1 Introduction

Gravity-driven granular flows, such as debris flows or pyroclastic density currents, are extremely sensitive to topographic changes during transport (Fisher 1990, 1995; Branney and Kokelaar 2002; Sulpizio and Dellino 2008). Break in slopes, obstacles, and ravine deviations are among the main factors in changing flow trajectories and run-out distances. In hazard assessment, computer routines can successfully reproduce past events and provide a hazard zonation based on inundation limits of simulated flows over a natural terrain. At present, several computational routines (i.e., TITAN2D, LAHARZ, FLOW3D) (Kover 1995; Schilling 1998; Patra et al. 2005) are currently used to simulate gravity-driven flows over topographies obtained from stereo coverage of satellite imagery (e.g. ASTER—Advanced Spaceborne Thermal Emission and Reflection Radiometer), radar data (e.g. SRTM—Shuttle Radar Topography mapping Mission), laser altimetry (e.g. LIDAR—Light Detection and Ranging), and generic vector data (e.g., contour lines from topographic maps). Despite the variability of the input parameters required by each routine in order to perform a numeric simulation, a Digital Elevation Model (DEM) is a common input for simulation algorithms. Previous works already evidenced the importance of DEM spatial resolution on the accuracy of numeric simulations, for example, for the LAHARZ routine used to delineate lahar hazard inundation zones (Stevens et al. 2002; Hubbard et al. 2007; Davila et al. 2007).

The aim of our work is to quantify how DEM resolution and accuracy affect the simulation results. For this purpose, we use TITAN2D, a modeling algorithm developed by Patra et al. (2005) for simulating granular flows. In particular, we use TITAN2D to reproduce past block-and-ash flows (BAFs) at Colima volcano in Mexico. The Colima volcano is a prominent volcanic edifice with an altitude of 4,260 m a.s.l., located 100 km south from Guadalajara city and 30 km north from Colima city (Fig. 1). It is one of the most active volcanoes in the world nowadays, presenting Merapi and Soufriere dome type collapses, Vulcanian and Plinian explosive eruptions, and extrusions of lava flows and domes (Macias et al. 2006; Saucedo et al. 2005; Luhr et al. 2010).

Fig. 1
figure 1

ASTER image showing the Colima volcano and surrounding area. The inset shows a sketch map of the Trans-Mexican Volcanic Belt with the location of the CVC and other active volcanoes: 1 San Juan; 2 Ceboruco; 3 Tancitaro; 4 Jocotitlán; 5 Nevado de Toluca; 6 Popocatépetl; 7 Ixtaccíhuatl; 8 La Malinche; 9 Pico de Orizaba; 10 Cofre de Perote; 11 San Martín; 12 El Chichón

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TITAN2D routine has already been extensively applied to evaluated small-scale pyroclastic flows, BAFs and rock avalanches for several volcanoes, such as the Popocatépetl, Nevado de Toluca, Chichón and Colima in Mexico (Pitman et al. 2003; Bursik et al. 2005; Rupp et al. 2006; Capra et al. 2008; Macias et al. 2008; Murcia et al. 2010; Sulpizio et al. 2010), Merapi volcano in Indonesia (Charbonnier and Gertisser 2009), Mt. Taranaki, New Zealand (Procter et al. 2010), Little Tahoma Peak in the USA (Stinton et al. 2004; Sheridan et al. 2005), and Mount Etna in Italy (Norini et al. 2008). The outcomes of these studies have highlighted the uncertainties in objectively defining input parameters for realistic simulations of volcaniclastic flows. These studies focus mostly on the incertitude of those parameters that govern flow mobility, such as internal and basal friction angle (i.e., Charbonnier and Gertisser 2009; Procter et al. 2010). Recently, Sulpizio et al. (2010) have introduced constrains on the effect of DEM resolution over the basal friction angle (ϕb), one of the main parameters that determine the flow runout. The ϕb corresponds to the minimum slope over which a granular material starts to move from its static position. The generic approach is to simulate past flows using a range of basal friction angles which depends on the DEM resolution. In fact, for low-resolution DEMs, the slope is smoothed and granular flows travel farther downstream even for low basal friction angles. In this work, we study in a systematic way how different DEM resolutions influence the output of simulations reproducing past pyroclastic flows, especially where topography is characterized by sudden changes in slope or close curves in deep ravines. We also investigate the range of different basal friction angles that accurately reproduce the same flow event over DEMs with different resolutions.

2 Methodology

2.1 TITAN2D

TITAN2D code was originally designed to simulate dry granular flows from an initial source area of collapse over a natural terrain (Pitman et al. 2003; Patra et al. 2005), but it is also highly suited to simulating BAFs formed from the collapse of large portions of lava domes. The code is based on a model for an incompressible Coulomb flow adapted from the work of Savage and Hutter (1989) using a “shallow-water” depth-averaged approximation (Iverson and Denlinger 2001). Mass and momentum conservation equations are solved with a Coulomb friction term for the interface between the granular material and basal surface and also for the internal friction of the flowing media (Pitman et al. 2003). The resulting hyperbolic system of equations is solved using a parallel adaptive mesh (Patra et al. 2005). In this study, we used the 2.0.1. release to perform all the parallel numeric simulations. Terrain data are entered into the algorithm via GRASS GIS (Geographic Resources Analysis Support System).

The main input parameters for running simulations areas follows: (1) the volume of the collapsed mass; (2) the basal friction angle (φb); and (3) the internal frictional angle (φi). The initial conditions required by TITAN2D are the following: the coordinates of the initiation point, the elongation and orientation of the collapsed mass, the initial velocity and direction, and the duration and mass flux rate of the initial pile of material (Sheridan et al. 2005; Charbonnier and Gertisser 2009; Procter et al. 2010). After the simulated flow starts, a criterion for determining its stop is crucial for an accurate assessment of the runout, since in numerical simulations a flow never stops if its basal friction angle is lower than the slope angle. This is an unrealistic condition for real flows, for which other physical parameters concur in flow stopping (e.g., aspect ratio of the sliding pile, kinetic and potential energy of the moving flow, shape of the valley, or channel for confined flows). Yu et al. (2009) proposed a stopping criterion for simulated flows to determine their maximum runout. For this global stopping criterion, a dimensionless average velocity is defined as:

$$ V_{2}^{ * } = {\frac{{V_{\text{ave}} \alpha^{0.5} }}{{\sqrt {gH_{0} \tan \varphi_{\text{bed}}^{0.5} } }}} = 0.03 $$
(1)

where V ave is the simulated average flow velocity, α is the transient aspect ratio of the sliding pile, g is the gravity acceleration, H 0 is the initial pile height, and φb is the basal friction angle (Yu et al. 2009). Based on this relation, it is possible to determine the minimum acceptable average flow velocity (V ave) below which the flow stops. Yu et al. (2009) proposed an V 2 * average value of 0.03 which represents the best fit of constant values of V 2 * that a simulated flow reaches after a dimensionless time of t* = t/(2H 0/g)0.5. Sulpizio et al. (2010) have shown that the V 2 * value is highly sensible to the φb value used to perform the simulation and actually dominates the other variables in Eq. (1) such as the V ave and the H 0, suggesting to evaluate the average velocity for each individual simulation.

2.2 Digital elevation models

The DEMs we used for the simulations are obtained from a LiDAR coverage of the Colima volcano with a 5-m horizontal resolution (Davila et al. 2007). The original 5-m raster was resampled to obtain four new DEMs with coarser horizontal resolutions of 10, 30, 50, and 90 m, respectively, using a nearest neighbor algorithm. These DEMs were then exported in ASCI file format compatible with GRASS in order to obtain the file format required by TITAN2D. Coarser resolutions were chosen because they represent the most common available formats, such as the 30-m DEM from SRTM data for the USA (1 arc second) and 90-m (3 arc second) for the rest of the world, the 30-m DEM obtained from ASTER image, or the 50-m DEM used in Mexico interpolated from 1:50,000 scale topographic maps (with 10 or 20 m contour lines) by INEGI (Instituto Nacional de Estadistica, Geografía e Informática).

Figure 2 shows the main differences among several topographic profiles performed on a series of DEMs from LiDAR (5, 10, 30, 50, and 90 m) and from SRTM (90 m), ASTER (30 m), and INEGI (50 m) data. The SRTM 90-m DEM was obtained from the CGIAR Consortium for Spatial Information web portal (http://srtm.csi.cgiar.org/). The ASTER data were obtained from a 2007 ASTER image acquired from the USGS-EROS web site (http://glovis.usgs.gov/). The ASTER 30-m DEM was extracted from the 1A 3 N and 3B VNIR bands using the SILCAST application (Sensor Information Laboratory Inc.).

Fig. 2
figure 2

aShadow representation of the 5-m LiDAR DEM showing the location of topographic profiles obtained from the different DEMs resolutions as shown in figures b, c, and d. These profiles are evidencing the difference in accuracy in obstacle altitude and surface roughness, which represent one of the main factors in controlling flow simulations

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Profiles from LiDAR DEMs (Fig. 2b–d) show that when the resolution decreases the topography is smoothed and the maximum elevation of obstacles and lateral terraces is reduced with a difference of up to 15 m in height. These variations in altitude have the potential to significantly modify the behavior of a simulated flow. In fact, depending on the topographic barrier heights, the flow could be able to overrun the obstacle or to turn around it. Comparing the LiDAR DEMs at 30-, 50-, and 90-m spatial resolution with the ASTER 30 m, INEGI 50 m, and SRTM 90 m, respectively, several significant differences can be easily identified (Fig. 2). Despite the evident difference in the volcano summit altitude (Fig. 2b, profile a-a’), the SRTM 90-m is similar to the LiDAR 90-m with a smoothed profile. However, the ASTER 30-m does not reproduce the complex topography visible in the LiDAR 30-m, and it is more similar to the SRTM 90-m DEM. The observed inaccuracy in altitude could be fixed if direct measurements are available, but the overall topography will maintain a low detail. Finally, the INEGI 50-m DEM has better altitude accuracy, and it is comparable with the LiDAR 50-m DEM.

2.3 High-performance computing cluster (HPCC)—HORUS

In 2008, our research group decided to take a challenge to the Computational Geodynamics Laboratory (CGL)—Geosciences Center at UNAM and start building a HPCC named Horus, where TITAN2D was compiled with MPICH v. 1.2.7. In its present configuration, Horus includes AMD Opteron (64 bit) 2.7 GHz dual- or quad-core processors and Intel 2.7 GHz (64 bit) quad-core processors. Horus has 43 compute nodes, each with 250 GB as local storage and 2 or 4 GB/core (Table 1).

Table 1 Hardware configuration of Horus
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In total, Horus contains 228 processing cores, 600 GB RAM memory, and a total of 30 TB for data storage. The cluster has a single 1000T centralized 48 ports switch that handles MPI traffic and is connected to a master node. We use as operating environment the open source Rocks Cluster Distribution-5.1

In order to test the performance of our HPCC against TITAN2D, several trials were ran with 2, 4, 8, 16, 32, 64 and 128 processors with fixed input parameters (Table 1) and a simulated time of 1,200 s. We also changed the number of computational cells from 5 to 10 and 15 for each DEM’s resolution for a total of 105 runs. The number of cells is used by TITAN2D to determine the maximum permitted level of grid-cell refinement for the parallel adaptive mesh, which has direct implications on computational time and disk space storage.

2.4 Data representation

TITAN2D offers different output formats (tecplot/mshplot/GMFG Viz/Web Viz/Paraview or grass_sites). Here, we used the grass-sites format that can be easily imported into ARCGIS@ after the application of a simple script that automates the post-processing of all files generated during the numeric simulations performed on Horus (Appendix). The post-processed files contain for each site the latitude, longitude, flow thickness, and velocity, among others parameters. For each simulated step, where the time interval is set at the beginning of the simulation, TITAN2D produces a file in the grass-sites format. Therefore, it is possible to represent a single step of a simulation after a determined time interval, or the entire flow path using all steps. The methodology employed to visualize the entire flow path is to merge in ARCGIS (ESRI) all steps in a single file and then performing a point statistic analysis (maximum flow thickness around each point over a specified neighborhood) to obtain the maximum flow depth for each point around a minimum square area.

2.5 Input data

To determine how DEM resolution affects the basal friction angle, several trials were performed with basal friction angles ranging from 5° to 25° at a 5° interval. The other input parameters, such as the pile dimension, starting point, and the internal friction angle, were considered fixed for all the simulations (Table 2). In particular, the internal friction angle was set at 30°, a common value used in previous works (i.e. Sheridan et al. 2005; Rupp et al. 2006; Capra et al. 2008). The input volume and starting point correspond to the 2004 BAF at La Lumbre ravine, already simulated by Sulpizio et al. (2010). This pyroclastic flow is particularly suitable for our benchmark, because it is characterized by several important changes in curvature of ravines and breaks in slope (Fig. 2a). These morphological changes are useful to better quantify the importance of DEM resolution to simulate past events and to assess the hazard posed by volcanic granular flows.

Table 2 Input parameters used to perform simulations
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3 Results

3.1 Computational efficiency

In Fig. 3, the relation between the number of cores and the duration of simulation after 1,200 s of flow is shown. Our computational system has the best performance using 16–32 processors, while using 2 or 128 processors the simulation takes almost the same time or even longer for 128 processors. This is caused probably by heavy data traffic between computing nodes. At 5-cells refinement (Fig. 3a), simulations have a longer duration as DEM resolution increases, but at 20-cells duration is fairly similar independently of the DEM resolution (Fig. 3c).

Fig. 3
figure 3

Diagrams showing the simulated time for different DEMs resolutions as a function of processing cores, and for different cell numbers at a 5, b 10, and c 20. The vertical axis is wall time (s), the real time spent by the simulation on Horus

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3.2 Stopping velocity

Stopping velocity is an important element recently introduced by Yu et al. (2009) and is used as a criterion to define at which point of the simulation the flow stops. As already mentioned, the adimensional average velocity (V 2 *) mostly depends on the initial pile dimension, the transient pile aspect, and the internal friction angle. Yu et al. (2009) proposed an average value of 0.03 for V 2 * in confined and unconfined flows. In our work, several trials were performed to illustrate how V 2 * changes with both, the basal friction angle and DEM resolution. In particular, the DEM resolution modifies the width and depth of the ravines (Fig. 3), with direct consequence on the transient pile aspect (α) and consequently on the V 2 * value. Figure 4 shows the results obtained by maintaining fixed the DEM resolution while changing the basal friction angle. In all DEMs, at lower φb the V 2 * increases with higher values for DEMs with enhanced resolution (Fig. 4). When comparing the V 2 * value at fixed φb and different DEM resolutions, we encountered a similar pattern (Fig. 5). In conclusion, lower DEM resolutions determine higher V 2 * values, being more evident as φb decreases.

Fig. 4
figure 4

Diagrams showing the variation of the stopping velocity V*2 versus the dimensionless time t* at a fixed DEM resolution and varying the basal friction angle. The DEM#_#° abbreviation in legend (i.e., DEM90_1°0) stands for D# DEM resolution and A# value of the basal friction angle

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Fig. 5
figure 5

Diagrams comparing the stopping velocity V*2 versus the dimensionless time t* at a fixed basal friction angle and varying the DEM resolution. The DEM#_#° abbreviation in legend (i.e., DEM90_10°) stands for DEM# DEM resolution and #° value of the basal friction angle

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3.3 Basal friction angle versus DEM resolution

To better understand the effect of the DEM resolution on the inundation limits of simulated flows, the same input parameters (i.e. φb of 13°) are used (Table 2). Figure 6 shows the visualization for the first 60 s of the flow and at its final depositional steps (calculated with the V 2 * stopping velocity criterion). Our results show that as DEM resolution decreases, the flow reaches longer runout. Only with the fine resolution DEMs of 5 and 10 m (Fig. 6b, c), the flow is confined by the caldera rim and enters into the main ravine. For the other coarser DEMs, the flow overruns the topographic barrier and spreads over the flank of the volcano. Figure 7a shows the trajectory of the simulated flows through their central flow path. Whereas for the 5- and 10-m DEMs the trajectory is deviated by the caldera rim, for the other cases, the flow path reassembles a fairly rectilinear path. It is also worth observing the behavior of the maximum average velocity of simulated flows (Fig. 7b). For the 5- and 10-m DEMs, where the flow is confined by the caldera rim, the flow gradually decelerates after the average velocity reaches its maximum. In contrast, for coarser DEMs resolution, the flow decelerates when it overruns the caldera rim and then reaccelerate to reach a second peak of the maximum average velocity (Fig. 7b). To better demonstrate this effect, the 2004 BAF emplaced in the La Lumbre ravine is here reproduced over the 5-m DEM. The input parameters are chosen as those used by Sulpizio et al. (2010) to reproduce the runout of this pyroclastic flow (Fig. 8a; Table 2). Then, the same simulation is run over the other DEMs. After 60 s, the simulated flow reveals that only with the 5-m DEM the flow keeps confined into the La Lumbre ravine (Fig. 8a). In the other cases (Fig. 8b–d), using a φb of 11° is inappropriate, since the flow overflows the caldera rim and subsequently subdivides into more ravines reaching greater runout and lower thickness. With the 90-m DEM, it is impossible to run the same simulation, since the φb of 11° is to low and the pile completely spread on the topography. To reproduce the 2004 flow on DEMs with lower resolution, it is necessary to use a greater φb value. For example, for the 50-m DEM a φb of 16° was suitable in order to obtain comparable results with the 5-m DEM simulation (Fig. 8e, f) but only for the runout. However, the thickness of the simulated flow does not reproduce the observed 3-m-thick front of the deposit, which is well reproduced only by the simulation with 5-m DEM (Fig. 8c).

Fig. 6
figure 6

a 3D view of the La Lumbre ravine. Note how the caldera rim acts as a topographic barrier to pyroclastic flows. From b to f: representation of the final depositional step (based on the V2* value) for a simulated flow with a constant basal friction angle of 10° and for different DEM resolutions

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Fig. 7
figure 7

a Trajectories for the center lines of flows simulated for different DEM resolutions, and b diagram showing the simulated average flow velocity (m/s) with time (t)

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Fig. 8
figure 8

Simulation results showing the trajectory and thickness of the 2004 pyroclastic flow emplaced in the La Lumbre ravine. a, b, c, and d modeling results of the flow after 60 s, over the 5, 10, 30, and 50 m DEM resolutions, respectively. e Numeric simulation that best fits the 2004 pyroclastic flow (black line, Sulpizio et al. 2010) with a basal friction angle of 11° for the 5-m DEM. f: numeric simulation that best fits the 2004 pyroclastic flow (black line, Sulpizio et al. 2010) using a basal friction angle of 16° and a 50-m DEM

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4 Discussion

Numerous GIS-enabled software packages (TITAN2D, Patra et al. 2005; FLOW3D, Malin and Sheridan 1982; Kover and Sheridan 1993; and LAHARZ, Schilling 1998) have been developed for either predicting the path and extent of volcanic flows or visualizing their effects and potential impacts to civil infrastructures. It has been already demonstrated that LAHARZ, a routine to determine lahar inundation zones, is largely sensitive to the spatial resolution of the digital elevation data (i.e., Stevens et al. 2002; Davila et al. 2007; Hubbard et al. 2007). High-resolution DEMs can keep confined lahars using small flow volumes. On the other hand, for low-resolution DEMs larger volumes are needed to reach the same runout. Our modeling results show that the DEM resolution is fundamental for BAF simulations with TITAN2D. Low-resolution DEMs, such as the 90-m DEM, are inappropriate to reproduce BAF over irregular topographic path where obstacles and abrupt turns can suddenly diverge the simulated flow. Inaccuracy in both, altitude of topographic obstacles and depth of ravines, made impossible to reproduce past flows with similar φb for different DEM resolutions. The LiDAR data we used in this study are probably one of the best sources for topographic data, even better than SRTM or ASTER DEMs when resampled at the same resolution. The first main implication of this result resides in the impossibility to compare input data such as φb used to simulate BAF in different volcanoes or for the same event reproduced by other authors using different topographic data sources. In this sense, it is quite unrealistic to propose fixed φb values for some type of BAFs, since it will strongly depend on the DEM resolution. Our simulations performed with the 5- and 10-m DEMs are fairly similar, the same for the 30- and 50-m DEMs (Figs. 6, 8). However, for the 90-m DEM results are largely different and often inappropriate to accurately reproduce past flows. The same situation stands for the stopping velocity, since this parameter is largely dependent on the basal friction angles used to perform the simulations (Sulpizio et al. 2010). Even though DEM quality evaluation is not the main purpose of this work, the comparison of SRTM, ASTER, INEGI, and LiDAR data (Fig. 2) revealed that vector data obtained from topographic maps, as those available in Mexico, represent a good source to acquire a DEM with an appropriated resolution. It is worth noting that the free availability of SRTM data makes this source to be largely used in our scientific community. Even with a 90-m in spatial resolution, the SRTM DEM seems to be better than the DEM generated from ASTER images. However, the resolution of such DEM is probably too low to reproduce past flow over complicated topography such as the case for the 2004 La Lumbre BAF.

5 Conclusions

In this paper, we show that TITAN2D parallel code represents a useful tool to perform simulations of BAFs trajectory and maximum runout. As several works have already shown (i.e. Charbonnier and Gertisser 2009; Procter et al. 2010), the importance of input parameters accuracy needed by this algorithm to reproduce past flows is crucial. Our work represents a significant contribution, evidencing that the source data, used to obtain the DEM over which the flow is simulated, represent a key input parameter that controls the simulation results. We strongly recommend determining the right set of input parameters by several trials in order to acquire characteristic values for the basal friction angle. In the particular case of rugged topography, a DEM with high resolution (i.e., 5–10 m) should be acquired in order to obtain confident simulation results.