Zone Allocation Model

Zone Allocation Model Description of Modeling Approach With Examples From TFL-48 Review Draft Mark Boyland and Ralph Wells 1 1 Centre For Applied Conservation Research, Faculty of Forestry University of British Columbia, 2424 Main Mall, Vancouver BC, V6T1Z4, Canada Revision: March 31, 2005

Contents Introduction... 1 Defining Zones... 1 Preservation and Habitat Objectives... 1 Spatial Targets... 2 Priority Objectives... 2 Methods... 3 Simulated Annealing Algorithm... 3 Objective Function... 5 Data Preparation... 8 Preliminary Results... 13 Literature Cited... 18 Figures Figure 1. ZAM simulated annealing algorithm flow chart.... 4 Figure 2. Penalty cost functions... 7 Figure 3: TFL-48 BEC zones (used for representation targets)... 8 Figure 4: TFL-48 age class used for age priority objective... 9 Figure 5: TFL-48 operability used for operability priority objective.... 10 Figure 6: TFL-48 productivity used for productivity priority objective... 11 Figure 7. Hexagonal grid overlaid with resultant map.... 12 Figure 8: ZAM sample output for TFL-48.... 14 Figure 9: ZAM result for representation and snakeweight = 0.... 15 Figure 10: ZAM result for representation and snakeweight = 1000.... 16 Figure 11: ZAM result for representation and snakeweight = 10,000.... 17 i

Introduction This report describes the Zone Allocation Model (ZAM) and outlines database development and model runs undertaken for Tree Farm Licence (TFL) 48. TFL-48 is held by Canfor Ltd, and is located in the western portion of the Dawson Forest District, in northeastern British Columbia. TFL-48 incorporates Engelmann Spruce-Subalpine Fir (ESSF), Sub-Boreal Spruce (SBS), Boreal White and Black Spruce (BWBS) and Alpine Tundra (AT) biogeoclimatic (BEC) zones (Meidinger and Pojar 1991). TFL-48 is approximately 644,000 ha, of which 89% are productive forest (Baker 2001). The TFL currently has an annual allowable cut of 580,000 cubic metres (Baker 2001) ZAM is a spatial, forest-level planning tool designed to help planners zone forest activities (Boyland et al. 2004; Boyland and Moy 2005). All lands are allocated into one of three zone types, with a Simulated Annealing (SA) algorithm that searches for allocations that meet objective targets. Multiple objectives are included in the model through an objective function that sums the deviances from objective targets into a solution score. The basic objectives are to (i) distribute lands into zones according to zone percent targets, (ii) organize the zones into regions with basic size and shape objectives, (iii) allocate land into zones according to ecological representation targets, and (iv) allocate land into zones according to priority objective targets. Defining Zones While any number of zone types could potentially be accommodated, a three zone system can incorporate most objectives (Binkley 1997; Beese et al. 2001). The three zones include harvesting and ecological preservation to varying degrees. A Timber zone includes mostly economic objectives, maintaining a major portion of the landbase at or below a maximum sustained yield rotation age. Biodiversity values are included within an Old Growth zone, where maintaining old forests is a primary goal. A third, Habitat zone, is included to incorporate those objectives that are not fully achieved in the other two zones. Examples of activities appropriate for the Habitat zone are wood products from long rotations or modified partial harvesting systems designed to retain habitat structures (e.g., large snags, large downed wood). These three zones are designed to separate the truly competing practices (harvest vs. preservation) in the Timber and Old Growth zones while grouping mildly competing objectives in the Habitat zone. The sections below present the zone criteria and rationale for zone building, and are listed in order of importance. Preservation and Habitat Objectives 1. Ecosystem representation. Preservation and habitat objectives require that each ecosystem type be equally represented in the Old-Growth zone. This prevents important habitats from being completely allocated into the Timber zone, where intensive harvesting would not provide adequate habitat needs. 1

Ecosystem representation is modelled by distributing each Biogeoclimatic Ecological Classification (BEC; Green and Klinka 1994) variant into zone types proportional to the zone percentage target. For example, if the Old-Growth zone target is 10% of the landscape, 10% of each BEC variant should be allocated into the Old-Growth zone. Spatial Targets 2. Zone region Size and Shape. Each zone may be allocated to the landbase in several different areas for example, there could be ten different Old-Growth regions that sum to meet the 10% Old-Growth zone target. The minimum size of regions is set to 1,000 ha or larger to prevent intermixing of incompatible uses in small pockets across the landscape. The shape of zone regions is also controlled, to prevent long winding regions that have little interior area. Priority Objectives Priority objectives are areas identified as having special value for a certain zone types. Where possible within the overall objectives of ecological representation into zone regions of appropriate size and shape, the priority is to allocate these areas into particular zones. 3. Forest age. The Old-Growth zone requires older stands to provide habitats not found in the Timber zone. Age class distribution is controlled by forcing more area of mature stands into the Old-Growth zone. 4. Operability. Some areas in the forest tenure are difficult to access due to terrain and other factors and are consider inoperable. These areas should be assigned to the Old- Growth zone wherever possible. 5. Productivity. Low site index stands are often only marginally economically viable, however present good habitat opportunities after developing old growth features through many years of growth. Allocating these stands to the Old-Growth zone improves the economic opportunities in the Timber zone, while retaining habitat within the Old- Growth zone. 2

Methods Simulated Annealing Algorithm The Simulated Annealing algorithm works by slightly modifying an existing solution, and considering whether to keep the modified solution or return to the previous solution (Figure 1). If the new solution is better than the old solution, the new solution is always accepted, and the cycle is repeated with the next modification. If the new solution is worse than the old solution, the new solution could still be retained, subject to a random selection process. The algorithm randomly chooses whether to accept or reject worse solutions based on a random selection process weighted by how much worse the new solution is compared to the old solution and also the current point in the annealing process. Small negative steps are accepted more frequently than large negative steps, and negative steps are accepted more frequently in the beginning of the annealing process than at the end. In the zoning model, modifications to the solution are made by randomly selecting a tile, and assigning it a new, random zone type. By accepting better solutions, the algorithm migrates the solution towards lower scores (better solutions). However, situations often occur where there are no possible changes to a single tile that could make the solution better, however there exists completely different solutions that are better. In order to find the better solutions, many tiles must be changed that temporarily result in worse scores, but will eventually result in improvements. By sometimes allowing changes that result in worse solutions, the algorithm can cross over poorly scored areas to find (hopefully) the best solution. 3

Create initial solution; Calculate score Make small change to solution Re-calculate score NO New solution better? YES Accept anyway? Keep solution NO YES Remove change, return to previous solution. Stop? NO Figure 1. ZAM simulated annealing algorithm flow chart. 4

Objective Function The objective function sums penalties for deviations from objective targets. Objective penalties are created by multiplying the objective deviation by a penalty weight. Individual objective penalty calculations are explained in eqs. 3 10. The objective function is defined by eq. 2. [ 2] K Minimize = 6 Z s P h h= 1 Z s is the penalty function score for solution s. P h is the penalty associated with objective h. zones regions T A =, k= 1 i= 1 Tk = 0, k ki [ 3] K P 1 wk ( Tk Aki ) A ki < Tk [ 4] K P1 A ki Tk The P1 penalty controls the size of each region, penalizing regions smaller than a minimum size parameter (eqs. 3 and 4). Each zone type has a target region area, with no penalty assigned for regions that exceed the target (eq. 4). T k is the target size for each region of zone type k; A ki is the area of each region i of zone type k; w k is the weight assigned to the size objective for each zone k. An example region size penalty cost function is illustrated in Fig. 3a. This curve shape was used to strongly discourage very small zones. zones [ 5] K P2 = k= 1 regions i= 1 B w k A ki ki The P2 penalty controls the shape of each region, penalizing regions as they depart from a circular shape (eq. 5). B ki is the area of the bounding circle of region i of zone type k; w k is the weight assigned to the shape objective for each zone k. The bounding circle is the smallest circle that completely encompasses the region. Regions with a linear shape will have a higher ratio of bounding circle to region area compared with regions with globular shapes. This objective does not produce circular regions but does pressure regions away from linear shapes. As with all the objective penalty scores, the magnitude of the pressure is controlled by the weight parameter. A shape penalty cost function is shown in Fig. 3b. The P3 penalty controls the distribution of representation objectives (eq. 6). T kj is the area target for each representation target j of zone type k; w kj is the weight assigned to the representation objective for each representation target j of zone type k. Representation targets have symmetrical penalty functions, accruing penalties as the achieved amount diverges in either direction from the target (Fig. 3c). 5

[ 6] K P zones priorityobjs 3 = wkj ( Akj Tkj ) k= 1 j= 1 The P3 penalty controls the distribution of representation objectives (eq. 6). T kj is the area target for each representation target j of zone type k; w kj is the weight assigned to the representation objective for each representation target j of zone type k. Representation targets have symmetrical penalty functions, accruing penalties as the achieved amount diverges in either direction from the target (Fig. 3c). [7] K zones priorityobjs P 4 = wkm ( T A ), km km A km< T km k= 1 m= 1 [ 8] K P4 = 0, A km T km P4 controls the distribution of priority objectives (eqs. 7 and 8). A km is the area for each priority objective target m of zone type k; T km is the area target for each priority objective target m of zone type k; w km is the weight assigned to the priority objective, for each region, for each priority objective target m of zone type k. P4 = 0 if the area is greater than or equal to the target (eq. 8). Priority objectives have asymmetrical penalty functions, accruing a penalty only when a minimum target is not achieved (Fig. 3d). There are four priority objectives in this analysis: age, preservation proposals, tenure, and production. [ 9] K P zones 5 = wk ( Tk Ak ) k= 1 P5 controls the total amount of area in each of the zone types (eq. 9). A k is the area in each zone type k; T k is the area target of each zone type k. Like the representation targets, the zone area penalty function is symmetrical around the target (Fig. 3e). P5 = 0 if the target proportion of the landbase is allocated to each of the zones. [ 10] K P userid 6 = wn ( Tn An ) n= 1 P6 controls a user-designated zone type for specified areas (eq. 10). The userid field marks lands that must be allocated to a certain zone type; A n is the area in userid n; T n is the target for userid n; wn is the weight assigned for userid n.by assigning a very high weight, the model will always allocate the land into the desired zone type. 6

M ax a) M ax b) Penalty cost 0 50 100 150 200 Shape ratio (%total) M ax c) M ax d) Penalty cost Penalty cost 0 25 50 75 100 Re pre se ntation (% total) 0 25 50 75 100 Priority obje ctives (%total) M ax e ) Penalty cost Penalty cost 0 5 10 15 20 Re gion size (x10000 ha) 0 25 50 75 100 Zone allocation (% total) Figure 2. Penalty cost functions: (a) region size penalty (10 000-ha target shown); (b) region shape penalty; (c) representation penalty (25% target shown); (d) priority objective, the x axis is scaled to 100% (65% target shown); and (e) zone allocation (65% target shown). 7

Data Preparation A resultant database was prepared for TFL-48 that included attribute data required to model zone objectives. Preliminary zone objectives were based on BEC zones (for ecosystem representation), age class, operability and productivity (Figures 4 8). Figure 3: TFL-48 BEC zones (used for representation targets). 8

Figure 4: TFL-48 age class used for age priority objective. 9

Figure 5: TFL-48 operability used for operability priority objective. 10

Figure 6: TFL-48 productivity used for productivity priority objective. 11

Tiles To create tiles, a hexagon grid was overlaid on the GIS resultant database, with the resulting polygons grouped by hexagonal tile. While the resultant dataset appears similar to raster cells, there is no loss of precision that often occurs during a raster conversion. Raster maps use a single data description for each cell, often taking an average value of polygons comprising the cell. The tiles used in this dataset are containers for the polygons within their boundaries, holding the exact areas and data fields for each polygon within the tile (Figure 7). Figure 7. Hexagonal grid overlaid with resultant map. Changing the size of the hexagons changes the resulting number of hexagons containing polygons. Reducing the number of tiles reduces the number of possible moves, and the problem complexity. However, increasing the number of tiles increases flexibility. It is expected that better solutions are possible with a finer spatial resolution, however this could be offset by the increased complexity of larger problems. For TFL-48, a tileset was created using a 42 ha tile size that resulted in approximately 14,000 tiles. 12

Results In order to evaluate ZAM algorithms and assess their relative influence on the spatial distribution of zones, initial runs were undertaken for TFL-48 (Figure 8). In one example of evaluation runs, we set representation targets only (65% Timber Zone; 35% Habitat Zone; 10% Old-growth Zone targets for each BEC zone). Initially the snakeweight algorithm was applied (see Boyland and Moy 2005 for description of snakeweight), with an increasing snakeweight factor (Figure 9 10). It was found that the snakeweight factor could be increased to 10,000 (but no further) without appreciably affecting the representation target penalties. Further runs are planned to continue the evaluation of spatial algorithms intended to control the size and shape of zone patches (described in more detail in Boyland and Moy 2005). In addition, runs will incorporate other priority objectives (including age, operability, productivity and representation). After these evaluative runs are completed, priority functions will be reviewed and revised, in consultation with Canfor partners, to reflect management objectives and local conditions. 13

Figure 8: ZAM sample output for TFL-48. Conservation zone is analogous to old-growth zone. 14

Figure 9: ZAM result for representation (65% Timber Zone; 35% Habitat Zone; 10% Old-growth Zone) and snakeweight = 0. 15

Figure 10: ZAM result for representation (65% Timber Zone; 35% Habitat Zone; 10% Old-growth Zone) and snakeweight = 1000. 16

Figure 11: ZAM result for representation (65% Timber Zone; 35% Habitat Zone; 10% Old-growth Zone) and snakeweight = 10,000. 17

Literature Cited Baker, K. 2001. Tree Farm Licence 48 Canadian Forest Products Ltd Rationale for Annual Allowable Cut (AAC) Determination. British Columbia Ministry of Forests. Victoria, B.C. Beese, W.J., Dunsworth, G., and Perry, J. 2001. The Forest project: three-year review and update. Ecoforestry 16(4):10-17. Binkley, CS. 1999. Ecosystem management and plantation forestry: new directions in British Columbia. New Forests 18:75-88. Boyland, M, J. Nelson and F.L. Bunnell. 2004. Creating land allocation zones for forest management: a simulated annealing approach. Can. J. For. Res. 34:1669-1682. Boyland, M and A. Moy. 2005. ZAM Software Manual. Centre for Applied Conservation Research, Vancouver, B.C. Meidinger, D. and J. Pojar (eds.). 1991. Ecosystems of British Columbia. B.C. Ministry of Forests, Research Branch Special Report, Series 6. Victoria, B.C. Nelson, J.D. 2001. Assessment of harvest blocks generated from operational polygons and forest cover polygons in tactical and strategic planning. Can. J. For. Res. 31:682-693 18