Refining the input space of plausible future debris flows using noisy data and multiple models of the physics

Abstract

Forecasts of future geophysical mass flows, fundamental in hazard assessment, usually rely on the reconstruction of past flows that occurred in the region of interest using models of physics that have been successful in hindcasting. The available pieces of data, are commonly related to the properties of the deposit left by the flows and to historical documentation. Nevertheless, this information can be fragmentary and affected by relevant sources of uncertainty (e.g., erosion and remobilization, superposition of subsequent events, unknown duration, and source). Moreover, different past flows may have had significantly different physical properties, and even a single flow may change its physics with respect to time and location, making the application of a single model inappropriate.

In a probabilistic framework, for each model M we define (M, P_M), where P_M is a probability measure over the parameter space of M. While the support of PM can be restricted to a single value by solving an inverse problem for the optimal reconstruction of a particular flow, the inverse problem is not always well posed. That is, no input values are able to produce outputs consistent with all observed information. Choices based on limited data using classical calibration techniques (i.e. optimized data inversion) are often misleading since they do not reflect all potential event characteristics and can be error prone due to incorrectly limited event space. Sometimes the strict replication of a past flow may lead to overconstraining the model, especially if we are interested in the general predictive capabilities of a model over a whole range of possible future events.

In this study, we use a multi-model ensemble and a plausible region approach to provide a more predictionoriented probabilistic framework for input space characterization in hazard analysis. In other words, we generalize a poorly constrained inverse problem, decomposing it into a hierarchy of simpler problems. We apply our procedure to the case study of the Atenquique volcaniclastic debris flow, which occurred on the flanks of Nevado de Colima volcano (Mexico) in 1955. We adopt and compare three depth-averaged models. Input spaces are explored by Monte Carlo simulation based on Latin hypercube sampling. The three models are incorporated in our large-scale mass flow simulation framework TITAN2D. Our meta-modeling framework is fully described in Fig.1 with a Venn diagram of input and output sets, and in Fig. 2 with a flowchart of the algorithm. See also for more details on the study. Our approach is characterized by three steps: (STEP 1) Let us assume that each model Mj is represented by an operator: f_Mj in R^d, where d is a dimensional parameter which is independent of the model chosen and characterizes a common output space. This operator simply links the input values to the related output values in Rd. Thus we define the global set of feasible inputs. This puts all the models in a natural meta-modeling framework, only requiring essential properties of feasibility in the models, namely the existence of the numerical output and the realism of the underlying physics. (STEP 2) After a preliminary screening, we characterize the codomain of plausible outputs: that is, the target of our simulations – it includes all the outputs consistent with the observed data, plus additional outputs which differ in arbitrary but plausible ways. For instance, having a robust numerical simulation without spurious effects, and with meaningful flow dynamics, and/or the capability to inundate a designated region. Thus, the specialized input space is defined as the inverse image of palusible outputs. (STEP 3) Furthermore, through more detailed testing, we can thus define the subspace of the inputs that are consistent with a piece of empirical data Di. For this reason those sets are called partial solutions to the inverse problem. In our case study, model selection appears to be inherently linked to the inversion problem. That is, the partial inverse problems enable us to find models depending on the example characteristics and spatial location.