1. Background

Forest fires are a significant disturbance and damage agent in both natural and plantation forests. This is most evident in sub-tropical and arid regions that include some of the world’s most important plantation growing areas. Plantations in these regions are at considerable risk of fire damage. This is due to local climatic conditions, the evolutionary history of the plantation forestry species regularly used, and an increasing rural human population (Strydom and Savage, 2016). A myriad of other factors including global heating, the pre-eminence of invasive vegetation, and the variable quality of land management means that the probability of significant fires is likely to be increasing (Forsyth et al., 2010). The impact of the changing climate means that managers in areas that until recently would have deemed the risk of fire damage to be insignificant are now finding that they are faced with an increased threat of forest fires that can be existential for some commercial forestry businesses.

The area affected by forest fires each year varies significantly and the occurrence of large-scale extreme events that can lead to significant economic damage and loss of life is notably difficult to predict. So how should forest managers and valuers address the risk of forest fires and take action to manage the risk in an informed manner?

Given the timeframes of plantation forestry this exercise inevitably requires some kind of future forecasting of the likelihood of forest fires. Forecasting can take many forms and ranges in complexity from simply assuming an annual loss of a certain proportion of the estate to attempting to model future fire events based on detailed climatic models and site factors.

Catastrophic forest fires are rare events with a myriad of complex and interacting causal factors. This means that they are unlikely to be adequately predicted through a simplistic approach and attempts to predict their occurrence based on sophisticated climatic and environmental data are unlikely to provide practical predictions for managers. Alternative approaches make use of statistical methods to model the probability of fires of different sizes so that managers and valuers can better understand the probable range of impacts of the losses associated with possible future forest fires.

Researchers have developed several methods to better understand and predict the reoccurrence of rare and extreme events. The most appropriate modelling method depends on the availability of historical data. In this research we focused on methods that can quantify the likelihood of rarer extreme forest fire events occurring based on the patterns observed in forest fires in previous years. This was because the forest manager in our case study had maintained a historical register of forest fires occurring in their plantations covering several decades. With these data in-hand we believed that attempting to model the likelihood of extreme events could be a valuable exercise if an appropriate modelling approach could be identified.

Large fires are inevitably found within the upper tail of a cumulative probability distribution of a forest fire database or register. This means that conventional statistical approaches based on the mean or variance are inadequate for modelling these extreme events and alternatives must be used. Methods for modelling extreme values have been developed in a field of statistics called extreme value theory (EVT) (Coles et al., 2001) and have been applied to modelling forest fires and wind damage. Popular modelling techniques include the generalised extreme value (GEV) distribution, the generalised Pareto distribution (GPD), and the GEV with a Poisson point process (PP). Generally, the objective of extreme statistical model development is to estimate the m-year fire return level defined as the burned area that is expected to exceed a threshold once in a region, on average, during the m-year period. Coles et al. (2001) offers a detailed technical description of the topic.

Numerous examples can be found in the forest science literature of the application of these models to modelling forest fire and storm damage. We examined all published English language studies available on this topic and provide a list of some of the most relevant papers in the bibliography. Based on the findings of our review we selected the GPD as a suitable tool for analysis of forest fire datasets. This method offers practical information on return periods and expected future fires sizes.

2. Case study

To investigate the utility of GPD methods to describe a forest fire regime and to develop methods to integrate model outputs into a forest valuation framework we initiated a case study in partnership with one of our clients. The case study forest encompassed ~35 000 ha across more than 10 plantations in a highly fire prone environment^[1]. The forest manager maintained a record of fire damaged areas between 2000 and 2021. In total ~50 000 ha was burned in this period with fires ranging in size between 0.5 ha and ~10 000 ha.

2.1 Extreme Fire Model Development

The case study dataset was used to develop a GPD model. This model type is widely used in various fields for predicting how often catastrophic events are likely to occur (the return period) and the magnitude of catastrophic event for a given return period (the return level).

When fitting a GPD, a threshold (in our case fire size (ha)) must be defined above which events are included during GPD fitting. This decision is subjective but general guidance from EVT is to choose a threshold that is the initial point at which further increasing the threshold does not significantly alter the model’s scale and shape parameters. However, the threshold cannot be too high as this will limit the amount of data available for model fitting. Based on a graphical analysis, and a review of model diagnostics, a threshold of 150 ha was chosen for model development. There was convincing evidence that the shape and scale parameters stabilised at this level and sufficient fire events exceeded this threshold and could be used for model fitting across the 21-year time span. The model fitted the observed exceedance data reasonably well and the maximum likelihood estimates for shape and scale parameters (with standard errors) were 666.23 (179.8) and 0.42 (0.23). These values were used to provide an estimated average of the annual size of fire damage, assuming a fire event larger than the selected threshold, of 815.8 ha.

The model developed could be used to estimate the probability of exceedance for fires resulting in different amounts of damage (Figure 2‑1). The model estimated that the probability of fire damage of 250 ha or larger was 0.86, the probability for exceedance of 500, 1 000, 2 000, 5 000, and 10 000 ha were 0.62, 0.35, 0.15, 0.03, and 0.009 respectively. The estimated return period for a fire exceeding 5 000 ha was 16 years and for fires exceeding 7 000, 8 000, 9 000, and 10 000 ha were 30, 39, 50.5, and 62.8 years respectively.

Figure 2‑1. Estimated Probability of Exceedance for Fires of Different Sizes

2.2. Modelling the Distribution of Fire Damage

Estate woodflow models are typically used as a key input into forest valuations that use a discounted cash flow (DCF) methodology. Woodflows for the estate were modelled using the Remsoft Woodstock software. Woodstock uses linear programming to schedule harvesting and provide an approximation of wood availability based on a forest description, stand growth rates, and a set of constraints specified by the user. The software also provides a mechanism for simulating silvicultural events (e.g. thinning and pruning) and can be configured to model other events based on a user designated schedule.

The GPD model provided a probability of fires exceeding a certain threshold within a given simulation period. However, it remains impossible to know where and when a catastrophic forest fire might occur in the future. One way of examining the likely consequences of the expected catastrophic fire regime is to repeatedly simulate fire damage to the estate at the levels indicated by the GPD model whilst incorporating a random component introduced using a random number generator. This was then used to review the distribution of randomly assigned fire events on the estate woodflow and the subsequent impact on economic value of the estate.

We developed a method to simulate a fire regime whereby stands were randomly selected to experience fire damage guided by the output from the GPD model. To recognise the random nature of fire damage and the uncertainty of predicting future events we iteratively selected stands to be burned many times with stand selection based on a random number generator. In this manner our approach was similar to a Monte Carlo type simulation. This method is useful for examining the quantitative risk to an asset and is designed to evaluate the range of possible outcomes given a set of probabilities and constraints.

The developed GPD model indicated that the probability of fires damaging 100 ha in a given year had a probability of 1 and so it was assumed that in each year in all simulations fires of this magnitude would be recorded. The GPD model also showed that for a return period of 25 years (approximately equivalent to the rotation length) the return level was 6 332 ha. Therefore, it was assumed that a catastrophic fire of this magnitude would occur once per simulation with the timing and the location of the fire designated at random.

2.3. Incorporating Fire Damage into a Valuation Model

The woodflow model introduced in the previous section was used to estimate the effect of the fire regime on estate woodflow and value. The impact on forest value is of interest as this provides a measure of the financial losses associated with the fire regime. Using data provided by the forest manager, we created a Woodstock model designed to produce a set of woodflows and cash flows for the purpose of obtaining a valuation of the tree crop.

Estate yields were modelled using a set of stand-level and generic yield tables provided by the forest manager. Once a stand had been selected for burning by the simulator the yield tables were adjusted to reflect the reduced recovery following a canopy burning event. Yield losses were greatest in younger stands than in older stands. This was because more mature stands can more readily be salvaged following burning compared to younger stands which may be completely lost.

Incorporating the GPD model into Woodstock was complex. Linear programming is a deterministic modelling approach where no randomness is involved in the development of future model states. Several approaches to include future fire regimes into the model were tried, many of which had unforeseen and undesirable consequences. We found that the best approach was to run the Woodstock model without any simulated fires, giving an age-class distribution for each year of the model after all planting, thinning, and clearfelling activities had been simulated. The GPD model was used to select from these age-class distributions stands to create a schedule of future fire events. This schedule was then converted into a set of Woodstock actions and constraints to force the estate model to burn the specific stand in the year as determine by the GPD model.

The model was run 100 times for each different burning scenario outputted from the GPD model. This was designed to help us understand the impact of the inherent randomness of forest fires on the woodflow and cash flow generated from the Woodstock model. The model took around four minutes per run, so to complete the 100 runs for each scenario took approximately 6.7 hours of computer time.

2.4. Results

The forest manager’s costs and yield assumptions were used to quantify the impact of the fire regime on estate Net Present Value (NPV), and this was compared to a scenario where losses associated with fires were not simulated. This meant that we could draw some inference about the possible range of impacts on forest value by observing the distribution of NPV values produced. In addition to the mean NPV values the extremes of the distribution are also of significant interest. The value lost to fire in our simulations ranged between 2.4% and 11.1% of total NPV.

The simulation year in which the catastrophic fire occurred had a significant impact on estate value loss (Figure 2‑2). There is a clear trend showing that the value loss was least when the catastrophic fire occurred towards the end of the simulation. This finding was expected and is the result of the integration of the fire regime model with a discounted cash flow value methodology. Under this approach fire losses incurred early in the simulation have a disproportionately large effect on the estate value loss compared with those that occur towards the end of the simulation. This finding highlights why the Monte Carlo type simulation approach was required and why the development of a single deterministic model is inadequate.

Figure 2‑2: The Simulation Year Containing the Catastrophic Fire Event and the Percentage Estate Value Loss

The other component of the variation in estate value loss is the random allocation of stands selected to be burned. The effect of this component can be observed in Figure 2‑3.

Figure 2‑3: Density Plot Showing the Distribution of Estate Value Loss (%) for Each Grouping

3. Scenario analysis

The modelling framework described in the previous sections enabled the impact of the current fire regime on estate woodflows and value to be examined. By adjusting the input variables in a logical manner, the impact of various scenarios on the estate can be investigated.

To test this the following hypothetical scenarios were developed.

Scenario 1: Through doubling the amount spent on fire protection and firefighting the return level of catastrophic fires experienced during the simulation period can be reduced from a one in twenty-five-year fire (6 332 ha) to a one in ten-year fire (3 852 ha). All other inputs remain the same and the annual fire loss also remains.

Scenario 2: Through doubling investment in forest fire prevention the catastrophic fire regime cannot be changed but the annual fires (100 ha per year) can be mostly eliminated. Spending on fire prevention would remain the same as the base case under this scenario.

Scenario 3: Due to climatic changes the return level of the catastrophic fire experienced during the simulation period increased from a one in twenty-five-year fire (6 332 ha) to a one in forty-year fire (8 026 ha). The annual small-scale fire regime and the spending on fire prevention and firefighting remain the same as for the base case scenario.

Scenarios 1–3 were simulated alongside the “base” scenario which reflects the current fire regime. The distribution of the simulated changes in estate value associated with each scenario are shown in Figure 3‑1.

As expected, Scenarios 1 and 2 led to an increase in NPV as the amount of fire damage was reduced when compared to the historical fire regime (base)—shown by the leftward shift in the kernel density plots for Scenarios 1 and 2.

Under Scenario 3 an increase in the amount of damage caused by the fire regime resulted in significantly greater impacts on estate value. The rightward shift in the kernel density plot for Scenario 3 supports this. The long tail of Scenario 3’s density plot towards the higher values also indicates that for a substantial subset of simulations the estate value sacrifice was significantly higher than for the other scenarios. This would be expected if the catastrophic fire event occurred early in the simulation cycle and resulted in the loss of the more valuable components of the crop.

Figure 3‑1: Distribution of Percentage Change in Estate Value for Each Scenario

4. Conclusions

In this project the statistical methods of extreme value theory were used to develop a GPD model that describes the fire regime experienced for a plantation estate in a fire prone region. The model was used to describe the likely return period and return level for catastrophic fires within the estate. The GPD model was then used to stochastically simulate a fire regime that could be integrated within a woodflow model that underpins forest valuation in a realistic manner. Methods were developed for integrating the fire regime within the Remsoft Woodstock estate modelling software so that the impact of forest fires on estate value could be calculated and compared to a scenario where no value was lost due to fires. Scenario analysis was then used to compare the impacts of adjusting the fire regime on forest value.

Substantial progress has been made towards developing methods for quantifying the impact of forest fires on estate value in a probabilistic manner where the range of potential impacts can be explored. However, the approach developed has the following limitations.

1. The base model is based on historic fire patterns and so the fitting dataset is deemed to be stationary. This is probably not realistic due to the effects of climate change and changing vegetation management patterns. Future fire regimes may not be the same as those experienced in the past.
2. Integration of the GPD fire model with Woodstock was a challenging, complex, and time-consuming process. The stochastic nature of fire modelling and deterministic nature of Woodstock linear programming solution techniques means that it is probably not the most suitable tool for this type of work.

For more information on this research contact Margules Groome.

The full text of the article was previously published by the New Zealand Institute of Forestry in the New Zealand Journal of Forestry, Volume N.Z.J.For. 2023, Issue N.Z.J.For. 68(1) 2023, pp 10-16, May 2023.

5. Bibliography

Bermudez, P. de Z., Mendes, J., Pereira, J.M.C., Turkman, K.F., Vasconcelos, M.J.P., 2009. Spatial and temporal extremes of wildfire sizes in Portugal (1984–2004). Int. J. Wildland Fire 18, 983–991. https://doi.org/10.1071/WF07044

Coles, S., Bawa, J., Trenner, L. and Dorazio, P., 2001. An introduction to statistical modeling of extreme values, Springer series in statistics. Springer, London ; New York.

Ferro, C.A.T., Segers, J., 2003. Inference for clusters of extreme values. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 65, 545–556. https://doi.org/10.1111/1467-9868.00401

Forsyth, G.G., Kruger, F.J. and Le Maitre, D.C., 2010. National Veldfire Risk Assessment: Analysis of Exposure of Social, Economic and Environmental Assets to Veldfire Hazards in South Africa 102.

Gilleland, E., Katz, R.W., 2016. extRemes 2.0: An Extreme Value Analysis Package in R. Journal of Statistical Software 72, 1–39. https://doi.org/10.18637/jss.v072.i08

Hefferman, J.E., Stephenson, A.G., 2018. ismev: An Introduction to Statistical Modeling of Extreme Values.

Holmes, T.P., Huggett, R.J., Westerling, A.L., 2008. Chapter 4 Statistical Analysis of Large Wildfires. In The Economics of Forest Disturbances. Springer, London ; New York.

Jiang, Y., Zhuang, Q., 2011. Extreme value analysis of wildfires in Canadian boreal forest ecosystems. Canadian Journal of Forest Research. 41, 1836–1851. https://doi.org/10.1139/x11-102

Joseph, M.B., Rossi, M.W., Mietkiewicz, N.P., Mahood, A.L., Cattau, M.E., Denis, L.A.S., Nagy, R.C., Iglesias, V., Abatzoglou, J.T., Balch, J.K., 2019. Spatiotemporal prediction of wildfire size extremes with Bayesian finite sample maxima. Ecological Applications 29, e01898. https://doi.org/10.1002/eap.1898

Moore, J.R., Manley, B.R., Park, D., Scarrott, C.J., 2013. Quantification of wind damage to New Zealand’s planted forests. Forestry: An International Journal of Forest Research 86(2), 173–183. https://doi.org/10.1093/forestry/cps076

Strydom, S. and Savage, M.J., 2016. A spatio-temporal analysis of fires in South Africa. South African Journal of Science, 112(11-12), pp.1-8. https://doi.org/10.17159/sajs.2016/20150489

[1] The study location is not referenced here to protect client confidentiality.