Quantitative effects of antecedent effective rainfall on ID threshold for debris flow

8 Studies have shown that the antecedent effect precipitation ( AEP ) is closely related to rainfall 9 intensity-duration (ID) threshold of debris flow. However, the quantitative relationship between 10 the AEP and ID threshold is still undetermined. In this study, a hydrological process based 11 numerical model (Dens-ID) that can derive the ID threshold curve is adopted to address this issue. 12 (JJG) in District of was chosen as the study area, 13 Dens-ID was used to derive a series of ID threshold curves corresponding to different AEP . Based 14 on calculated data sets including AEP , ID curves, parameters of ID curve equation ( α and β ), and 15 debris flow density, the influence of AEP on the ID threshold curve is deeply explored. We found 16 that although solid materials and runoff are the two necessary conditions for the formation of 17 debris flow, the specific roles played in which are different: the volume of loose solid sources 18 provides a basal condition for debris flow and determines the scale of debris flow, while the runoff 19 volume will have a sudden change during the rainfall process, which is a key factor promoting the 20 formation of debris flow. In the condition of AEP ranging from 20 mm to 90 mm, AEP and α can 21 2 be described by the equation α =-0.0078 AEP 2 +0.68 AEP +6.43, and β shows a linear change law 22 with AEP . The error of the two equations were evaluated using 45 historical rainfall data that 23 triggered debris flows, which is equal to 37.85% and 11.1%. Due to the two functions, the ID 24 threshold curve can regularly move in the I-D coordinate system rather than a conventional 25 threshold curve stay the same regardless of AEP variation, it is beneficial to improve the 26 prediction capacity of the ID threshold. functions that were verified using the ID curves fitted using historical rainfall data for JJG. The 509 of the relationships between AEP and α and β is, Dens-ID of AEP α and β compared to those 511 indicated by historical rainfall data. can be attributed to limitations on the ability of 512 Dens-ID to describe debris flow formation, the uncertainty of the input parameters of Dens-ID, 513 and the suitability of rain gauge data for detecting rainstorm centers.


Introduction 28
Precipitation that affects debris flow formation includes triggering rainfall and antecedent 29 effective precipitation (AEP) before the event (Chen et al., 2015;Chen et al., 2018;Oorthuis et al., 30 2021). AEP is precipitation that remains in soil before a debris flow occurs; it reflects the degree 31 of soil saturation (Zhang et al., 2015). Increased AEP, and thus increased moisture content, has 32 been shown to enhance surface rainfall-induced runoff in various environments (Tisdall, 1951; which is referred to as a rainfall parameter set. Each data point [Ii, Di] corresponds to a unique 158 value of ρmix within the density set; thus, the correlation between the rainfall parameters and debris 159 flow density can then be established by Dens-ID. An ID curve can then be fitted through the 160 collected [Ii, Di] data to show the relationship between I and D. Each fitted ID curve corresponds 161 to a unique ρmix within the density set, which is also considered to be the isodensity line (Zhang et 162 al., 2020). Two values close to the left and right boundaries are chosen from the density set as ρmix, 163 and the ID threshold curve corresponding to these two density values can represent the lower and 164 upper boundaries for debris flow formation. The ID curves corresponding to a density value ρmix 165 are fitted as follows: 166 Step 1: Assign values of 1.2 and 2.2 g/cm 3 to . 167 Step 2: Assign a value to the AEP. In nature, the AEP represents the antecedent soil moisture 168 before the rainfall process that may trigger a debris flow. In Dens-ID, the natural debris flow gully 169 is divided into a series of grid cells, and the AEP represents the soil moisture content of each grid 170 cell before rainfall infiltration. Using the initial hydrological conditions represented by the AEP, 171 Dens-ID simulates hydrological processes such as runoff and infiltration during the triggering 172 https://doi.org/10.5194/hess-2022-57 Preprint. Discussion started: 16 February 2022 c Author(s) 2022. CC BY 4.0 License.
precipitation process. To quantitatively analyze the effect of AEP on the ID threshold curve, AEPi 173 was assigned values of 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, and 120 mm. 174 Step 3: Assign a value to Ii, which generally represents the average rainfall intensity of a 175 rainfall process that can trigger a debris flow and is held constant until the calculations in Step 4 176 are complete. The initial value of Ii is set to 1 mm/h. When Step 4 is complete, Ii is increased by 177 0.5 up to Imax. At Imax, a debris flow with density ρmix can be triggered in the gully when D = 1. 178 Step 4: Under constant Ii, the calculation time of the model starts at t = 1 h and increases by 1 179 h at each calculation step until t = Di, where Di represents the rainfall duration required to trigger a 180 debris flow with density ρmix. After t = Di, the model calculation for a given Ii is complete. 181 Step 5: Repeat Steps 3 and 4 and collect the Ii and Di values at which Dens-ID outputs the 182 pre-set ρmix. When the rainfall intensity Ii increases to Imax, the calculation for a given AEPi is 183 complete. Thus, the data set of Ii and Di for a certain AEPi is obtained, and the corresponding ID 184 threshold curve can be fitted using these data. 185 Step 6: Repeat Steps 2, 3, 4, and 5, and collect the Ii and Di values. When AEP reaches 120 186 mm, the calculation for a given ρmix is complete. Station. The spatial resolution of the DEM is 0.5 m, and the data were obtained in December 2017 219 by aerial photogrammetry using an unmanned aerial vehicle. A DEM with a grid size of 10 m was 220 generated from the original terrain data using the resampling tools in ArcGIS, which were used to 221 derive the geometrical parameters of JJG such as slope length, gradient, and river channels. 222 ◆ Data necessary for hydrological simulation 223 Three main soil types (Table 1) occur in the JJG: dry red soil, red-yellow soil, and gravelly 224 soil. Gravelly soil is widely distributed upstream in JJG and is the main source of solid material 225 for debris flow. The hydrological parameters listed in Table 1 were obtained from the National 226 Soil Database. The grid size of the land use map is 250 m, and its parameters, such as the 227 normalized difference vegetation index, were obtained from the Moderate Resolution Imaging 228 Spectroradiometer database. These data related to hydrological parameters were converted into a 229 map with an accuracy comparable to that of the DEM using the resampling tool in ArcGIS. 230 ◆ Soil mechanical parameters 232 Eq. 7 (section 4.3) can be used to determine two soil mechanical parameters, soil cohesion c 233 and internal friction angle φ, by direct shear tests of soil samples from JJG. Most of the solid 234 material for debris flows in JJG originates from gravelly soil; therefore, three groups of soil 235 samples were taken from several typical slopes covered by a gravelly soil mass, and one sample 236 each was taken from the red-yellow and dry red soil. As shown in Table 2 Table 2 can be assigned to each grid cell of the DEM according to the 240 distribution of soil types in JJG. 241 Table 2 Cohesion c and internal friction angle φ of soil samples from JJG 242

Soil samples
Soil mechanical parameter To validate the quantitative relationship between the AEP and the ID threshold curves of 244 debris flows, data for 45 debris flow events in JJG and the triggering rainfall processes were 245 collected. Rainfall events must be separated from long-term rainfall sequences to identify the 246 rainfall processes that triggered the 45 debris flow events. The inter-event time (IET) was defined 247 as a measure of the minimum time interval between two consecutive rainfall pulses (Adams et al., The AEP was calculated as the weighted sum of rainfall periods before a debris flow (Long et 255 al., 2020) and is expressed as follows: 256 where the AEP is the antecedent effective rainfall; K is the attenuation coefficient, which is equal 258 to 0.78 according to a field test in JJG (Cui et al., 2003); and n is the number of days preceding the 259 debris flow. Table 3 lists the calculated AEP, average rainfall intensity (I), and rainfall duration (D) 260 of each debris event. The calculated AEP values in the third column of Table 3      In addition, Fig. 4 shows that the distance between the two ID threshold curves becomes 291 larger with increasing AEP, indicating a higher occurrence probability of rainfall-induced debris 292 flow. A database including all the data sets, including [I, D], the fitted curves, and AEP (Table 4) 293 was used to quantitatively analyze the effect of AEP on the threshold curve. 294 Note that Dens-ID cannot collect sufficient [Ii, Di] data for fitting the ID threshold curve for a 296 density of 2.2 g/cm 3 and AEP = 10 mm. At this low AEP value, the supply rate of solid material is 297 lower than the runoff rate; thus, it is difficult to trigger a high-density debris flow in JJG. By   cause the debris flow in JJG to quickly become hyperconcentrated. Therefore, the red dotted line 327 in Fig. 5 also shows that debris flows generally begin suddenly but quickly reach Stage 3 because 328 of the rapid increase in runoff. 329 The black dashed line in Fig. 5 represents the variation of . The hydrological conditions 330 represented by AEP = 20 mm induce shallow landslides in JJG before rainfall begins. In the initial 331 stage of the rainfall process, the supply rate of solid material is higher than the runoff rate in JJG; 332 however, as the rainfall process continues, the supply rate is overtaken by the runoff rate, and the 333 total volume stabilizes at a maximum value.  The parameters of the ID threshold curve of debris flow, and β, determine the position of 361 the fitting curve in I-D coordinates. Therefore, it can be deduced that α and β depend on AEP. In 362 this section, the data sets from Dens-ID are used to derive the functional relationships between 363 AEP and these two parameters. First, it is necessary to clarify the physical meaning of α and β. 364 Under the numerical simulation conditions of this study, the variation interval of the independent 365 variable D in the formula I = αD β is [1, Dmax], and the variation interval of I is [Imax,1]. According 366 to the formula, when D is equal to 1 h, I = α. When D = 1, the rainfall duration required to trigger 367 a debris flow is 1 h, and the rainfall intensity I reaches the maximum value, Imax. Therefore, the 368 combination of D and I under these conditions represents high-intensity rainfall. According to this 369 analysis, is numerically equal to the value of Imax, and thus this parameter represents the critical 370 rainfall intensity required to trigger a debris flow for D = 1 h. 371 Before the physical meaning of β is discussed, the expression I = αD β needs to be written 372 logarithmically, as follows: 373 By denoting logI as YI, logD as XD, and logα as Bα, Eq. 7 can be rewritten as follows: intensity with increasing rainfall duration, that is, the rate of decrease from Imax to 1 mm/h. The α 382 and β values in Table 4 can be classified into two groups according to debris flow density (1.2 or 383 2.2 g/cm 3 ). The and values in the two groups show similar variation with AEP. Thus, one data 384 group (Table 5) corresponding to a density of 2.2 g/cm 3 was selected to examine the effect of AEP 385 on α and β. 386 Effect of AEP on : The effect of AEP on α is described by the following equations, which 388 were fitted using the AEP and α values in Table 5: 389 The condition for α = Imax is D = 1, and the combination of D = 1 and represents a high-391 intensity, short-duration rainfall process. As shown in Fig. 8, Eq. 9 is used to quantify the rainfall 392 intensity threshold at which this type of rainfall process triggers a debris flow for different AEP 393 https://doi.org/10.5194/hess-2022-57 Preprint. Discussion started: 16 February 2022 c Author(s) 2022. CC BY 4.0 License. values. In Fig. 8, α (or Imax) represents parabolic variation with AEP. Interestingly, α does not 394 always decrease with continuously increasing AEP. When AEP ≤ 50 mm, the α values necessary 395 for triggering a debris flow increase simultaneously with AEP; when AEP > 50 mm, α decreases 396 with increasing AEP, but the decrease does not continue indefinitely with increasing AEP, because 397 for AEP > 90 mm, α is constant at 6.8 mm (Table 5). The key variables Vs and Vw are used to explain the quantitative evolution described by Eq. 9. 401 To facilitate the analysis, the Vs and α values calculated by Dens-ID were normalized, and they are 402 plotted versus AEP (AEP-Vs and AEP-α) in Fig. 9. Vs increases continuously for AEP < 50 mm, 403 at which it reaches a maximum. As Vs increases with increasing AEP, a larger volume value of 404 runoff (Vw) is required to bring the debris flow density (ρmix) to a fixed value of 2.2 or 1.2 g/cm 3 , 405 which requires stronger hydrodynamic conditions, and thus a higher hourly rainfall intensity. 406 Before point P1 in Fig. 9, the rainfall intensity (or α) at which a debris flow occurs for D = 1 is 407 positively correlated with AEP. Although AEP no longer contributes to the variation of Vs after 408 AEP reaches 50 mm, the soil water content can still increase with continuously increasing AEP, 409 reducing the surface infiltration rate and increasing the runoff volume generated from rainfall. Under these hydrological conditions, the rainfall intensity Imax (or α) required to trigger a debris 411 flow with a fixed density value decreases gradually; thus, α is negatively correlated with AEP. 412 When AEP exceeds 90 mm (P2 in Fig. 9), α stops gradually decreasing and remains constant, 413 indicating that at AEP = 90 mm, the loose solid material in JJG become saturated. Under these 414 hydrological conditions, α has a constant value of 6.8 mm and does not change with AEP. 415 Therefore, for the two inflection points P1 and P2 in Fig. 9, AEP is the external driving factor and 416

420
Effect of AEP on β: The effect of AEP on β is described by the following equations, which 421 were fitted using the AEP and β values in Table 5. 422

Validation of quantitative relationship 432
Using the historical rainfall data in Table 3

444
The curves fitted using historical rainfall data and Dens-ID for the same AEP were drawn in 445 separate graphs, where each graph corresponds to a different AEP value between 15 and 90 mm. 446 As shown in Table 2, only one debris flow event each was collected from the observation station 447 for AEP values of 60 and 90 mm. In Fig. 12(e) and (f), the single points at which the I and D data 448 in Table 3

477
As shown in the average error is approximately 11.10%. According to the physical meaning of α and β, the 481 error of Eq. 9 (approximately 37.85%) indicates that Dens-ID overestimates the triggering rainfall 482 intensity (Imax) for D = 1. Additionally, the calculated β values, which represent the deceleration 483 rate of rainfall intensity with increasing rainfall duration, have a smaller error than the α values. 484  is the main trigger for debris flow. The historical rainfall data in Table 3 were obtained at the 495 rainfall station represented by a red circle in Fig. 2, which is approximately 2 km from Menqian 496 Gully. Because of this spatial deviation, the rain gauge may be unable to detect the center of 497 rainstorms, and thus the measured rainfall data may be smaller than the actual values. 498

Conclusions 499
Rainfall simulations using Dens-ID were employed to construct a database of ID threshold 500 curves under different AEP conditions, and this database was used to thoroughly examine the 501 quantitative effect of AEP on the ID threshold curves. The following conclusions are drawn. 502 (1) The ID threshold curve obtained using Dens-ID can be expressed by a power function, 503 and the R 2 values of the fitted power functions are all larger than 96%. The fitted curves from our 504 model are all consistent in shape with the threshold curve obtained from the statistical model, 505 indicating that the model can reflect the hydrological process of rainfall-induced debris flow with 506 high reliability. 507 (2) The relationships between AEP and the parameters α and β can be described by 508 https://doi.org/10.5194/hess-2022-57 Preprint. Discussion started: 16 February 2022 c Author(s) 2022. CC BY 4.0 License. functions that were verified using the ID curves fitted using historical rainfall data for JJG. The 509 errors of the relationships between AEP and α and β are approximately 37.85% and 11.10%, 510 respectively. That is, Dens-ID overestimates the effects of AEP on α and β compared to those 511 indicated by historical rainfall data. This result can be attributed to limitations on the ability of 512 Dens-ID to describe debris flow formation, the uncertainty of the input parameters of Dens-ID, 513 and the suitability of rain gauge data for detecting rainstorm centers. 514 (3) The two derived equations can clarify the variation of debris flow ID curves with AEP. 515 The conventional ID threshold curve remains the same regardless of AEP once it is determined. 516 However, the AEP can significantly affect the determination of the ID curve. The effects of AEP 517 on α and β cause the originally static ID curve to become a variable threshold in the I-D 518 coordinate system. Consequently, the ID curves fully reflect the effects of AEP when they are used 519 to predict debris flow. Our study may improve the prediction precision of ID curves.