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Parameter identification for simulating debris flows using cellular automata

Kristina Georgievaruhr university bochum

Petyo Gadzhanovruhr university bochum

Abstract: The increase in temperature due to the climate change and the human intervention in natural systems increase the risk of debris flows. In densely populated areas debris flows cause severe damage to people and infrastructure. Therefore, it is important to analyze and predict debris flows realistically. In the last years, cellular automata have been applied to simulate the behavior and movement of debris flows. The main challenge in applying these automata is the appropriate specification of global parameters for calculating new cell states. This paper presents the implementation of a parameter identification framework for debris flow simulation based on the SCIDDICA cellular automata model (D’Ambrosio, 2003). To demonstrate the performance of the framework, validation tests have been conducted, based on controlled experiments.

Keywords: cellular automata, debris flow, simulation, parameter identification

1. introductionDebris flows are one of the most common types of landslides, which occur mostly in mountain and glacial areas. Activated by addition of moisture, erosion, wildfires or volcanic eruptions, debris flows damage infrastructure and endanger humans. Various attempts have been made to thoroughly assess the physics of debris flows. However, invasive techniques to collect slope data, such as installing sensors in boreholes, often change the state of the debris. Therefore, models to numerically simulate debris flows have been developed. SCIDDICA is a family of deterministic models based on the Cellular Automata (CA) computational paradigm, specially developed for debris flow simulations (D’Ambrosio, 2003).

CA represent a mathematical idealization of complex systems and describe the dynamical behavior of natural phenomena in terms of interactions between system particles. Patterns of complex behavior are thus generated by discretizing space and time to process a large

Kristina Georgieva, Petyo Gadzhanov

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number of identical, simple, locally interacting components (Ilachinski, 2011). The components are referred to as cells in the literature. Cells interact only with cells in their neighborhood and update their current state according to a uniquely defined set of rules called transition function. By taking into account the states of the cells in the neighborhood, cell generations evolve from the initial state of the system until a termination condition is fulfilled. CA have gained popularity in diverse scientific areas, because they allow to process the interactions between the cells in a fully parallel manner. As a result, computationally efficient modeling of natural phenomena is possible.

SCIDDICA is a two-dimensional hexagonal CA model for simulating debris flows. In SCIDDICA, the cell state can be decomposed in sub-states, namely the altitude of the bedrock, the thickness of landslide debris, the depth of erodible soil cover, and the run-up height. These cell sub-states are computed at each simulation step for each cell using a transition function. In addition, global parameters account for the general characteristics of the model. The global parameters include:

• apothem of the cell pa,

• adhesion (debris thickness that cannot leave the cell) padh,

• friction threshold for debris outflows pf,

• relaxation rate of debris outflows pr,

• run-up loss at each step prl,

• activation threshold for mobilization of the soil cover pmt, and

• factor of progressive erosion per.

The global parameters are initialized at the beginning of the simulation and remain constant during simulation execution. Finding appropriate values for the global parameters is the main challenge in applying SCIDDICA or similar CA models. One way to obtain appropriate global parameter values is to calibrate the model against debris flows in an analogous geological context (D’Ambrosio, 2003). For that purpose, historical debris flows are compared with simulations. Parameter values are manually assigned and modified with respect to the result of the simulation to continuously improve the set of parameters. However, the described process is computationally intensive and requires human evaluation for each simulation execution. Therefore, an approach to automatically identify optimal values of global parameters is needed.

Iovine, D’Ambrosio and Di Gregorio have presented an automated technique for identification of optimal global parameter values based on genetic algorithms (Iovine, 2004). An average debris

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flow simulation with SCIDDICA requires at least 15 000 CA steps. To evaluate the best set of global parameters, the simulation is run multiple times with different global parameter values. However, the computation is carried out sequentially. As a consequence, the evaluation of the parameters requires days when performed on a single computer. To overcome this limitation, a high-performance CA simulation framework for SCIDDICA, capable to automatically identify optimal parameter values, has been developed.

2. Case studyThe parameter identification framework has been validated against an experiment conducted at the USGS debris flow flume in Oregon, USA (Iverson, 2010). The flume is a rectangular concrete channel 95 meters long and 2 meters wide, which slopes 31 degrees and ends with a planar runout surface (Figure 1). To create a debris flow, a mixture of 9.4 m3 water-saturated gravel, sand and mud is placed behind a headgate at the top of the channel, and then released. Time-series data with high resolution is collected at three measurement cross sections downslope from the headgate at 33, 66 and 90 meters. The aggregated data implies mean peak thickness values of the flow of ~0.2 m and standard deviations of ~0.05 m.

Figure 1: Flume geometry (Iverson, 2010)

3. Inverse problemA simulation is a forward problem in terms that the result of a measurement is predicted. In contrast, using the result of some measurement to estimate parameter values is referred to as an inverse problem (Tarantola, 2005). Inverse problems are described by a model space and a data space. For the debris flow parameter identification problem they are defined as follows:

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• Model space is the search space for the parameter values. Attempts to assign rea-sonable values to the global parameters and execute the simulation have shown that the parameters with the biggest influence on the SCIDDICA simulation outcome are run-up loss, height threshold and adhesion. The factor of progressive erosion and the activation threshold for the mobilization of the soil cover are not varied, because no erosion can occur in a concrete channel. The value of each of the varied parame-ters must be within a certain range. Range bounds (Table 1) were adopted from the algorithm proposed by Iovine (Iovine, 2004). They were further restricted in accordan-ce to the validation experiment. The model space M is established by combining all feasible parameter valuese as a Cartesian product (Formula 2) such that

(1)

where the parameters can take any value within the range.

• The data space consists of all conceivable instrumental responses, i.e. the debris thickness at 33 and 66 meters approaching 0 and the thickness at the 90th meter approximating 0.15 after the debris stopped moving.

Table 1: Global parameters evaluated by the framework

Parameter Range Best value

Run-up loss prl 0.001 – 1 0.3149454410690523

Friction threshold pf 0.005 – 1.5 0.7957569406569751

adhesion padh 0.001 – 1 0.028156298331660425

4. Objective functionTo identify the optimal global parameter values, the least squared method is used. The least squared method is a standard approach to find the best fit to a data set. The idea is to compute the differences between the actual real case values and the simulated values and to minimize their squared sum. In this fashion, the absolute simulation error is minimized. Formula 2 expresses the objective function f to be minimized by the optimization framework, where tm denotes the measured debris thickness, ts - the simulated debris thickness, pm - the mean peak value of the measured thickness, ps – the simulated peak, dm - the standard deviation of the measured thickness, and ds – the simulated standard deviation. The differences are computed for the data collected at the three measurement cross sections of the flume.

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(2)

5. ImplementationTo simulate debris flow events with CA, a multi-module framework has been developed – the Debris Flow Simulation Framework (DFSF). A basic overview of its architecture and most important modules is represented by Figure 2. The entire implementation has been carried out in Java.

Figure 2: The Debris Flow Simulation Framework (DFSF) architecture overview

5.1. ca coreThe CA module provides fundamental entities and specifies basic transformation and processing behavior required to derive a cellular model. As shown in Figure 3, the CA core consists of the following building blocks:

• Cell defines base soil properties of a finite geographic area in a geometry indepen-dent manner. Thus, cells with different geometric form like triangle, square or hexa-gon are supported. Moreover, cells are the only atomic entity and do not depend on any other building block. A cell is able to detect changes in its own state.

• Grid is a set of a finite number of cells. The total number of cells defines the resolu-tion of a simulation. An increase of the grid resolution leads to improvement in the area representation, whilst degrading the processing speed. The grid is also able to track its own state based on the states of the cells it contains.

• Neighborhood is defined for each cell in the grid. It consists of the cell itself, also called “central cell”, and all adjacent cells. The number of neighbors depends on the

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geometric form of the cell. Thus, every neighborhood could be represented as a finite subset of the grid.

• Overlay describes an abstract layer applied atop the cellular grid at the beginning of the simulation. Its main purpose is to specify the topographic terrain on which the debris flow takes place by providing cells with data such as altitude of the bedrock, the thickness of landslide debris and the depth of erodible soil cover.

• Cell Transformation defines a function, which is used repeatedly by the CA to trans-form cells. A cell transformation could be considered analogous to a transition func-tion as defined in automata theory. Transformations may take into account not only the current state of a specific cell, but also the states of its neighbors. DFSF transfor-mations are designed to operate on a single cell at a time, therefore fully supporting functions applied only to a specific subset of grid cells.

• Grid Transformer is responsible for the execution of a set of transformations on each cell that is part of the grid. To do so, the transformer iterates sequentially over a predefined collection of transformations.

• Generation is a set of grid cells, the state of which had changed during processing of the specified transformations. A generation is created after the grid transformer had finished execution and right before the simulation time unit is advanced by 1.

• Processor is the top-most CA entity. It is designed to establish bindings between other modules, to control the interactions among them and to define a basic simu-lation processing behavior. Communication with modules external to the CA core is also performed by the processor. The implementation of the processor is strongly problem-oriented and highly dependent on the geometry form of the cells. Therefore, DFSF provides an abstract definition of the processor entity.

To ensure maximum problem detachment and ease of future extensibility most of the above introduced building blocks are implemented in abstract manner.

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Figure 3: CA core building blocks as they relate to each other

Figure 4: Cell neighborhood and the corresponding indexes with respect to the central cell

The CA core communicates with the other modules through the processor. The communication is accomplished by utilizing an event-based notification pipeline. The pipeline enables the propagation of messages and data upon occurrence of predefined events. Implementation is carried out following the publish-subscribe pattern to support multiple external modules as subscribers.

5.2. SCIDDICA implementationTo enable CA processing as specified by SCIDDICA, a new tangible DFSF model has been derived. The model is based on the CA core specified above. It utilizes hexagonal cells that expose soil and topographic properties as required by SCIDDICA.

The neighborhood of each cell consists of six directly adjacent cells. To provide position awareness, an indexing approach for neighbors is introduced with respect to the central cell. Starting from the north side, indexes from 0 to 5 are assigned to each adjacent cell in clockwise direction. An example is shown in Figure 4. If no immediate neighbor exists, i.e. if the central cell lies on a grid border, an empty reference is saved for the current index.

The transition function consists of four independently defined cell transformations. An adapted variant of the SCIDDICA S4a transformation rules are utilized by the latest version of the DFSF model. They are applied simultaneously to the whole grid by the grid transformer.

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5.3. Flume simulationTo recreate the structure of the Debris Flow Flume at H.J. Andrews Experimental Forest, a specific overlay is derived and applied atop the CA grid. Cell altitudes are altered to model the walls, headgate and slope of the flume channel. Debris thickness is specified only for the cells behind the gates, thus fulfilling the initial SCIDDICA conditions. The position of the headgate is considered a reference point, or 0, with respect to the channel length. Cross-sections are established downslope at 33, 66 and 90 meters. At these points measurement data such as mean thickness and peak thickness is collected and propagated for further processing and analysis.

To visually verify and present the debris flow simulation, a problem-specific Graphical User Interface (GUI) has been implemented. Based on the open source Visualization Toolkit (VTK) project (http://www.vtk.org), the GUI provides a real-time 3D representation of the changing cell states and thus of the whole simulated event, as shown in Figure 5. The visualization data required by the GUI module is obtained through the event-based notification pipeline that is exposed by the processor entity of the CA core.

Figure 5: The DFSF Graphical User Interface module

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A generic file Input/Output (I/O) module has also been implemented. Based on the generation entity, provided as part of the CA core, it allows the preservation of already completed simulations. Registering as a subscriber enables the I/O to receive notifications when new generations had been created. Utilizing binary serialization on selected only cell sub-states leads to fast write access speed and space-efficient preservation, due to the lack of data overhead. Reading a simulation file allows the visual recreation of a simulation, thus eliminating the mostly heavy computational effort.

6. OptimizationTo identify the optimal values of global debris flow parameters, the flume simulation is coupled with the Multi-method Optimization Package (MOPACK).

6.1. MOPACKMOPACK is a Java-based framework, which provides interfaces and complete implementation of miscellaneous optimization algorithms by utilizing several optimization strategies and deterministic and stochastic methods (Nguyen, 2012). In addition, MOPACK provides a flexible distribution concept to carry out simulations simultaneously in a distributed computing environment by exploiting parallel computing paradigms. The distribution concept is based on the high performance cloud computing framework GridGain, which relies on a very fast MapReduce implementation (http://www.gridgain.com).

Particle Swarm Optimization (PSO) has been applied to find optimal global parameter values. PSO is a stochastic population-based technique, in which, a swarm of potential solutions, called particles, is used (Kennedy and Eberhart, 1995). The particles do not depend on each other. As a result, PSO is well suited for execution in distributed environments. MOPACK offers an implementation of PSO, which is able to operate in parallel, thus providing an opportunity to accelerate optimization.

For parameter identification, a MOPACK debris flow optimization problem is defined. It consists of an objective function and a set of global debris flow parameters. For all parameters minimum, maximum and initial values are specified. By varying the given parameters within their min-max range, numerous solutions are evaluated during the optimization process. Parameters, which lead to a fully minimized objective function, are considered identified and thus a solution of the debris flow optimization problem.

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6.2. ResultsAs is often the case with optimization problems, no single optimal solution exists. After several refinements of the initial minimum-maximum range of the global parameters, an absolute error approaching 0.2 has been reached. To achieve similar values an average of ~320 MOPACK iterations were required. Figure 6 presents the absolute error devolution as a function of the optimization iterations. The graphic results from an optimization of a simulation with 8 cells horizontal resolution.

Figure 6: Absolute error devolution as a function of optimization iterations

7. ConclusionsThe design and prototype implementation of a parameter identification framework for debris flow simulation based on cellular automata have been presented in this paper. The framework comprises of a core CA module, an implementation of a debris flow model, and an integrated optimization framework. It has been shown, that the optimization framework, operating in a fully parallel manner, allows for high-performance identification of optimal values of global parameters. The framework, presented in this paper, has been validated against an experiment conducted at the USGS debris flow flume. Although the feasibility of the implemented parameter identification approach has been demonstrated, the framework can further be improved by extending the objective function to provide more accurate representation of the debris distribution. Future research also includes validation of the framework against real historical debris flows and execution of simulations in analogous geological contexts.

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LiteraturD’Ambrosio D., Di Gregorio S., and Iovine G. (2003) Simulating debris flows through a hexagonal cellular automata model: SCIDDICA S3–hex, Nat. Hazards Earth Syst. Sci., Vol. 3, pp. 539-544

Ilachinski, A. (2011) Cellular Automata: A Discrete Universe. World Scientific Publishing Co. Pte. Ltd., Singapore

Iovine, G., D’Ambrosio, D., and Di Gregorio, S. (2004). Applying genetic algorithms for calibrating a hexagonal cellular automata model for the simulation of debris flows characterised by strong inertial effects, Geomorphology, Elsevier Science , Vol. 66, Issue 1-4 , March , 2005, pp. 287-303

Iverson, R. M., Logan, M., LaHusen, R. G., and Berti, M. (2010). The perfect debris flow? Aggregated results from 28 large-scale experiments, Journal of Geophysical Research, Vol.115, Issue F3, 2010, pp. F03005

Kennedy, J. and Eberhart, R. (1995). Particle Swarm Optimization, Proceedings of IEEE International Conference on Neural Networks. IV. pp. 1942–1948

Nguyen, V.V., Hartmann, D., and König, M. (2012) A distributed agent-based approach for simulation-based optimization, Advanced Engineering Informatics, Vol.26, Issue 4, October 2012, pp. 814–832

Tarantola, Albert (2005). Inverse Problem Theory and Methods for Model Parameter Estimation, Society for Industrial and Applied Mathematics, Philadelphia

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