Martin Ambroz

In this contribution, we present a Lagrangean approach to forest fire modelling. The fire perimeter is represented by a three-dimensional discrete curve on a surface. Our mathematical model is based on empirical fire spread laws influenced by the fuel properties, wind, terrain slope, and shape of the fire perimeter with respect to the topography (geodesic and normal curvatures). The motion of the fire perimeter is governed by the intrinsic advection-diffusion equation. To obtain the numerical solution, we employ the semi-implicit scheme to discretize the curvature term. For the advection term, we use the so-called inflow-implicit/outflow-explicit approach combined with the implicit upwind scheme. A fast treatment of topological changes (splitting and merging of the curves) is also incorporated and briefly described .The propagation model is applied to artificial and real-world experiments. To adapt our model to wildfire conditions, we tune the model parameters using the Hausdorff distance as a criterion. Using data assimilation, we estimate the normal velocity of the fire front (rate of spread), the dominant wind direction and selected model parameters.

The spread of natural fires is a complex issue, as its mathematical modeling needs to consider many parameters. Therefore, the results of such modeling always need to be validated by comparison with experimental measurements under real-world conditions. Remote sensing with the support of satellite or aerial sensors has long been used for this purpose. In this article, we focused on data collection with an unmanned aerial vehicle (UAV), which was used both for creating a digital surface model and for dynamic monitoring of the spread of controlled grassland fires in the visible spectrum. We subsequently tested the impact of various processing settings on the accuracy of the digital elevation model (DEM) and orthophotos, which are commonly used as a basis for analyzing fire spread. For the DEM generated from images taken during the final flight after the fire, deviations did not exceed 0.1 m compared to the reference model from LiDAR. Scale errors in the model with only approximal WGS84 exterior orientation parameters did not exceed a relative accuracy of 1:500, and possible deformations of the DEM up to 0.5 m in height had a minimal impact on determining the rate of fire spread, even with oblique images taken at an angle of 45°. The results of the experiments highlight the advantages of using low-cost SfM photogrammetry and provide an overview of potential issues encountered in measuring and performing photogrammetric processing of fire spread.

We introduce a novel method for map registration and apply it to transformation of the river Ister from Strabo’s map of the World to the current map in the World Geodetic System. This transformation leads to the surprising but convincing result that Strabo’s river Ister best coincides with the nowadays Tauernbach-Isel-Drava-Danube course and not with the Danube river what is commonly assumed. Such a result is supported by carefully designed mathematical measurements and it resolves all related controversies otherwise appearing in understanding and translation of Strabo’s original text. Based on this result, we also show that Strabo’s Suevi in the Hercynian Forest corresponds to the Slavic people in the Carpathian-Alpine basin and thus that the compact Slavic settlement was there already at the beginning of the first millennium AD.

The NaturaSat software integrates various image processing techniques together with vegetation data, into one multipurpose tool that is designed for performing facilities for all requirements of habitat exploration, all in one place. It provides direct access to multispectral Sentinel-2 data provided by the European Space Agency. It supports using these data with various vegetation databases, in a user-friendly environment, for, e.g., vegetation scientists, fieldwork experts, and nature conservationists. The presented study introduces the NaturaSat software, describes new powerful tools, such as the semi-automatic and automatic segmentation methods, and natural numerical networks, together with validated examples comparing field surveys and software outputs. The software is robust enough for field work researchers and stakeholders to accurately extract target units’ borders, even on the habitat level. The deep learning algorithm, developed for habitat classification within the NaturaSat software, can also be used in various research tasks or in nature conservation practices, such as identifying ecosystem services and conservation value. The exact maps of the habitats obtained within the project can improve many further vegetation and landscape ecology studies.

In this paper, we present a mathematical model and numerical method designed for the segmentation of satellite images, namely to obtain in an automated way borders of Natura 2000 habitats from Sentinel-2 optical data. The segmentation model is based on the evolving closed plane curve approach in the Lagrangian formulation including the efficient treatment of topological changes. The model contains the term expanding the curve in its outer normal direction up to the region of habitat boundary edges, the term attracting the curve accurately to the edges and the smoothing term given by the influence of local curvature. For the numerical solution, we use the flowing finite volume method discretizing the arising advection-diffusion intrinsic partial differential equation including the asymptotically uniform tangential redistribution of curve grid points. We present segmentation results for satellite data from a selected area of Western Slovakia (Záhorie) where the so-called riparian forests represent the important European Natura 2000 habitat. The automatic segmentation results are compared with the semi-automatic segmentation performed by the botany expert and with the GPS tracks obtained in the field. The comparisons show the ability of our numerical model to segment the habitat areas with the accuracy comparable to the pixel resolution of the Sentinel-2 optical data.

In this paper we introduce mathematical model and real-time numerical method for segmentation of Natura 2000 habitats in satellite images by evolving open planar curves. These curves in the Lagrangian formulation are driven by a suitable velocity vector field, projected to the curve normal. Besides the vector field, the evolving curve is influenced also by the local curvature representing a smoothing term. The model is numerically solved using the flowing finite volume method discretizing the arising intrinsic partial differential equation with Dirichlet boundary conditions. The time discretization is chosen as an explicit due to the ability of real-time edge tracking. We present the results of semi-automatic segmentation of various areas across Slovakia, from the riparian forests to mountainous areas with scrub pine. The numerical results were compared to habitat boundaries tracked by GPS device in the field by using the mean and maximal Hausdorff distances as criterion.

We introduce two level-set method approaches to segmentation of 2D macrophage images. The first one is based on the Otsu thresholding and the second one on the information entropy thresholding, both followed by the classical subjective surface (SUBSURF) method. Results of both methods are compared with the semi-automatic Lagrangian method in which the segmentation curve evolves along the edge of the macrophage and it is controlled by an expert user. We present the comparison of all three methods with respect to the Hausdorff distance of resulting segmentation curves and we compare also their perimeter and enclosed area. We show that accuracy of the automatic SUBSURF method is comparable to the results of the semi-automatic Lagrangian segmentation.

This paper first gives a brief overview of the Lagrangian forest fire propagation model [Ambroz, M.—Balažovjech, M.—Medl’a, M.—Mikula, K.: Numerical modeling of wildland surface fire propagation by evolving surface curves, Adv. Comput. Math. 45 (2019), no. 2, 1067–1103], which we apply to grass-field areas. Then, we aim to estimate the optimal model parameters. To achieve this goal, we use data assimilation of the measured data. From the data, we are able to estimate the normal velocity of the fire front (rate of spread), dominant wind direction and selected model parameters. In the data assimilation process, we use the Hausdorff distance as well as the Mean Hausdorff distance as a criterion. Moreover, we predict the fire propagation in small time intervals.