In this work, we propose the general idea of using a path-finding algorithm to solve a variational problem. By interpreting a variational problem of finding the function that minimizes a functional integral as a shortest path finding, we can apply the shortest path-finding algorithm to numerically estimate the optimal function. This can be achieved by discretizing the continuous domain of the variational problem into a spatially weighted graph. The weight of each edge is defined according to the function of the original problem. We adopt the Moser lattice as the discretization scheme since it provides adjustable connections around a vertex. We find that this number of connections is crucial to the estimation of an accurate optimal path. Dijkstra’s shortest path-finding algorithm was chosen due to its simplicity and convenience in implementation. We validate our proposal by applying Dijkstra’s path-finding algorithm to numerically solve three famous variational problems, i.e., the optical ray tracing, the brachistochrone, and the catenary problems. The first two are examples of problems with no constraint. The standard Dijkstra’s algorithm can be directly applied. The third problem is an example of a problem with an isoperimetric constraint. We apply the Lagrangian relaxation technique to relax the optimization in the standard Dijkstra algorithm to incorporate the constraint. In all cases, when the number of sublattices is large enough, the results agree well with the analytic solutions. In all cases, the same path-finding code is used, regardless of the problem details. Our approaches provide more insight and promise to be more flexible than conventional numerical methods. We expect that our method can be useful in practice when an investigation of the optimal path in a complex problem is needed.

Accurately estimating aboveground biomass (AGB) remains a challenge due to the limitations of various measurement methods. While destructive sampling provides precise results, it is costly, labor-intensive, and environmentally unsustainable. Advances in remote sensing technologies offer efficient alternatives for AGB estimation. Our study highlights that airborne laser scanning (ALS) is more effective in assessing forest structure than detailed biomass composition. Orthophoto analysis, while applicable for rapid assessments, tends to overestimate AGB and carbon stock, particularly in dense forests. Differences in tree height measurements between ALS, orthophoto extraction, and field methods emphasize the varying precision of remote sensing techniques. Integrating ALS and orthophotos may enhance accuracy by using orthophotos for tree crown identification and ALS for height measurements. This combined approach could reduce errors in biomass estimation, especially in dense forests where traditional methods are challenging. Our findings suggest that ALS is more suitable for large-scale biomass assessments, while orthophoto data may be more effective in open forests. Field calibration remains essential to improve accuracy, particularly for orthophotos, which are influenced by environmental factors such as light, shadow, and seasonal changes.

Stem diameter plays a pivotal role in forest mensuration, encompassing applications in growth assessment, stand density estimation, wood production, and carbon sequestration. Traditionally measured with diameter tape or calipers through ground-based methods, the advent of Terrestrial Laser Scanning (TLS) has introduced a non-destructive 3D modeling technology for precise measurements. In this study, we aimed to assess the accuracy and precision of eucalyptus stem diameter estimation using TLS in comparison to ground-based diameter tape measurement. The results showed an average diameter at breast height (dbh) of 13.840 cm and an average height of 18.5 m for the sample trees. Stem curves from TLS were automatically derived from the 3D Forest program analyses of standing eucalyptus trees. Our analysis reveals that the average Root Mean Square Error (RMSE) for estimated stem diameter using TLS was 4.34 cm for the first ten sections of the tree from the ground. The measurement of stem size in eucalyptus trees using TLS tends to show increased RMSE in the upper sections of the stem. Our research underscores the high accuracy achieved in estimating stem diameters, particularly for the lower portions of the trees, even under light wind conditions.