification accuracy was 95%, which is a satisfactory result.
ISPRS Int.J. Geo-Inf.2019, 8 FOR PEER REVIEW 12 shown in Equation (11).The fitting parameter, R In the above equations, P represents the probability of the occurrence of post-fire debris flows.Z is the sum of linear weights after superposition of variables.The probability of debris-flow occurrence (P) of an individual basin can be estimated as a function of: -Hypsometric integral value of each basin (HI); -total three-hour precipitation (PR, mm); -percent of the low-severity burned area in each basin (%Low-severity burn); and -percent of the high and moderate severity burned area in each basin (%Low-severity burn).
A receiver operating characteristic (ROC) curveis a graphical plot that illustrates the diagnostic ability of a binary classification system as its discrimination threshold is varied.The area under the curve is defined as the AUC (area under the curve, with a numeric range between 0.5 and 1).The classifier with a larger AUC value would be much better [65].In this research, 22 new watersheds (Figure 11a), which were different from the ones for conducting the model, were used to generate the ROC curve.Using the "ROC curve" tool in the SPSS software, taking the P value calculated using the above model as the "Test Variable", the actual occurrence of the debris flow as the "State Variable" ("Value of State Variable" is 1), the ROC curve could be drawn, and the AUC value was calculated automatically (Figure 11b).Fressard et al. defined five classes of accuracy using the parameter, AUC: Excellent accuracy for AUC values from 0.9 to 1.0; good accuracy for AUC values from 0.8 to 0.9; fair accuracy for AUC values from 0.7 to 0.8; poor accuracy, for AUC values from 0.6 to 0.7; and fail accuracy, for AUC values from 0.5 to 0.6 [66].In this paper, the AUC value was 0.876, which indicates a good accuracy of the prediction ability of the logistic regression model.We used 0.5 as the threshold of probability of the logistic regression model to judge the occurrence of debris flows.For the prediction of debris flow occurrence probability in the 22 watersheds, only one basin showed a negative result.The classification accuracy was 95%, which is a satisfactory result.

Model Application around Montecito
The occurrence probabilities of post-fire debris flows in different watersheds around Montecito at a different time were calculated (Figures 12 and 13) by putting burn severity, HI, and rainfall into the model (Equations ( 10) and ( 11)).The probability change of post-fire debris flows due to the change of rainfall could reflect the danger change of post-fire debris flows in a long time.
The probability of debris flow occurrence in five watersheds varied differently with the rainfall process, as shown in Figures 12 and 13.It is observed that HI, short-duration heavy rainfall, and burn severity played different roles in this process.
(1) The occurrence probability of post-fire debris flow of watershed 1 increased first at the time of 21:00 on January 8.At the same time, the occurrence probability of post-fire debris flows of watershed 2 was lower than watershed 1, even with a higher burn severity.This may be attributed to a relatively lower HI value in watershed 2. A high value of HI indicates a steep terrain, which could enhance debris initiation.In these five watersheds, it was found that both watershed 2 and 4 also had a low probability of debris flow because of their small HI values.
(2) For a single basin, with the increase of rainfall, the probability of post-fire debris flow increased.All five basins had a peak of occurrence probability when the maximum rainfall occurred during the rainfall process.They faced the greatest threat when the rainfall reached its peak at 6:00 am on January 9 when rainfall played a dominant effect (Figure 12).
(3) The influence of the burn severity for the post-fire debris flows could be reflected by comparing watershed 1 with 3. The two watersheds experienced the same rainfall process and had similar HI values.However, watershed 3 was calculated with a higher probability of debris flow occurrence than watershed 1.This was mainly attributed to the different burn severity of the fire.According to Formula (10), it was found that the percent of the low-severity burned area and high and moderate severity burned area in each basin had a similar coefficient (8.6 compared with 7.4) in the equation.This means a similar effect when using them as the input to calculate the P value-the occurrence probability of post-fire debris flows.The reason why watershed 3 had a higher probability of debris flows was that it had a higher proportion of fire burn (89.8%) than watershed 1 (82.7%).A higher proportion of fire burn, including low-severity burn, high, and moderate severity burn, meant more material sources for the debris flows.
ISPRS Int.J. Geo-Inf.2019, 8 FOR PEER REVIEW 13 The occurrence probabilities of post-fire debris flows in different watersheds around Montecito at a different time were calculated (Figures 12 and 13) by putting burn severity, HI, and rainfall into the model (Equations ( 10) and ( 11)).The probability change of post-fire debris flows due to the change of rainfall could reflect the danger change of post-fire debris flows in a long time.
The probability of debris flow occurrence in five watersheds varied differently with the rainfall process, as shown in Figures 12 and 13.It is observed that HI, short-duration heavy rainfall, and burn severity played different roles in this process.
(1) The occurrence probability of post-fire debris flow of watershed 1 increased first at the time of 21:00 on January 8.At the same time, the occurrence probability of post-fire debris flows of watershed 2 was lower than watershed 1, even with a higher burn severity.This may be attributed to a relatively lower HI value in watershed 2. A high value of HI indicates a steep terrain, which could enhance debris initiation.In these five watersheds, it was found that both watershed 2 and 4 also had a low probability of debris flow because of their small HI values.
(2) For a single basin, with the increase of rainfall, the probability of post-fire debris flow increased.All five basins had a peak of occurrence probability when the maximum rainfall occurred during the rainfall process.They faced the greatest threat when the rainfall reached its peak at 6:00 am on January 9 when rainfall played a dominant effect (Figure 12).
(3) The influence of the burn severity for the post-fire debris flows could be reflected by comparing watershed 1 with 3. The two watersheds experienced the same rainfall process and had similar HI values.However, watershed 3 was calculated with a higher probability of debris flow occurrence than watershed 1.This was mainly attributed to the different burn severity of the fire.According to Formula (10), it was found that the percent of the low-severity burned area and high and moderate severity burned area in each basin had a similar coefficient (8.6 compared with 7.4) in the equation.This means a similar effect when using them as the input to calculate the P value-the occurrence probability of post-fire debris flows.The reason why watershed 3 had a higher probability of debris flows was that it had a higher proportion of fire burn (89.8%) than watershed 1 (82.7%).A higher proportion of fire burn, including low-severity burn, high, and moderate severity burn, meant more material sources for the debris flows.

Discussion and Conclusions
As a special type of geological disasters, post-fire debris flow is significantly different from an ordinary debris flow.In particular, the threshold of rainfall required for the occurrence of post-fire debris flows is reduced because mountain fires produce a large number of material sources.When studying the probability of post-fire debris flow by constructing a model, the factors selected by different scholars are different.Determining the key factors and reducing factor quantities are not only conducive to reducing uncertainty, but also can improve the accuracy of the forecast model for post-fire debris flow.The burn severity, HI, and rainfall factors selected in this paper play an important role in the formation of post-fire debris flow.The characteristics of burn severity, HI, and rainfall factors in the post-fire debris flows of watersheds around Montecito were first analyzed.On a larger scale, 380 rainfall records corresponding to 380 events' records of 63 watersheds at different times were obtained, and a prediction model of post-fire debris flow was made by using a logistic regression model to represent the occurrence probability of debris flows in each watershed.The application of the model used for the five watersheds of Montecito had good results and also reflected the change of occurrence probability of post-fire debris flows around Montecito during a rainfall process.The change of occurrence probability due to changes in rainfall is very help for an early warning of post-fire debris flows.
On the other hand, one of the advantages of this model is that it is easy to use and practical.Input parameters are easy to obtain from the public source data.The HI value of each basin can be obtained before mountain fires.After the fire, the severity of the fire can be obtained by using remote sensing images quickly.Based on the weather forecast, the future rainfall can be known, and then by inputting the above variables into the model, the occurrence probability of future post-fire debris flows after fires can be obtained.It is helpful for disaster prevention and reduction.Nevertheless, the model is based on the mountain fire area in California, and the practicability of other regions needs further verification and the model parameters may need to be adjusted.

Figure 1 .
Figure 1.Disaster photos in which the areas affected by the debris flows are circled with yellow lines.(a) Homes and streets of a neighborhood affected by the Santa Barbara County mudslides in Santa Barbara, California on 9 January 2018, from the perspective of a Coast Guard MH-65 Dolphin helicopter crew involved in rescuing injured and stranded victims.(Modified from U.S. CoastGuard photo[30]); (b) San Ysidro Ranch and Casa de Maria damage taken on 11 January 2018 due to the catastrophic debris flow from heavy rains after the Thomas Fire, modified from[31].

Figure 1 .
Figure 1.Disaster photos in which the areas affected by the debris flows are circled with yellow lines.(a) Homes and streets of a neighborhood affected by the Santa Barbara County mudslides in Santa Barbara, California on 9 January 2018, from the perspective of a Coast Guard MH-65 Dolphin helicopter crew involved in rescuing injured and stranded victims.(Modified from U.S. Coast Guard photo [30]); (b) San Ysidro Ranch and Casa de Maria damage taken on 11 January 2018 due to the catastrophic debris flow from heavy rains after the Thomas Fire, modified from [31].
, and others.In this study, Landsat 8 data of 23 November 2017, Landsat 8 data of 9 December 2017, Sentinel-2A data of 13 December 2017, Landsat 7 data of 17 December 2017, Landsat 8 data of 25 December 2017, Sentinel-2A data of 12 January 2018, and Landsat 8 data of 26 January 2018 were downloaded to analyse the fire.The Shuttle Radar Topography Mission (SRTM) Digital Elevation Model (DEM) data with a 30-m resolution from United States Geological Survey (USGS) was used for watershed extraction.Rainfall data were obtained through the World Weather Online website[38].
, and others.In this study, Landsat 8 data of 23 November 2017, Landsat 8 data of 9 December 2017, Sentinel-2A data of 13 December 2017, Landsat 7 data of 17 December 2017, Landsat 8 data of 25 December 2017, Sentinel-2A data of 12 January 2018, and Landsat 8 data of 26 January 2018 were downloaded to analyse the fire.The Shuttle Radar Topography Mission (SRTM) Digital Elevation Model (DEM) data with a 30-m resolution from United States Geological Survey (USGS) was used for watershed extraction.Rainfall data were obtained through the World Weather Online website [38].

Figure 4 .
Figure 4. Technical route of this study.

1 .
The Spread of Mountain Fires and Burn Severity around Montecito Landsat 8 data on 9 December 2017 (Figure

Figure 4 .
Figure 4. Technical route of this study.

4 .
Data Interpretation and Model Construction 4.1.Data Interpretation 4.1.1.The Spread of Mountain Fires and Burn Severity around Montecito Landsat 8 data on 9 December 2017 (Figure

Figure 7 .
Figure 7. Hypsometric integral curve.The black line represents the three stages of the development of the watershed geomorphology.

4. 1
.3.Rainfall Post-fire debris flows generally are triggered by one of two processes: Surface erosion caused by rainfall-induced runoff, and slope failure caused by rainfall infiltration.Runoff-dominated processes are by far the most prevalent because fires commonly reduce the infiltration capacity of soils, which increases runoff and erosion [64].Rainfall conditions are often regarded as the

Figure 7 .
Figure 7. Hypsometric integral curve.The black line represents the three stages of the development of the watershed geomorphology.

Figure 7 .
Figure 7. Hypsometric integral curve.The black line represents the three stages of the development of the watershed geomorphology.

Figure 8 .
Figure 8. Rainfall in Montecito during the period from (a) 7 January to 8 January 2018 with a cumulative precipitation of less 13.7 mm; (b) 7 January to 8 January 2017 with a cumulative precipitation of 16.5 mm [38].

Figure 8 .
Figure 8. Rainfall in Montecito during the period from (a) 7 January to 8 January 2018 with a cumulative precipitation of less 13.7 mm; (b) 7 January to 8 January 2017 with a cumulative precipitation of 16.5 mm [38].

Figure 9 .
Figure 9. Watersheds selected to establish the prediction model to analyze the occurrence probability of post-fire debris flows.

Figure 10 .
Figure 10.Burn severity affected by the Thomas Fire.

Figure 9 .
Figure 9. Watersheds selected to establish the prediction model to analyze the occurrence probability of post-fire debris flows.

Figure 9 .
Figure 9. Watersheds selected to establish the prediction model to analyze the occurrence probability of post-fire debris flows.

Figure 10 .
Figure 10.Burn severity affected by the Thomas Fire.

Figure 10 .
Figure 10.Burn severity affected by the Thomas Fire.

Figure 11 .
Figure 11.Model accuracy verification.(a) Watersheds used to be verified; (b) Receiver operating characteristic (ROC) curve and area under the curve (AUC) value.

Figure 11 .
Figure 11.Model accuracy verification.(a) Watersheds used to be verified; (b) Receiver operating characteristic (ROC) curve and area under the curve (AUC) value.

Figure 12 .
Figure 12.Change of post-fire debris flow occurrence probability of each watershed around Montecito along with rainfalls at different times.

Figure 12 .
Figure 12.Change of post-fire debris flow occurrence probability of each watershed around Montecito along with rainfalls at different times.

Table 1 .
Watershed area and accumulation area of different debris flows.

Table 1 .
Watershed area and accumulation area of different debris flows.
2, of the model was 0.887, which indicates a good fitting degree.