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Info-Gap Decision Theory for Assessing the Management of Catchments for Timber Production and Urban Water Supply

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Abstract

While previous studies have examined how forest management is influenced by the risk of fire, they rely on probabilistic estimates of the occurrence and impacts of fire. However, nonprobabilistic approaches are required for assessing the importance of fire risk when data are poor but risks are appreciable. We explore impacts of fire risk on forest management using as a case study a water catchment in the Australian Capital Territory (ACT) (southeastern Australia). In this forested area, urban water supply and timber yields from exotic plantations are potential joint but also competing land uses. Our analyses were stimulated by extensive wildfires in early 2003 that burned much of the existing exotic pine plantation estate in the water catchment and the resulting need to explore the relative economic benefits of revegetating the catchment with exotic plantations or native vegetation. The current mean fire interval in the ACT is approximately 40 years, making the establishment of a pine plantation economically marginal at a 4% discount rate. However, the relative impact on water yield of revegetation with native species and pines is very uncertain, as is the risk of fire under climate change. We use info-gap decision theory to account for these nonprobabilistic sources of uncertainty, demonstrating that the decision that is most robust to uncertainty is highly sensitive to the cost of native revegetation. If costs of native revegetation are sufficiently small, this option is more robust to uncertainty than revegetation with a commercial pine plantation.

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Acknowledgments

Adrian Manning kindly assisted in the collection of water price and other data that underpinned the analyses presented in this article. We are grateful to Mark Burgman, Yakov Ben-Haim, and Colin Thompson for introducing us to info-gap decision theory, and to Yakov Ben-Haim, Yohay Carmel, Atte Moilanen, and an anonymous reviewer for comments on the manuscript. This research was supported by the Baker Foundation’s funding of ARCUE and grants by the Australian Research Council.

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Correspondence to Michael A. McCarthy.

Appendix

Appendix

Mathematical Detail of Info-gap Model

The uncertainty models for mean fire interval m and the increase in water yield w can be written as

$$ U_m(\alpha,{\tilde m})\,=\,\{m:\max[0,(1-\alpha/2){\tilde m}]\,\le \,m \le(1+\alpha /2){\tilde m}\}\,\alpha\,\ge 0, $$

and

$$ U_w(\alpha,{\tilde w})\,=\,\{w:(1-\alpha)\,{\tilde w}\,\le\,w\,\le\,(1+\alpha)\,{\tilde w}\}\,\alpha\,\ge\,0. $$
(3)

The expected value of the timber from the pine plantation was obtained from a simulation model of fire occurrence and forest harvesting over multiple rotations (Fig. 1). The expected value was expressed as the present net value (per hectare) by discounting future revenues and costs geometrically, such that the present value (pv) of a given revenue R (or cost if R is negative) that is obtained t years into the future is

$$ {\rm pv}\,=\,R\,/\,(1+r)^t, $$
(4)

Where r is the annual discount rate (we used values within the range 0.02–0.06). The expected costs and revenues over the next 100 years were evaluated and summed to determine the net present value of the two management options. The simulation model was then used to establish the relationship between the expected present value of timber and the average interval between fires (m).

The amount of water flowing from the catchment depended on the revegetation option. The annual flow of water (in mm equivalent rainfall) when the landscape was planted with pines was equal to A, whereas it was equal to A + w when planted with native vegetation, where w is the (uncertain) increase in water yield that is achieved revegetated with native plants (as defined above). Therefore, the extra volume of water obtained from the catchment when revegetated with native plants is 10w kL/ha/year. Given the value of water is $0.50/kL, the extra value of this water is $5w/ha/year. The flow of water given by A is obtained regardless of the management option that is employed, so it is ignored in subsequent calculations when comparing revegetation with pines and native plants.

Rather than trying to maximize the expected value, the info-gap approach determines robust management options that guarantee a minimally tolerable level of performance EVc. The optimal management strategy is chosen such that it maximizes the reliability of achieving this satisfactory outcome. This is achieved by determining, for each of the management options, how wrong we can be about the models (specifically the model parameters in this case) but still satisfy our performance requirement EVc. This maximal degree of error is termed robustness. The best management strategy is the one that has the greatest robustness (ie., lets us be maximally wrong).

Once the expected value of each management strategy (EV[pine] and EV[native]) and the uncertainty model for the parameters are defined, we can then write the robustness functions \( {\hat \alpha} \) for performance requirement EVc when revegetating with native plants or pines:

$$ {\hat \alpha}({\rm native,EV_c})\,=\,\max\left[\alpha:\mathop{\rm \min}_{w \in U_w(\alpha,{\tilde w})}\,{\rm EV}[{\rm native}]\ge {\rm EV_c}\right] $$
(5)
$$ {\hat \alpha}({\rm pine,EV_c})\,=\,\max\left[\alpha:\mathop{\rm \min}_{m \in U_m(\alpha,{\tilde m})}\,{\rm EV}[{\rm pine}]\ge {\rm EV_c}\right] $$
(6)

The robustness function \( {\hat \alpha}(j,{\rm EV_c}) \) (j = native or pine) is the maximum value of α (largest horizon of uncertainty) such that the minimum expected value of EV[j] given uncertainties in m and w is greater than or equal to the performance requirement EVc. The robustness function is the largest degree of uncertainty that still guarantees an acceptable outcome. We then choose the option (native or pine) such that the robustness is maximized for a given performance requirement. The options can be compared graphically by plotting the robustness versus the performance requirement.

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McCarthy, M.A., Lindenmayer, D.B. Info-Gap Decision Theory for Assessing the Management of Catchments for Timber Production and Urban Water Supply. Environmental Management 39, 553–562 (2007). https://doi.org/10.1007/s00267-006-0022-3

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