## 2 Materials and Methods  
The study site is a representative area of the type of failure to be analyzed. Along the coast of Ecuador, in such places as Montanita, Olon, and Ayangue, we find coastal cliffs with stability problems linked to the erosion of the waves and tides at the foot of the cliffs. To adequately analyze the failure mechanism, it is necessary to understand that many landslides are produced through stratification with tensile cracks in vertical slopes, but that the classic models do not fit reality well enough because they consider the slope in the form of a two-dimensional section. This presents a real problem in three dimensions. Conducting MFS slope studies using 3D models sometimes requires computer programs that are not yet accessible to many researchers. The methodology proposed in this work is accessible since it utilizes stereographic projection and reduces the problem to one less dimension. To show the effectiveness of the methodology, we have chosen a test site where the typology is easily identified and represents the difficult approach of [PERSON] et al. [5] in a very didactic way.  
The analysis of interactions between discontinuity planes and slopes is extremely complex to solve using 3D models and isometric projection but is considerably simplified with stereographic projection [1]. We refer the reader to the bibliography for information on stereographic and kinematic assessments of rock slopes, especially using the MFS approach as detailed in [PERSON] et al. [5] and [PERSON] et al. [1]. The main idea of the methodology is to use the usual criteria in kinematic analysis for locating poles or planes in some areas of the stereogram that meet the instability conditions. Any pole or crossing of planes that fall in this zone implies kinematic instability between these fractures and the slope. What the MFS methodology does is expand the area known as the \"restricted daylight envelope\" by considering the slope not as a single plane but as a curved surface that can be simplified as an envelope that spans from one tangent plane to another passing through the tip or \"nose\" of the concave slope.  
The back analysis methodology has been applied in the following way: the model must be equivalent to reality, and when a slope has failed, the model must indicate a safety factor \(\leq 1\). Normally, we assign joint strength parameters according to various approaches, whether empirical, laboratory, or a combination of both. In this case, we have reconstructed what slopes and detached blocks looked like in their initial situation before the failure. If the analysis or calculation indicates a safety factor greater than unity (or the stable zone of the stereogram), then that means that our calculation is not realistic, and we therefore have to modify the resistance parameters of cohesion or friction until the result comes out to a safety factor of 1, or within the limit of the unstable zone of the stereogram.  
The discontinuity friction angle is necessary to draw the friction circle; indeed, it is required in all kinematic analyses via stereographic projection. Friction (\(\varphi\)) is obtained from the Barton-Bandis failure criterion and onsite mapping [13; 14; 15].