Supplementary Files
Keywords
forest management
historical range and variation simulation
uncertain spatial data
How to Cite
Abstract
Modern forest management is often focused on minimizing wildfire risk by regaining fire-resiliency from historical conditions. This process is labor and resource intensive and it is usually infeasible to treat entire forests. Therefore, forest treatment allocation optimization is applied to help identify the best mitigation plan, with a key objective being the minimization of departure from historical landscape conditions. However, the vagueness and imprecision of historical landscape condition models and their corresponding departure indicators due to data limitations often necessitate the use of simulated substitutes. Since spatial optimization, integral for identifying optimal mitigation plans, relies on simulation outputs, addressing uncertainty becomes a crucial concern. We analyze optimization outputs for different historical condition scenarios and propose strategies to identify robust alternatives in two steps. First, we conduct a series of spatial optimizations using a weighted-sum method, systematically varying the importance assigned to different objectives for each scenario of historical conditions. This process helps us understand how changes in the assumptions about past forest conditions and the relative weights assigned to each assumption affect the optimal management plans. Second, we analyze the resulting solutions to identify patterns and overlaps among the optimal spatial configurations across scenarios. By focusing on the identification of common patterns, we develop a strategy to select solutions that are robust to uncertainty in historical condition models. We find that the strategy based on identifying commonalities among optimal solutions yields management plans that are, on average, 8.9% more robust compared to plans derived without explicit consideration of robustness. We conclude that the commonality seeking strategy in solution space alleviates risk in decision making. These insights highlight the importance of developing strategies for applying robust strategies in spatially explicit optimization problems.
References
Ager, A. A., Houtman, R. M., Day, M. A., Ringo, C., & Palaiologou, P. (2019). Tradeoffs between US national forest harvest targets and fuel management to reduce wildfire transmission to the wildland urban interface. Forest Ecology and Management, 434, 99–109. https://doi.org/10.1016/j.foreco.2018.12.003
Albrich, K., Rammer, W., Turner, M. G., Ratajczak, Z., Braziunas, K. H., Hansen, W. D., & Seidl, R. (2020). Simulating forest resilience: A review. Global Ecology and Biogeography, 29(12), 2082–2096. https://doi.org/10.1111/geb.13197
Austin, K. G., Baker, J. S., Sohngen, B. L., Wade, C. M., Daigneault, A., Ohrel, S. B., Ragnauth, S., & Bean, A. (2020). The economic costs of planting, preserving, and managing the world’s forests to mitigate climate change. Nature Communications, 11(1), 5946. https://doi.org/10.1038/s41467-020-19578-z
Bissonette, J. A., & Storch, I. (Eds.). (2007). Temporal dimensions of landscape ecology: Wildlife responses to variable resources (1st ed.). Springer US. https://doi.org/10.1007/978-0-387-45447-4
Bolouri, S., Vafaeinejad, A., Alesheikh, A. A., & Aghamohammadi, H. (2018). The ordered capacitated multi-objective location-allocation problem for fire stations using spatial optimization. ISPRS International Journal of Geo-Information, 7(2), Article 44. https://doi.org/10.3390/ijgi7020044
Boyer, O., Sai Hong, T., Pedram, A., Mohd Yusuff, R. B., & Zulkifli, N. (2013). A mathematical model for the industrial hazardous waste location-routing problem. Journal of Applied Mathematics, 2013, Article 435272. https://doi.org/10.1155/2013/435272
Brunckhorst, D. J., & Trammell, E. J. (2023). Future options redundancy planning: Designing multiple pathways to resilience in urban and landscape systems facing complex change. Urban Science, 7(1), Article 11. https://doi.org/10.3390/urbansci7010011
Bugmann, H. K. M. (1996). A simplified forest model to study species composition along climate gradients. Ecology, 77(7), 2055–2074. https://doi.org/10.2307/2265700
Burns, R. M., & Honkala, B. H. (Eds.). (1990). Silvics of North America: 1. Conifers; 2. Hardwoods (Vol. 2). United States Department of Agriculture, Forest Service.
Calflora. (2025). Calflora: Explore the wild plants of California. Calflora Database. Last accessed July 16, 2025, from https://www.calflora.org/
Christiansen, R. E., Lazarov, B. S., Jensen, J. S., & Sigmund, O. (2015). Creating geometrically robust designs for highly sensitive problems using topology optimization. Structural and Multidisciplinary Optimization, 52(4), 737–754. https://doi.org/10.1007/s00158-015-1265-5
Church, R. L., & Murray, A. T. (2008). Business site selection, location analysis and GIS. In Business Site Selection, Location Analysis and GIS (pp. 259–280). https://doi.org/10.1002/9780470432761
Cushman, S. A., & McGarigal, K. (2019). Metrics and models for quantifying ecological resilience at landscape scales. Frontiers in Ecology and Evolution, 7, Article 440. https://doi.org/10.3389/fevo.2019.00440
Deb, K. (2001). Multi-objective optimization using evolutionary algorithms. John Wiley & Sons.
Deb, K., & Gupta, H. (2005). Searching for robust Pareto-optimal solutions in multi-objective optimization. In C. A. Coello Coello, A. Hernández Aguirre, & E. Zitzler (Eds.), Evolutionary Multi-Criterion Optimization (pp. 150–164). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-31880-4_11
Dolanc, C. R., Safford, H. D., Dobrowski, S. Z., & Thorne, J. H. (2014). Twentieth century shifts in abundance and composition of vegetation types of the Sierra Nevada, CA, US. Applied Vegetation Science, 17(3), 442–455. https://doi.org/10.1111/avsc.12079
Leitmann, G. (1996). One method for robust control of uncertain systems: Theory and practice. In J. Doležal & J. Fidler (Eds.), System Modelling and Optimization (Vol. 32, Issue 1, pp. 64–74). Springer US. https://doi.org/10.1007/978-0-387-34897-1_6
Dong, S., Wang, H., Mostafavi, A., & Gao, J. (2019). Robust component: A robustness measure that incorporates access to critical facilities under disruptions. Journal of The Royal Society Interface, 16(157), 20190149. https://doi.org/10.1098/rsif.2019.0149
Druckenbrod, D. L., Dale, V. H., & Olsen, L. M. (2006). Comparing current and desired ecological conditions at a landscape scale in the Cumberland Plateau and Mountains, USA. Journal of Land Use Science, 1(2–4), 169–189. https://doi.org/10.1080/17474230601079480
Estes, B., & Gross, S. (2021). Stanislaus National Forest Climate Summary. Last accessed July 16, 2025, from https://acconsensus.org/wp-content/uploads/2021/08/StanisalusNationalForest_ClimateSummary__2021.pdf
Finney, M. A. (2001). Design of regular landscape fuel treatment patterns for modifying fire growth and behavior. Forest Science, 47(2), 219–228. https://doi.org/10.1093/forestscience/47.2.219
Finney, M. A. (2005). The challenge of quantitative risk analysis for wildland fire. Forest Ecology and Management, 211(1–2), 97–108. https://doi.org/10.1016/j.foreco.2005.02.010
Fotheringham, A. S., Densham, P. J., & Curtis, A. (1995). The zone definition problem in location-allocation modeling. Geographical Analysis, 27(1), 60–77. https://doi.org/10.1111/j.1538-4632.1995.tb00336.x
Gabrel, V., Murat, C., & Thiele, A. (2014). Recent advances in robust optimization: An overview. European Journal of Operational Research, 235(3), 471–483. https://doi.org/10.1016/j.ejor.2013.09.036
Gurobi. (2025). Gurobi Optimizer. Gurobi Optimization, LLC. Last accessed July 16, 2025, from https://www.gurobi.com/
Grüne, C. (2024). The complexity classes of Hamming distance recoverable robust problems. In J. A. Soto & A. Wiese (Eds.), LATIN 2024: Theoretical Informatics (pp. 321–335). Springer Nature Switzerland.
Hann, W. J., Shlisky, A., Havlina, D., Schon, K., Barrett, S. W., DeMeo, T. E., Pohl, K., Menakis, J. P., Hamilton, D., Jones, J., Levesque, M., & Frame, C. K. (2008). Interagency Fire Regime Condition Class (FRCC) Guidebook (Version 1.3.0). National Interagency Fuels Technology Team. Last accessed July 16, 2025, from https://npshistory.com/publications/fire/frcc-guidebook-2008.pdf
Hassani, H., Khodaygan, S., & Ghaderi, A. (2023). Bayesian reliability-based robust design optimization of mechanical systems under both aleatory and epistemic uncertainties. Engineering Optimization, 55(4), 543–563. https://doi.org/10.1080/0305215X.2021.2014828
Hengl, T., Mendes de Jesus, J., Heuvelink, G. B. M., Ruiperez Gonzalez, M., Kilibarda, M., Blagotić, A., Shangguan, W., Wright, M. N., Geng, X., Bauer-Marschallinger, B., Guevara, M. A., Vargas, R., MacMillan, R. A., Batjes, N. H., Leenaars, J. G. B., Ribeiro, E., Wheeler, I., Mantel, S., & Kempen, B. (2017). SoilGrids250m: Global gridded soil information based on machine learning. PLOS ONE, 12(2), e0169748. https://doi.org/10.1371/journal.pone.0169748
Henne, P. D., Elkin, C. M., Reineking, B., Bugmann, H., & Tinner, W. (2011). Did soil development limit spruce (Picea abies) expansion in the Central Alps during the Holocene? Testing a palaeobotanical hypothesis with a dynamic landscape model. Journal of Biogeography, 38(5), 933–949. https://doi.org/10.1111/j.1365-2699.2010.02460.x
Hildemann, M., & Verstegen, J. A. (2021). Quantifying uncertainty in Pareto fronts arising from spatial data. Environmental Modelling & Software, 141, 105069. https://doi.org/10.1016/j.envsoft.2021.105069
Jankowski, P. (1995). Integrating geographical information systems and multiple criteria decision-making methods. International Journal of Geographical Information Systems, 9(3), 251–273. https://doi.org/10.1080/02693799508902036
Keane, R. E. (2012). Creating historical range of variation (HRV) time series using landscape modeling: Overview and issues. In Historical Environmental Variation in Conservation and Natural Resource Management (pp. 113–127). Wiley. https://doi.org/10.1002/9781118329726.ch8
Keane, R. E., Holsinger, L. M., & Pratt, S. D. (2006). Simulating historical landscape dynamics using the landscape fire succession model LANDSUM version 4.0. https://doi.org/10.2737/RMRS-GTR-171
Keane, R. E., Hessburg, P. F., Landres, P. B., & Swanson, F. J. (2009). The use of historical range and variability (HRV) in landscape management. Forest Ecology and Management, 258(7), 1025–1037. https://doi.org/10.1016/j.foreco.2009.05.035
Keane, R. E., Holsinger, L., & Parsons, R. A. (2011). Evaluating indices that measure departure of current landscape composition from historical conditions. https://doi.org/10.2737/RMRS-RP-83
Labbé, M., Thisse, J.-F., & Wendell, R. E. (1991). Sensitivity analysis in minisum facility location problems. Operations Research, 39(6), 961–969. https://doi.org/10.1287/opre.39.6.961
LANDFIRE. (2016). LANDFIRE Vegetation Departure (VDep). LANDFIRE, US Department of the Interior; US Department of Agriculture. Last accessed July 16, 2025, from http://www.landfire/viewer
Lempert, R. J. (2019). Robust decision making (RDM). In Decision Making under Deep Uncertainty (pp. 23–51). Springer International Publishing. https://doi.org/10.1007/978-3-030-05252-2_2
Lempert, R., Popper, S., & Bankes, S. (2003). Shaping the next one hundred years: New methods for quantitative, long-term policy analysis. RAND Corporation. https://doi.org/10.7249/MR1626
Maier, H. R., Guillaume, J. H. A., van Delden, H., Riddell, G. A., Haasnoot, M., & Kwakkel, J. H. (2016). An uncertain future, deep uncertainty, scenarios, robustness and adaptation: How do they fit together? Environmental Modelling & Software, 81, 154–164. https://doi.org/10.1016/j.envsoft.2016.03.014
Marchau, V. A. W. J., Walker, W. E., Bloemen, P. J. T. M., & Popper, S. W. (Eds.). (2019). Decision making under deep uncertainty. Springer International Publishing. https://doi.org/10.1007/978-3-030-05252-2
Marler, R. T., & Arora, J. S. (2010). The weighted sum method for multi-objective optimization: New insights. Structural and Multidisciplinary Optimization, 41(6), 853–862. https://doi.org/10.1007/s00158-009-0460-7
McGarigal, K., & Marks, B. J. (1995). FRAGSTATS: Spatial pattern analysis program for quantifying landscape structure. https://doi.org/10.2737/PNW-GTR-351
McGaughey, R. J., & Carson, W. W. (2003). Fusing LIDAR data, photographs, and other data using 2D and 3D visualization techniques. USDA Forest Service, Pacific Northwest Research Station. Last accessed July 16, 2025, from https://www.fs.fed.us/pnw/olympia/silv/publications/opt/488_McGaugheyCarson2003.pdf
Moran, P. A. P. (1950). Notes on continuous stochastic phenomena. Biometrika, 37(1/2), 17–23. https://doi.org/10.2307/2332142
Murray, A. T. (2003). Site placement uncertainty in location analysis. Computers, Environment and Urban Systems, 27(2), 205–221. https://doi.org/10.1016/S0198-9715(02)00016-9
Murray, A. T., & Gottsegen, J. M. (1997). The influence of data aggregation on the stability of p‐median location model solutions. Geographical Analysis, 29(3), 200–213. https://doi.org/10.1111/j.1538-4632.1997.tb00957.x
Murray, A. T., Church, R. L., & Pludow, B. A. (2022). Location analytics for transitioning to fire resilient landscapes. In T. X. Bui (Ed.), Proceedings of the 55th Hawaii International Conference on System Sciences (HICSS-55) (pp. 5725–5733). University of Hawaii at Manoa. https://hdl.handle.net/10125/79989
NASA. (2020). NASADEM Merged DEM Global 1 arc second V001 [Data set]. NASA Jet Propulsion Laboratory; NASA Land Processes Distributed Active Archive Center (LP DAAC). https://doi.org/10.5067/MEASURES/NASADEM/NASADEM_HGT.001
Nayak, S. (2020). Linear programming. In Fundamentals of Optimization Techniques with Algorithms (pp. 9–70). Elsevier. https://doi.org/10.1016/B978-0-12-821126-7.00002-4
Nonaka, E., & Spies, T. A. (2005). Historical range of variability in landscape structure: A simulation study in Oregon, USA. Ecological Applications, 15(5), 1727–1746. https://doi.org/10.1890/04-0902
Ramesh, R., & Zionts, S. (2013). Multiple criteria decision making. In Encyclopedia of Operations Research and Management Science (pp. 1007–1013). Springer US. https://doi.org/10.1007/978-1-4419-1153-7_653
Rubio-Sanchez, M., Raya, L., Diaz, F., & Sanchez, A. (2016). A comparative study between RadViz and Star Coordinates. IEEE Transactions on Visualization and Computer Graphics, 22(1), 619–628. https://doi.org/10.1109/TVCG.2015.2467324
Sakawa, M., & Yano, H. (1990). Trade-off rates in the hyperplane method for multi-objective optimization problems. European Journal of Operational Research, 44(1), 105–118. https://doi.org/10.1016/0377-2217(90)90319-7
Schumacher, S., Bugmann, H., & Mladenoff, D. J. (2004). Improving the formulation of tree growth and succession in a spatially explicit landscape model. Ecological Modelling, 180(1), 175–194. https://doi.org/10.1016/j.ecolmodel.2003.12.055
Shavazipour, B., Kwakkel, J. H., & Miettinen, K. (2021). Multi-scenario multi-objective robust optimization under deep uncertainty: A posteriori approach. Environmental Modelling & Software, 144, 105134. https://doi.org/10.1016/j.envsoft.2021.105134
Shirabe, T. (2005). A model of contiguity for spatial unit allocation. Geographical Analysis, 37(1), 2–16. https://doi.org/10.1111/j.1538-4632.2005.00605.x
Snell, R. S., Peringer, A., & Bugmann, H. (2017). Integrating models across temporal and spatial scales to simulate landscape patterns and dynamics in mountain pasture-woodlands. Landscape Ecology, 32(5), 1079–1096. https://doi.org/10.1007/s10980-017-0511-1
Steele, B. M., Reddy, S. K., & Keane, R. E. (2006). A methodology for assessing departure of current plant communities from historical conditions over large landscapes. Ecological Modelling, 199(1), 53–63. https://doi.org/10.1016/j.ecolmodel.2006.06.016
Tanaka, K., & Sugeno, M. (1998). Introduction to fuzzy modeling. In Fuzzy Systems (pp. 63–89). Springer US. https://doi.org/10.1007/978-1-4615-5505-6_3
Tušar, T., & Filipič, B. (2015). Visualization of Pareto front approximations in evolutionary multiobjective optimization: A critical review and the prosection method. IEEE Transactions on Evolutionary Computation, 19(2), 225–245. https://doi.org/10.1109/TEVC.2014.2313407
USAFacts. (2022). Climate in Stanislaus County, California. USAFacts. Last accessed July 16, 2025, from https://usafacts.org/issues/climate/state/california/county/stanislaus-county
Zhang, J. (2021). Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey. WIREs Computational Statistics, 13(5), e1539. https://doi.org/10.1002/wics.1539
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