Calculation of lahar transit times using digital elevation data

Volcanic debris flows, or lahars, represent a significant hazard because of their rapid emplacement, long runout distances, and tremendous destructive potential. Accurate knowledge of advance rates and flow depths is therefore critical to hazard mitigation in lahar-prone areas. With the increasing availability of digital topographic data, models of lahar transit can now be applied to better representative descriptions of natural topography, with the expectation of improved hazard predictions. Using an elementary model of lahar transport dynamics, we have developed a mathematical treatment of lahar propagation over digital topography. Our results show that flow depths and transit times depend on the cumulative effects of changes in local slope along the flow path. These cumulative effects can be significant in predicting lahar advance rates and transit times. In adapting the continuum model to digital topography, we treat the changes in elevation between each adjacent pair of elevations as contiguous inclined planes and establish new boundary conditions sequentially for all downstream intervals to satisfy both local and total volume conservation requirements. Thus, we are able to derive transit times and the evolution of flow velocity and depth profile as a function of distance, x, and time, t. To illustrate the usefulness of this approach, we have applied the model using parameters typical of large volume (106 to 107 m3), long runout (25–50 km) lahars at Mt. Ruapehu, New Zealand. By discretizing an analog topography model into N equal-sized intervals, we demonstrate that as N increases, the predicted transit time over the digital topography decreases asymptotically. Several hundred intervals are required to achieve convergence, thus yielding estimates of the minimum resolution of the digital topography (∼10 m vertical, ∼250 m horizontal) required to give accurate transit-time assessments. We also show that one cannot simply model irregular topography as a drop in elevation over a runout distance without risking significant errors in transit-time predications. The result that transit-time predictions decrease with increasing topographic resolution has significant implications for both hazard assessment and the ability to provide timely warnings to communities in the flow paths of lahars.