Investigating volcanoes behaviour With novel statistical approaches Igor Barahona a, Luis Javier Alvarez b, Antonio Sarmiento b, Octavio Barahona c a UCIM-UNAM - CONACYT b UCIM-UNAM. c Colegio de Morelos Guanajuato, México. June 17 th 1 of 2017
OUTLINE 1. Introduction 2. Objectives 3. Methodology 4. Results 5. Discussion 2
3 OUTLINE 1. Introduction 2. Objectives 3. Methodology 4. Results 5. Discussion
Popocatepetl volcano date back from the prehispanic civilizations. Aztec codices represented its eruptions on years 1363, 1509 and 1518. Popocatepetl means in Nahuatl language the smoking mountain It is at 5450 meters height Located 80 kilometres at south east from Mexico city Its one of the most active volcanoes in the country 4
Mexican Federal Government is responsible for monitoring the eruptive activity 12 monitoring stations were deployed around the volcano 5
Only four stations are equipped with HD cameras. Pictures are permanently taken (7 by 24) every minute. We analyse pictures taken at Tlamacas station. They correspond to the period from 5:17 to 7:38 hrs of January 6 th, 2016 6
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Data that describe volcanic activity is often highly volatile, irregular, and random appearance. Irregular data is composed by two major sources of uncertainty. Huffaker et al (2016) (1) Real uncertainty due to the inherent randomness and natural variation of real-world processes (2) Perceived uncertainty due to decision-maker's limited perceptions of reality 8
Key point is to reduce the gap between real and perceived Perceived Real
Three perspectives on modelling Real uncertainty due to the inherent randomness and natural variation of real-world processes Perceived (accuracy limited) Modelling partially helpful for decision making Wrong perception. Modelling uselessness for decision making
Two types of alternatives for modelling Deterministic Deterministic models are believed to ignore the uncertainty. Mostly incapable of accurately describe the behaviour of the observed phenomena Stochastic Stochastic models are believed to capture much of the uncertainty. Provide a more accurate description of the observed phenomena. 11
A process moves successively through a set of states s0,s1,...,sns0,s1,...,sn, e.g. the change of scenario's due to the passing of events/time Introduction Objectives Methodology Results Discussion Stochastic or Deterministic?. Basic definitions A state is a tuple of variables which are given to a specific value. Typically representing a real-world phenomena. A process moves progressively through a set of states S 1, S 2,, S n. For instance, the change is due to the passing of the time. A machine determines how a process moves through one state to another, e.g the natural or social phenomena. The machine can only be at one state at a time A deterministic machine. For a particular state S n and action α, processes is moved to only one successor state S n+1 A stochastic machine. For a particular state S n and action α, there might be set of S n+1, S n+2, S n+k of possible successors Where P(X = s k ) is the probability of being in the state s k. If there are k k possible states, then the sum of probabilities is P(X = s i ) i=1
So What? 13
Stochastic modelling is required when realworld processes are physically random (Indeterministic). Causal relationships are not available to support deterministic formulations 14
Key question on investigating volcanoes is to find out the type of process is generating this observed signal Fourier spectrum was believed to be the most suitable tool in order to characterize volcanoes behaviour (Hilborn, 1994) Nonlinear Time Series famously documented by Lorenz (1963) offer a plausible alternative to traditional modelling. This structure takes the form of orbits moving in an m-dimensional Euclidean space. It represents the evolution of the states of the system under study. (Konstantinou, et al. 2013). 15
OUTLINE 1. Introduction 2. Objectives 3. Methodology 4. Results 5. Discussion 16
Objectives of this research. 1. Apply the concepts of Non Linear Time Series (NLTS) and Dynamic Systems (DS) on the analysis of Popocatepelt s digital images. 3 objectives were settled 2. Apply Singular Spectrum Analysis (SSA) to decompose time series in 2 components: Signal and Noise 3. Provide evidence to ascertain whether time series mimics a stochastic dynamic process 17
OUTLINE 1. Introduction 2. Objectives 3. Methodology 4. Results 5. Discussion 18
Decompose each on R-B-G colours
Nonlinear time series A linear stochastic process is a succession of random variables ordered by a time index. Small & Tse (2003) Let x(t)=[x t+1, x t+2,, x t+n ] be a temporal stochastic process, denoted by n consecutive elements Then, it is said to be stationary if the joint probability is independent of the period t, regardless of the sample size n. Hilborn (2000) 20
Singular Spectrum Analysis Singular Spectrum Analysis (SSA) is a signal processing technique to separate an observed time series into signal (structured variation) and noise (unstructured variation) SSA is a data-adaptive signal processing approach that can accommodate highly irregular oscillations in signals Noisy data increase the difficulty of detecting oscillatory patterns. Signal processing isolates the structured variation in data. 21
Singular Spectrum Analysis Signal Observed Noise 22
Autocorrelation and mutual information Given a time series {x t : 1 t N} 1. Autocorrelation function is given by: C k = i=1 N k (x i x)(x i+k i=1 N (x i x) 2 x) x i is the mean of the time series 2. The average mutual information is given by: I k = N k i=1 P(x i, x i+k )log 2 P(x i, x i+k ) P x i P(x i+k ) P(x) is the probability of observing x = x i ó x = x i P(x, y) is the joint probability of observing x = x i e y = x i+k
Embedding dimension 3. Phase Space Reconstruction from Time Series Data X 1 = (x 1, x 1+T, x 1+2t,..., x 1+ d 1 T ) X 2 = (x 2, x 2+T, x 2+2t,..., x 2+ d 1 T ) X m = (x m, x m+t, x m+2t,..., x m+ d 1 T ) m= the biggest entire, given m + (d-1)t T N 4. Analysis of Singular Values Cov= 1 N i=1 N (X i X)(X i X) T N X = 1 N i=1 X i
Testing for low-dimensionality The shadow attractor reconstructed from delayed copies of x(t) preserves essential mathematical properties of the original system. Small and Tse, (2002) The number of delay-coordinate vectors required to define the shadow phase space is termed the embedding dimension (M), which represent the minimum system dimensionality required to contain a reconstructed attractor Huffaker et. al (2016). Statistical methods are used to select an embedding delay (d) and embedding dimension (M). Kantz & Schreiber (1997) 25
Complete framework 1. Decompose pictures on R-B-G components 2. Create the time series based on R-B-G vectors 3. Decompose R-B-G vectors on Signal & Noise 4. Test for low-dimensional nonlinear structure 5. Estimate parameters (M) and (d) Concluding remarks 26
OUTLINE 1. Introduction 2. Objectives 3. Methodology 4. Results 5. Discussion 27
A total of 560 pictures were decomposed on RBG vectors Corresponding to the period from 5:17 to 7:30 hrs of January 6 th, 2016. A total of 560 pictures were integrated to the time series Three vectors of order c(560,1) were obtained, each one of them correspond to the R-B-G components 28
Summary of descriptive statistics 29
Summary of descriptive statistics 30
Dynamic representation
Observed Values
Signal reconstructed values
Noise reconstructed values
Estimated delay period 5 5
Noise reconstructed values Following the algorithm proposed by Huffaker et. al (2016), we reconstructed an attractor for Red signal With parameters M=4 and d=5, the reconstructed attractor offers preliminary empirical evidence of deterministic low-dimensional nonlinear system dynamics
OUTLINE 1. Introduction 2. Objectives 3. Methodology 4. Results 5. Discussion 37
A novel form for investigating eruptive activity on volcanoes is proposed Analysing digital data can bring better understanding of complex natural systems Similar studies with images from other volcanoes are required, in order to compare results This analysis can be used for modelling the dynamic of future eruptive episodes of Popocatepetl volcano This is only an illustrative example. More research is required for obtaining suitable results for decision making and policy making 38
References Hilborn RC. Chaos and nonlinear dynamics: an introduction for scientists and engineers. Oxford University Press: Oxford, UK, 2000. Huffaker R, Muñoz-Carpena R, Campo-Bescós MA, Southworth J. Demonstrating correspondence between decision-support models and dynamics of real-world environmental systems. Environmental Modelling & Software 2016; 83: 74-87. DOI http://dx.doi.org/10.1016/j.envsoft.2016.04.024 Kantz H, Schreiber T. Nonlinear time series analysis. Cambridge university press, 2004. Konstantinou KI, Perwita CA, Maryanto S, Surono, Budianto A, Hendrasto M. Maximal Lyapunov exponent variations of volcanic tremor recorded during explosive and effusive activity at Mt Semeru volcano, Indonesia. Nonlinear Processes in Geophysics 2013; 20(6): 1137-1145. DOI 10.5194/npg-20-1137-2013 Lorenz EN. Deterministic nonperiodic flow. Journal of the atmospheric sciences 1963; 20(2): 130-141 Small M, Tse CK. Applying the method of surrogate data to cyclic time series. Physica D: Nonlinear Phenomena 2002; 164(3 4): 187-201. DOI http://dx.doi.org/10.1016/s0167-2789(02)00382-2 40
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