re 7. The conditional probability at three rainfall intensity ranges for each station and total (considering all landslide events and the higher precipitation value reported in the study region).
<0.01. The number is low, because most rain events in the region are <30 mm and very little movement arose during those days. For daily intensities of >30 to 60 mm, the probability is higher but still <0.1. In contrast, when the intensity is >60 mm, the probability is almost 0.5.
However, this one-dimension calculation omits the duration of the precipitation, a limitation overcome by a two-dimensional calculation.
3.2.2. Two Dimension
Bayes’ equation allows incorporation of the two variables of interest within the precipitation for the joint probability of triggering a MRP  . Other variables can be used in place of as previous rain; however, this will depend on prior interpretation to show a better explanatory power. Previous visual analysis of the data concerned duration and intensity of rain (Figure 8). Conventional thresholds are generally obtained in this way, establishing a lower limit of precipitation events that trigger movement, or by statistical formulas that result in a linear trend   . The threshold lower envelope proposed by Jibson as 
where I is the intensity (mm/day) and D is the duration (days) defines a line below which the values do not trigger any MRP (Figure 9). However, a global threshold may have more uncertainty. A global threshold proposed by Caine as 
Figure 8. Analysis of two-dimensional Log-I, Log-D. Puebla, proposed for the Sierra Norte de Puebla visually designed threshold, lower envelope (Equation (2)) and world global threshold (Equation (3)).
Figure 9. Probability calculation for intensity and duration of rain. The vertical axis indicates the probability calculated for each of the scenarios. The dotted line indicates the threshold that increases the likelihood of triggering movement in the Sierra Norte de Puebla. The white boxes indicate the lack of data for calculating probability and the red boxes the null probability of landslide occurrence.
clearly defines a zero probability of movement below the threshold but has the limitation of generating very general intensity values corresponding to <100 mm for one day. Therefore, it is proposed to raise the threshold for the Sierra Norte de Puebla, in order to approach the likelihood of triggering.
The equation (Equation (4)) was calculated for each of the stations with reference back to the joint base. There were nine ranks, 3 × 3: durations of 0 - 10 days, >10 to 20 days and >20 days; and intensity of 0 - 30 mm, >30 to 60 mm and >60 mm, resulting in nine combinations for conditional calculation (Table 1).
P(A) is the a priori probability; the number of MRPs for total rainfall events.
P(B,C) is the marginal probability; the number of rain events in the first intensity range (0 to 30 mm/day, corresponding to the first quadrant) and duration of 1 to 10 days by total rainfall events during that period.
P(B,C|A) is the conditional probability in 1 to 10 days by the total number of movements recorded in that period.
P(A|B,C) is the posterior or conditional probability; the number of registered movements when a rain event had an intensity of 0 - 30 mm and duration of 1 to 10 days, the total rainfall events of such intensity and duration that occurred in the period (Table 1).
The Bayes probability is calculated for each conditional scenario (Figure 9),
Table 1. Two dimensional probability matrixes for the Sierra Norte de Puebla. B, C indicate the joint probability of having a certain value (or range of values) of the two variables; the rainfall magnitude and the rainfall duration. The P(B,C|A) is the conditional probability of B given A and C, in this case the probability of having been a rainfall event of magnitude B and duration C when a landslide occurs; P(A) is the prior probability of A (or simply prior), in this case the probability that a landslide occurs regardless of whether a rainfall event of magnitude B and duration C occurs; P(B,C) is the marginal probability of B, in this case the probability of having been a rainfall event of magnitude B regardless of whether a landslide occurs; and P(A|B,C) is the conditional probability of A given B and C, in this case the probability of a landslide when a rainfall event of magnitude B and duration C occurs.
with higher probability of MRP occurrence when the rainfall exceeds 60 mm and from 1 to 10 days; in turn, the probability decreases with precipitation of 0 to 30 mm and from 1 to 10 days. Also, probabilities are nearly 0 for rainfall >30 mm at 20 to 30 days. This result could help decision makers when there is a high probability of precipitation of over >60 mm for >1 day. Also, a proposed new threshold is determined for the Sierra Norte de Puebla that best defines the probability of landslide according to the intensity and duration of rainfall.
The relative ease of the method allows incorporation of variables other than precipitation, such as soil type.
4. Warning System
The most important step in disaster risk management is prevention, so the present results should be incorporated in a warning scheme based on that proposed by Brunetti  . Five probability ranges are defined as guidance to the decision makers in the region (Figure 10).
First, in those regions of the sierra that increase in risk after a warning of a tropical cyclone, or in the rainiest months (particularly in La Niña years), or just with a forecast (numerical weather prediction, satellite images) of steady rain, the rainfall for the next 24, 48, 72 and 96 hours is calculated for each of the most significant 15 stations selected for their homogeneous distribution and reliability. According to the intensity thereby deduced for each time, the conditional probability is determined and classified into one of 5 categories: 1) well below the threshold, 0 < 0.029; 2) under the threshold, 0.0291 < 0.25; 3) on the threshold, 0.251 < 0.39; 4) above the threshold, 0.391 < 0.5; and 5) well above the threshold, 0.51 < 1.0. Each site is assigned a class number corresponding to a greater or lesser MRP risk. In those sites with values above or well above the
Figure 10. Example of landslide alert bulletin for the Sierra Norte de Puebla. Top left, location of 15 selected stations. Low left corner, classes of probability of a mass removal process. Top right, prediction of rain for the next few hours for each station selected; ↑ (increase), ↓ (decrease) and the correspondent assigned class of probability.
threshold, a higher risk of disaster is expected and an alert bulletin is issued to the appropriate authorities. This process can be performed automatically to work in real time with the information provided by the National Weather Service.
5. Discussion and Conclusions
The advantages of the Bayes method are as follows: application to almost any natural phenomenon in which a risk is involved; versatility to incorporate into a study the required variables; ability to develop appropriate thresholds at the scale required; and the relative ease of the method to integrate the uncertainty. Although the method has been applied in several countries with high reliability, in Mexico there have been no studies using this approach.
In this study, the two most important variables (intensity and duration of rain) in rainfall-related triggers of mass removal were considered in order to obtain a priori, marginal and largely conditional probabilities, since the uncertainty is crucial in the development of more suitable disaster prevention measures.
In general, the main limiting threshold factor is the obtaining of information and choice of variables. Inclusion of other relevant physical variables (slope, soil type, soil humidity, etc.) in probability calculation could enhance the study. However, the probability calculation is more complex because the additional information is often not available. Also, the scale can be seen as a limiting factor; although it would more accurate to determine a local threshold for each municipality, or regional one for the Sierra Norte and Sierra Nororiental, however there are no enough weather stations with reliable data especially in Puebla, despite their long history of disasters caused by rains. A better option may be to use precipitation data from remote sensing sources (satellite or radar).
Future studies could include more ranges of intensity and duration of precipitation and information about slopes, areas of population, vegetation, and soil mechanics inter alia, performing a multi-criteria evaluation.
The location of the Sierra Norte de Puebla among three physiographic provinces as well as its proximity to the Atlantic Ocean provided the conditions that increase the risk of MRPs and disaster for a highly vulnerable population. Extreme hydro-meteorological events during La Niña years intensify and prolong the precipitation; this is strongly correlated with MRP triggering so obtaining thresholds of these variables is of great importance. The Bayesian method allows conditional probability thresholds to predict future landslides according to the precipitation forecast, and these can be applied in warning systems in the region. The threshold classification can be modified to improve accuracy and thereby to prevent material and human losses.
This research was performed in 2012-2015 while the first author was preparing the undergraduate thesis at the Institute of Geography, UNAM, and was partially sponsored by the María Teresa Gutiérrez de MacGregor Fellowship. Dr Matteo Berti is gratefully acknowledged for making software available for checking our results and plotting Figure 9. Special thanks are owed to M. Sc. Luís Galvan for his help with GIS in producing some plots and slope calculation and M. Sc. Armenia Franco for precipitation data.
Cite this paper
González, A. and Caetano, E. (2017) Probabilistic Rainfall Thresholds for Landslide Episodes in the Sierra Norte De Puebla, Mexico. Natural Re- sources, 8, 254-267. https://doi.org/10.4236/nr.2017.83014