Advances and emerging issues in national forest inventories

Abstract

National forest inventories (NFIs) have a long history, although their current major features date only to the early years of the twentieth century. Recent issues such as concern over the effects of acid deposition, biodiversity, forest sustainability, increased demand for forest data, international reporting requirements and climate change have led to the expansion of NFIs to include more variables, greater diversity in sampling protocols and a generally more holistic approach. This review focuses on six selected topics: (1) a brief historical review; (2) a summary of common structural features of NFIs; (3) a brief review of international reporting requirements using NFI data with an emphasis on approaches to harmonized estimation; (4) an overview of inventory estimation methods that can be enhanced with remotely sensed data; (5) an overview of nearest neighbors prediction and estimation techniques; and (6) a brief overview of several emerging issues including carbon inventories in developing countries and use of lidar data. Although general inventory principles will remain unchanged, sampling designs, plot configurations and measurement protocols will require modification before they can be applied in countries with tropical forests. Technological advances, particularly in the use of remotely sensed data, including lidar data, have led to greater inventory efficiencies, better maps and accurate estimation for small areas.

Keywords:

• estimation
• harmonization
• interference
• k-nearest neighbor
• lidar
• remote sensing

Introduction

Strategic national forest inventories (NFIs) are conducted by at least 40 countries representing 2.4 billion hectares of forest, more than half the forested area of the earth (Tomppo et al., 2010a, Preface). Thus, a comprehensive review of the history, advances and emerging issues for the entire international forest inventory community is an impossible task for a relatively short journal article. As a result, this review focuses on only selected areas: (1) a brief historical review with emphasis on recent issues that have shaped current approaches to implementing NFIs; (2) a summary of common, structural features of NFIs, albeit with a large diversity of operational implementations; (3) a brief review of international reporting requirements using NFI data with emphasis on approaches to harmonized estimation; (4) an overview of inventory estimation methods that can be enhanced with remotely sensed data; (5) an overview of nearest neighbors prediction methods; and (6) a brief overview of several emerging issues.

History

The history of forest inventories traces back to at least the end of the Middle Ages, when intensive use of forest resources led to wood shortages and forced users to begin forest planning (Loetsch & Haller, 1973; Gabler & Schadauer, 2007). These inventories focused on forest area and crude estimates of what is now characterized as growing stock volume. In addition, they were usually local in scope, they emphasized assessments of timber resources for specific purposes, and they were often conducted by timber users (Loetsch & Haller, 1973; Davis et al., 2001). However, such inventories were soon acknowledged to be inadequate for compiling national forest information for purposes of formulating national forest policy; thus, NFIs were initiated. NFIs have different histories in different countries, but some type of forest information has been collected in both European and North American countries since the nineteenth century (von Berg, 1858; LaBau et al., 2007). Early NFIs were very much commodity driven with emphasis on acquiring information on forest area, volume and increment of growing stock, and the amount of timber. In addition, they addressed issues of age, size and species structures of forests, silvicultural status, and necessary cutting and silvicultural regimes (Ilvessalo, 1927). Their primary purpose was to provide information for forestry authorities, timber users and planners who developed national forest policies.

Multiple issues have emerged in recent decades to cause re-evaluation of the traditional role of forest inventories. First, as a result of declines in sensitive forest ecosystems in the 1980s, forest health monitoring programs were developed, some of which were integrated into NFIs. Second, the role of forests in providing non-wood goods and services such as wildlife habitat, recreational opportunities and contributions to water quality has received increased attention in recent years, particularly in urbanized societies. Third, human-induced habitat losses and tropical deforestation have accelerated the rate of species extinctions and led to international regimes such as the Convention on Biological Diversity (CBD, 2009) which, in turn, led to forest biodiversity monitoring programs in the early and mid-1990s. Fourth, processes such as the Ministerial Conference on the Protection of Forests in Europe (MCPFE, 2009) and the Montréal Process (2005) were established with goals of assessing forest sustainability criteria and indicators. Fifth, the combined effects of the use of fossil fuels, deforestation in the tropics and farming have produced increased levels of atmospheric carbon dioxide (CO₂) and other greenhouse gases. In response, many countries joined the United Nations Framework Convention on Climate Change (UNFCCC) to consider actions to reduce these effects and to cope with the expected global temperature increases (UNFCCC, 2009). Of particular importance is that forest is a land-use category whose resources can be managed to produce a positive effect on the greenhouse gas balance (Cienciala et al., 2008). The combined effects of all these issues have cast a new light on the role of forests and created new expectations for both forestry and forest inventories.

Importantly, the changing roles of forests have substantially altered demands for forest information. Because NFIs are the primary source of forest information for all these purposes, the scope of NFIs has broadened accordingly, resulting in the introduction of a wide variety of new variables and corresponding design modifications. Today, NFIs reflect a wider and more holistic perspective on global forest resources, their management and uses. This perspective is expressed via seven broad thematic elements of sustainable forest management: (1) extent of forest resources and their contribution to the global carbon cycle; (2) forest health and vitality; (3) forest biological diversity; (4) productive functions of forests; (5) protective functions of forests; (6) socioeconomic functions of forests; and (7) legal, policy and institutional frameworks related to forests (Tomppo et al., 2010a).

Structural features of national forest inventories

Two NFI approaches are used, one based on data acquired using statistical sampling, and the other based on data aggregated from stand-level management inventories. Although the evidence is now clear that the latter approach tends to underestimate forest area and growing stock volume (Tomppo et al., 2001; Adermann, 2010), systematic forest assessments based on statistical sampling methods began only in the twentieth century. The first sample-based NFIs were initiated in the Nordic countries: 1919 in Norway, 1921 in Finland and 1923 in Sweden. In the United States of America sample-based inventories were initiated in 1928. Today, the NFIs of most countries are sample based. A brief summary of the sampling design, plot configuration and inventory variable features of NFIs based on the detailed results reported by Tomppo et al. (2010a, Chapter 2) follows.

Sampling designs and plot configurations

NFIs have exhibited a progressive evolution towards statistical sampling techniques, with the majority of countries now using probability sampling designs (Tomppo et al., 2010a, Table 2.3). With these designs, each population unit has a positive, known probability of selection into the sample, and a randomization procedure is used to select the particular units for the sample. Probability sampling designs are of multiple forms. Most NFIs use systematic sampling based on two-dimensional grids whose starting points are randomly selected. Grid spacings range from 0.5 to 20 km, although most are in the range 1.4–4.0 km. Other countries use tessellated sampling designs in which large areas are tessellated with regular polygons and plot locations are randomly selected within the polygons. In addition, stratified sampling is often used when different sampling intensities are desired for different land uses and land covers, different climatic and topographical regions, or regions of different interest or priority. Strata are most often defined using maps constructed using a combination of political, biophysical and remotely sensed data, and forest attribute predictions such as proportion forest cover, forest type and volume.

NFIs now generally use detached sample plots of which all, or at least some, are permanent to facilitate estimation of change and trends as required for sustainability assessments. In addition, the plots are often configured in clusters for purposes of logistical efficiency. The number of plots per cluster is often four, although it may be as large as 18. The shapes and sizes of sample plots vary enormously among NFIs. The inventories of nearly all countries with boreal and temperate forests use circular plots with multiple concentric components for which smaller trees are measured only on the smaller components. However, square and rectangular plots are also used, and for some variables transect and Bitterlich sampling are used.

The diversity of sampling approaches and plot sizes and shapes has increased recently to accommodate the need for more diverse forest information. For example, transects may be established within large circular plots for assessing deadwood, and soil pits may be used to assess soil carbon. Thus, NFIs whose sole objectives previously related to estimation of timber resources have given way to NFIs that encompass a host of outputs including ground vegetation, deadwood, biodiversity and soil information, in addition to timber information.

Inventory cycles are typically either 5 years or 10 years. Traditionally, inventory cycles were achieved by conducting complete periodic inventories over 1–2 years and then repeating the inventories 5–10 years later. Increasingly, however, inventory cycles in Europe and North America are achieved by measuring 10–20% of plots each year. If the 10–20% of plots measured annually are systematically distributed over entire regions, then current estimates can be obtained in any year and the effects of catastrophic events such as hurricanes, wind and ice storms, and pest and disease outbreaks can be rapidly assessed.

Inventory variables

The two most common NFI variables for which population parameters are estimated are forest area and growing stock volume (Tomppo et al., 2010a, Tables 2.1 and 2.2) Priority is given to estimates of the current status of these parameters and change in these parameters between inventories. All countries assess forest area, and most base their definitions of forest on percentage canopy cover and minimum patch area, with many also using minimum area width and minimum tree height at maturity. Canopy cover thresholds range from 10% to 50%, with the smaller threshold being the most common. Methods for assessing forest area fall into four categories: (1) assessment of ground plots or ground sampling points; (2) interpretation of plot locations using aerial photographs; (3) counting forest and non-forest points on systematic grids superimposed on aerial photographs; and (4) assessment of maps. Assessment of ground plots is the most common method, is increasing in frequency of use and is probably the most accurate method.

National definitions of growing stock volume differ considerably with respect to multiple components. Nearly all countries base growing stock volume on tree diameter over bark. However, beyond this criterion, definitions of growing stock are quite diverse. For example, the height at which diameter is measured, often characterized as breast height, varies from 1.3 to 1.5 m, the minimum tree diameter at breast height (dbh) varies from 0.0 to 12.7 cm, and the minimum top height diameter varies from 0.0 to 10.1 cm for trees used to estimate volume. In addition, countries are split between those that do and those that do not include stump volume, branches and recently dead trees in growing stock volume. Most countries estimate volume for productive forest land, but others also estimate it for poorly productive forest land and other wooded land. Although volumes for these land classes are usually estimated separately, the definitions for the classes vary considerably.

International reporting and harmonization

Numerous international agreements require that countries report national estimates of parameters related to forest area, growing stock volume and their change. Parties to the UNFCCC (1992) are required to produce annual reports of greenhouse gas emissions and removals by sources and sinks. The CBD (2009) requires that countries identify and monitor components of biological diversity for purposes of conservation and sustainable use. Both the Montréal Process (2005) and the MCPFE (2009) require that member countries report on sustainability and biodiversity indicators.

Extensive and comprehensive data on the status of forests available for reporting under international agreements are provided by NFIs. Although NFIs share a primary objective of conducting forest resource assessments to assess the sustainable yield and management of forests, the previous section clearly documents that they do not assess common sets of variables or use common definitions of variables. Further, inventory sampling designs, plot configurations, measurement protocols and analytical methods vary considerably among countries. An effect of these differences is a lack of comparability in estimates submitted for international reporting by individual countries using their own definitions, thresholds and measurement protocols. Traub et al. (1997) reported that the pan-European estimate of forest area decreased by 6% when the United Kingdom's definition of forest was used and increased by 3% when Luxembourg's definition was used. Cienciala et al. (2008) reported that when using a definition proposed by the Food and Agriculture Organization (FAO) of the United Nations rather than the Finnish national definition, Finland's forest area estimate increased by 10.6%, its growing stock volume estimate increased by 2.7% and its annual volume increment estimate increased by 1.8%. If a minimum dbh of 10.4 cm were used for Finnish forests instead of 0.0 cm, the volume estimate would decrease by 14%, the volume increment estimate would decrease by 25% and the carbon sink estimate would decrease by 26%.

Comparability of inventory estimates submitted for international reporting may be achieved using two approaches, standardization and harmonization. However, because NFI features such as sampling designs and plot configurations for individual countries have been developed over time to accommodate their unique topographies, climates, forest types and commercial interests, standardization is often not a viable option. Harmonization acknowledges that individual countries have developed the unique features of their NFIs for specific purposes and are justified in their desire to maintain them and, therefore, focuses on developing methods for producing comparable estimates despite the lack of standardization.

The issue of harmonization has received increased international attention in recent years by the Intergovernmental Panel on Climate Change (IPCC), the Global Forest Resources Assessments (FRA) conducted by FAO, and the 2000 Temperate and Boreal Resource Assessment (TBFRA), as a part of FRA 2000. The most comprehensive effort directed toward harmonization of NFI estimates was conducted by COST Action E43, Harmonization of National Forest Inventories in Europe: Techniques for Common Reporting (COST, 2009; COST Action E43, 2008; Tomppo et al., 2010a). The approach to harmonization proposed by COST Action E43 entailed two steps: (1) construct common reference definitions; and (2) construct bridges that facilitate estimation in accordance with the reference definition using data collected using national definitions.

Vidal et al. (2008) described the process used by COST Action E43 to develop a reference definition for forest land. The most commonly used variables as indicated by the proportions of European countries that use them and the areas these countries represent are minimum area, minimum tree crown cover, minimum area width and minimum tree height (Table I). Thus, the proposed reference definition for forest land is based on these four variables. A reference definition must not only identify the relevant variables but also specify thresholds for these variables. For a tree crown cover threshold of 10%, data are currently available for European countries representing more than 65% of the total area and nearly 75% of their total forest area (Table II). Thus, the proposed reference definition for forest land includes a tree crown cover threshold of 10%. Similar analyses were used to select a threshold of 0.5 ha for minimum area, 5 m for minimum tree height at maturity and 20 m for minimum area width. Thus, the COST Action E43 reference definition for forest land was (Tomppo et al., 2010a):

Forest is land spanning more than 0.5 ha with trees higher than 5 metres and with tree crown cover of at least 10%, or able to satisfy these thresholds in situ. For tree rows or shelterbelts, a minimum width of 20 m is required. It does not include land that is predominantly agricultural or urban land use.

This definition is very similar to the 2005 FRA definition (FAO, 2005).

Table I. Variables used in national definitions of forest land.^a

	Proportion^b
Variable	Countries	Area of countries	Forest area of countries
Minimum area	0.96	0.99	0.99
Minimum tree crown cover	0.81	0.69	0.56
Minimum width	0.74	0.66	0.54
Minimum tree height	0.59	0.47	0.42
Note: ^aMcRoberts et al. (2009); ^bproportions for 27 European countries participating in COST Action E43.

Table II. Tree crown cover threshold used in national definitions of forest land.^a

	Cumulative proportion^b
Tree crown cover threshold	Country area	Forest area
0.05	0.12	0.12
0.10	0.67	0.73
0.20	0.79	0.81
0.30	0.84	0.87
0.50	0.94	0.95
Not used	0.06	0.05
Note: ^aMcRoberts et al. (2009); ^bproportions for 27 European countries participating in COST Action E43.

Bridges for producing estimates in accordance with reference definitions using data collected using national definitions can take multiple forms. Crucial factors affecting construction of bridges are the variables and corresponding thresholds in the national definitions under which data are acquired. The nature of these data is the primary factor that distinguishes three kinds of bridge: reductive, expansive and neutral bridges. Reductive bridges are used when the national definition is broader in scope than the reference definition, in which case some of the national data can be simply excluded. When the scopes of the reference and national definitions are the same, neutral bridges are appropriate. Expansive bridges are necessary when the scope of the reference definition is broader than that of the national definition, in which case data are missing for estimation based on the reference definition and must be supplied via prediction, imputation or other methods. Examples of bridges for harmonization purposes are found in McRoberts et al. (2009) and Tomppo et al. (2010a).

Using remotely sensed data to enhance inventory sampling and estimation

Although remotely sensed data are increasingly used to enhance inventories, they cannot completely replace ground sample data. The expansion of inventories to address issues such as sustainability and biodiversity requires information on variables such as deadwood, lichens and soil carbon. In addition, remotely sensed data are not currently sufficient for producing species-level estimates, particularly in tropical countries such as Brazil, which is estimated to have more than 11,000 tree species that reach diameters of 10 cm and more than 3000 species of which there are more than a million individuals (Hubbell et al., 2008). Nevertheless, the statistical sophistication of methods related to the use of remotely sensed data has increased substantially in recent years (Mc Roberts & Tomppo, 2006). The two most important conclusions, as described and discussed in the following sections, are that remotely sensed data have now been incorporated into operational forest inventories, and that estimates obtained from remote sensing-based maps can now be expressed in forms similar to sample-based estimates.

Sampling

The efficiencies of inventory sampling designs have been evaluated using maps constructed using satellite imagery that depict forest attributes such as volume by tree species. A map can be considered a model of forests and land use which can then be used to assess measurement and travel costs using Geographical Information System (GIS) techniques. This approach has been used operationally to plan sampling designs since 1992 for the eight inventories of northern Finland conducted as part of the Finnish NFI and subsequently for the entire country. Standard errors of population parameters were estimated using map predictions based on satellite image and field data, as well as sampling simulations (Tomppo, 2009). In the very north of Finland, where the road network is sparse and travel costs are large, a double sampling for stratification design was used. A similar approach focusing on optimal sample allocation (Cochran, 1977) was used in a collaborative effort among the Tanzanian government, FAO and the Finnish Forest Research Institute (Tomppo et al., 2010b) to plan the sampling design for the new Tanzanian NFI. Advantages of this approach are that greater sampling intensity can be allocated to areas with greater variation in forest variables and resources can be allocated using sound statistical principles.

Expansion factors

When using plot observations as the basis for estimates of inventory variables for subpopulations such as municipalities, counties and provinces or states, the per unit area observation for each plot must be multiplied or expanded by the area the plot represents to obtain an estimate of the total for the subpopulation. For example, if 100 trees are observed on a 0.5 ha plot, then the observation converts to a per unit area observation of 200 trees ha⁻¹. For a sampling intensity of one plot per 1000 ha, the plot expansion factor is 1000, in which case the total number of trees represented by the plot observation is 200,000 trees. More accurate area estimates may be obtained when plot expansion factors are derived from landscape features. For example, if the plot whose observation converts to 200 trees ha⁻¹ is the only plot on a particular soil type and there are 1500 ha of this soil type in the population, then a plot expansion factor of 1500 would produce a better estimate of the total number of trees than would the 1000 expansion factor.

The Finnish NFI has developed an innovative method for estimating expansion factors using remotely sensed data that can be used to improve small area estimates (Tomppo & Halme, 2004). Whereas NFI plot data are routinely used to estimate population parameters for areas on the order of 100,000s of hectares, a typical small area such as a municipality is approximately 10,000 ha and lacks sufficient numbers of plots to produce acceptably precise estimates using only plot data (Tomppo, 1996; Tomppo et al., 2008). The approach relies on a synthetic estimator that uses plot data for a larger region such as a Landsat scene to produce estimates for a smaller area such as a municipality. For application to the small area, the expansion factor, c _j , for the jth plot in the larger region is calculated using the k-nearest neighbors (kNN) technique as:

where w _j,i is a measure of the similarity of the ith pixel in the small area to the pixel containing the center of the jth plot in the larger region. Thus, the plot expansion factor is a function of the proportion of pixels in the small area that are similar to the pixel containing the center of the plot in the larger area (Tomppo, 1996).

Stratified estimation

NFIs that use probability-based sampling designs typically also use approaches to inference that are characterized as design based. Properties of design-based estimators derive from the probabilities of selection of population units into the sample. Design-based inference is based on two assumptions: a probability sample and a constant value, apart from measurement error, for each population unit. Estimators for population parameters are derived to correspond to the sampling designs and are typically unbiased, although the estimate obtained with any particular sample may deviate considerably from the true value of the population parameter.

Stratified estimators are a subclass of design-based estimators that are necessary for cross-strata estimation when sampling methods and intensities differ by strata. Stratified estimation can also be used to increase the precision of estimates and thereby enhance inferences, regardless of whether stratified sampling is used. The essence of stratified estimation is to assign sampled population units to homogeneous classes or strata, calculate within stratum sample means and variances, and then calculate a weighted average of the within stratum estimates where the weights are proportional to the stratum sizes. If the stratification is effective, the stratified variance estimate will be less than the variance estimate based on an assumption of simple random sampling. Stratified estimates are calculated using estimators described by Cochran (1977).

Historically, stratified estimation for inventory applications was implemented in northern Finland, the USA and elsewhere using a modification of a double sampling for stratification approach (Bickford, 1952; Poso, 1972; Poso & Kujala, 1978). In the first phase, an initial, dense, systematic grid of photoplots was laid over aerial photographs of an inventory area. The photoplots were interpreted with respect to the land use and the size, density and species of trees on forest land and grouped into homogeneous classes or strata. The relative areal extent of each stratum was estimated as the proportion of first phase photoplots assigned to the stratum. The second phase sample was drawn from the photoplots in the strata using either systematic or random sampling. Field plots were established with centers at the centers of the second phase photoplots and were measured by field crews. The data for the two phases were combined using stratified estimators (Cochran, 1977).

In recent years, satellite imagery has replaced aerial photography as a source of information for constructing strata in many countries (McRoberts et al., 2002a, b, 2006; Nilsson et al., 2005). With these approaches, the satellite imagery is often classified with respect to selected forest attributes, and the strata are either the classes or aggregations of the classes. The weight for each stratum is easily calculated using GIS techniques as the proportion of pixels with centers in the stratum. McRoberts et al. (2006) proposed an approach to stratification in which strata are based on maps of classes of proportion forest cover; e.g. 0.000–0.099, 0.100–0.499, 0.500–0.899 and 0.900–1.000. A similar approach is now in operational use in the USA. The utility of stratifications is often assessed using relative efficiency, defined as

where is the estimate obtained under the assumption of simple random sampling and is the estimate obtained using the stratified estimator. For forest area and volume in the north central USA, RE = 5.87 and RE = 2.71, respectively, were obtained. RE is equivalent to the factor by which sample sizes would have to be increased to achieve the same precision using simple random sampling estimators without stratification as is achieved using the stratified estimator. Thus, RE = 2.0 indicates that in the absence of stratification, the sample size would have to be doubled to achieve the same precision, a prohibitively expensive proposition for most NFIs.

Model-assisted estimation

Model-assisted estimators are a second subclass of design-based estimators in which a model is used to describe the population, particularly the point scatter (Särndal et al., 1992). Relative to stratified estimators, model-assisted estimators make greater use of the same ancillary data because they use the complete range of the data rather than grouping them into classes. Although model-assisted estimators use model predictions for population units, they rely on the probability sample for validity. Model-assisted estimators can take multiple forms including the familiar ratio, regression and difference estimators.

Särndal et al. (1992) defines the model-assisted difference estimator which, for equal probability samples, can be expressed as:

with variance that can be approximated as:

where y _i is an observation, is a model prediction, n is the sample size, and N is the population size. The primary advantage of model-assisted estimators, in general, is that they capitalize on the relationship between the sample observations and their model predictions to reduce the variance of the population parameter estimate. A particularly appealing feature of the difference estimator is that any model or prediction approach may be used to obtain the population unit predictions. Despite their flexibility and the reductions in variances they produce, model-assisted estimators suffer the effects of small sample sizes when used for small area estimation.

Recent applications of model-assisted techniques include Opsomer et al. (2007), who used a model-assisted approach with non-parametric regression to estimate multiple forest attributes in the western USA from forest inventory and Landsat data. In addition, Baffetta et al. (2009) estimated timber volume in Italy using the model-assisted difference estimator with forest inventory and Landsat data and the kNN technique; Andersen et al. (2009) used a model-assisted approach with a regression model to estimate biomass in Alaska, USA, from forest inventory and lidar (light detection and ranging) data; and McRoberts (2010) used the model-assisted difference estimator with a logistic regression model to estimate proportion forest in the USA from forest inventory and Landsat data.

Model-based estimation

The assumptions underlying model-based inference differ considerably from those underlying probability-based inference. First, the observation for a population unit is a random variable whose value is considered a realization from a distribution of possible values, rather than a constant as is the case for probability-based inference. Second, the basis for a model-based inference is the model, not the probabilistic nature of the sample as is the case for probability-based inference. McRoberts (2010) provides a comparison of design-based and model-based approaches.

With model-based inference, a model,

is fitted to the sample data, where X _i is the vector of ancillary variables for the ith population unit, β is a vector of parameters to be estimated, and ε _i is the random residual. The model prediction, , for the ith population unit serves as an estimate of the mean of the distribution of possible values for the unit; i.e. . Model-based estimators for the population parameter, µ, are:

and

where calculation of depends on the particular approach to prediction used. The primary challenges with model-based approaches are computational intensity, lack of fit assessment and variance estimation.

McRoberts et al. (2006, 2007McRoberts et al. (2010) used model-based approaches with the kNN technique with Landsat data to estimate proportion forest area, basal area, volume and stem density. Both Nelson et al. (2004) and Ståhl et al. (2010) used model-based approaches with lidar data to estimate above-ground biomass.

Nearest neighbors techniques

Within the international forest inventory community, nearest neighbors techniques have emerged as a class of useful and intuitive methods for predicting forest attributes from climatic, topographic, remotely sensed and other ancillary variables. With nearest neighbors techniques, population unit predictions are calculated as linear combinations of observations for the population units in the sample that are nearest or most similar in a space of ancillary variables to the unit requiring a prediction. Nearest neighbors techniques are appealing for multiple reasons: (1) they can be used for both univariate and multivariate prediction; (2) they are non-parametric in the sense that no assumptions regarding the distributions of response or ancillary variables are necessary; (3) they are synthetic in the sense that they can readily use information external to the geographical area for which an estimate is sought; (4) they are useful for map construction and small area estimation; (5) they can be used with a wide variety of data sets; and (6) when used to classify remote sensing data, they lend themselves better to statistical analysis than many other classification techniques.

The primary forest inventory applications of nearest neighbors techniques are in three areas: (1) imputing missing values for forest inventory and monitoring databases (Eskelson et al., 2009); (2) enhancing design-based inference; and (3) mapping, small area estimation and model-based inference. The Finnish NFI has used the kNN technique operationally with satellite image and NFI field plot data for small area estimation since 1990.

More recently, nearest neighbors techniques have gained considerable popularity and utility for use with forest inventory and satellite image data to estimate forest attribute response variables. Applications have truly been international, with published reports for Austria (Koukal et al., 2007), Canada (LeMay & Temesgen, 2005), China (Tomppo et al., 2001), Costa Rico (Thessler et al., 2008), Finland (Tomppo & Halme, 2004; Tomppo, 2006), Ireland (McInerney & Nieuwenhuis, 2009), Italy (Chirici et al., 2008; Baffetta et al., 2009), Japan (Kajisa et al., 2008), New Zealand (Tomppo et al., 1999), Norway (Gjertsen, 2007), Siberia (Fuchs et al., 2009), Sweden (Nilsson, 1997; Holmström & Fransson, 2003; Reese et al., 2003), the USA (Moeur & Stage, 1995; McRoberts et al., 2007) and Zambia (Maltamo & Eerikäinen, 2001). A bibliography of published peer-reviewed paper on nearest neighbors is available at: http://blue.for.msu.edu/NAFIS/biblio.html.

For notational purposes, let Y denote a possibly multivariate vector of response variables with observations for a sample of size n from a finite population of size N, and let X denote a vector of ancillary variables with observations for all population units. In the terminology of nearest neighbors techniques, the set of population units for which observations of both response and ancillary variables are available is designated the reference set; the set of population units for which predictions of response variables are desired is designated the target set; and the space defined by the ancillary variables, X, is designated the feature space. All elements of both the reference and target sets are assumed to have a complete set of observations for all feature space variables. For continuous response variables, the kNN prediction for the ith target set element is:

where is the set of response variable observations for the k reference set elements that are nearest to the ith target set element in feature space with respect to a distance metric, d, and w _ij is the weight assigned to the jth nearest neighbor with . For categorical variables the predicted class of the ith target set element is the most heavily weighted class among the k-nearest neighbors. Frequent choices for the weights are , where . Many distance measures may be expressed in matrix form as:

where i denotes a target set element for which a prediction is sought, j denotes a reference set element, X _i and X _j are the vectors of observations of feature space variables for the ith and jth elements, respectively, and M is a square matrix. Popular choices for M include the identity matrix which results in Euclidean distance, a non-identity diagonal matrix which results in weighted Euclidean distance, the inverse of the covariance matrix of the feature space variables which results in Mahalanobis distance (Kendall & Buckland, 1982). Other choices for M are based on canonical correlation analysis (Moeur & Stage, 1995; Maltamo & Eerikäinen, 2001; Temesgen et al., 2003; LeMay et al., 2008) and canonical correspondence analysis (Ohmann & Gregory, 2002; Ohmann et al., 2007; Pierce et al., 2009). Chirici et al. (2008) and Hudak et al. (2008) evaluated additional distance metrics.

Recent investigations of nearest neighbors techniques have shifted from simple descriptions of applications to more foundational work on efficiency and inference. McRoberts (2009) reported diagnostic tools for evaluating and enhancing nearest neighbors prediction of continuous, univariate response variables. Finley et al. (2006) and Finley and McRoberts (2008) investigated enhanced search algorithms for identifying nearest neighbors. Tomppo and Halme (2004) and Tomppo et al. (2009) used a genetic algorithm approach to optimize the weights for ancillary variables in the distance metric. However, the full potential of nearest neighbors techniques cannot be realized until they can be used for construction of valid statistical inferences, a process that requires variance estimates. Kim and Tomppo (2006) reported a method for estimating the uncertainty of predictions using a variogram approach; McRoberts et al., (2007) derived a kNN variance estimator for areal means from the conceptual assumptions underlying kNN estimation; and Magnussen et al. (2010b) investigated an estimator of mean square error.

Nearest neighbors techniques have been used with satellite imagery to support and enhance most inventory estimation methods. McRoberts et al. (2002a) used the kNN technique to construct strata for use with stratified estimation. The Finnish NFI uses the kNN technique operationally with satellite imagery to calculate expansion factors (Tomppo et al., 2008). Baffetta et al. (2009) described an application of the kNN technique for use with the model-assisted difference estimator. McRoberts et al., (2007) used the kNN technique for a model-based approach to inference for small area applications.

Emerging issues

Climate change

The United Nations-REDD (Reducing Emissions from Deforestation and forest Degradation) program is a collaborative partnership among FAO, the United Nations Development Program and the United Nations Environment Program (UN-REDD, 2009). UN-REDD was launched in September 2008 to create financial value for carbon stored in forests, to offer incentives for developing countries to reduce forest carbon emissions and to invest in low-carbon paths to sustainable development. The UN-REDD program supports countries in the development of cost-effective and robust measurement, reporting and verification (MRV) systems based on good science and available technology.

Because of the similarity between MRV programs and NFIs, some REDD countries are initiating new NFIs to support their MRVs. However, factors unique to these developing, often tropical, countries may require NFI features that deviate considerably from features of NFIs countries with boreal and temperate forests. First, although forest information requirements and expectations for developing countries are great, national financial resources are often limited. In addition, whereas information produced by NFIs in developed countries is used for strategic, national policy making, in developing countries forest policy decisions are often made locally, meaning that information for smaller areas is necessary. Thus, optimization of sampling designs using ancillary data such as satellite imagery is crucial, although satellite image-based classification may be more difficult in tropical countries because of the large number of species. Second, the combination of limited financial resources and large proportions of forest lands that are remote and difficult to access means that the sampling designs will be likely to feature strata with considerably different sampling intensities. Further, selection of the sampling intensities within strata may not be a function of desired precision as much as a function of an acceptable distribution of resources. Third, because of limited financial resources and limited access to remote regions, greater reliance on remotely sensed data may be expected. However, for tropical countries, acquisition of cloud-free imagery will be a challenge. In addition, methods for assessing the accuracy of satellite image-based maps and deriving estimates and inferences from them are necessary (McRoberts, 2010). Fourth, whereas radii as great as 12 m for circular NFI plots may be appropriate for less dense boreal and temperate forests, they may be too large for more dense tropical forests. For example, stem densities in tropical forests may make it impossible to determine accurate distances from plot centers to trees on the periphery of large radii plots. As a result, a larger number of smaller circular plots may be necessary or, as recommended by FAO, narrow, rectangular plots may be necessary (Saket et al., 2002). However, the latter plots are not without challenges such as dealing with four corner control points versus a single plot center control point for circular plots. In addition, the greater perimeter of rectangular plots for the same plot area means more boundary trees will have to be assessed in terms of whether they are or are not on the plot. Finally, facilitation of harmonized international reporting should be considered when defining forest and growing stock volume and when selecting minimum thresholds for variables used in these definitions such as patch size, crown cover, height at maturity, breast height, diameter at breast height and top height diameter. In particular, thresholds such as patch size of 0.1 ha and crown cover of 10% for use in the definition of boreal and temperate forest may be inappropriate or difficult to accommodate for tropical forests.

Lidar

Most remotely sensed data used by NFIs have been from passive sensors such as Landsat and SPOT that respond only to reflected sunlight. Data are now becoming widely available from active systems such as lidar that measure the elapsed time between emission of a short duration pulse of light and detection of the reflected pulse back at the sensor. The elapsed time is converted to distance which, for forestry applications, provides information that can be used to assess forest canopy height and vertical structure.

For inventory applications, data from small-footprint, discrete-return lidar systems with a ground footprint size of approximately 0.1–2.0 m have been the most intensively investigated. Such systems typically record two to five discrete echoes of each emitted pulse, often with one or more echoes reflected from the canopy and a last echo reflected from the ground. The first task with lidar data is usually to determine the terrain surface and then to estimate the relative height of the canopy echoes that are reflections from vertical positions within and on the outer surface of the forest canopy. When aggregated over a sample plot or forest stand, the collection of heights estimated from canopy echoes resembles the distribution of biological material in various vertical height layers. This information can be combined with field observations to construct statistical models for predicting height, growing stock volume, total biomass and other biophysical parameters (Nelson et al., 1988; Næsset 1997a, b).

Small-footprint applications date back to approximately 1980 (Solodukhin et al., 1977; Nelson et al., 1984) and focus on estimating cross-sectional areas and heights of individual tree canopies using profiling systems. Lidar data collected from profiling systems are only collected along a narrow strip underneath the aircraft and are used to produce two-dimensional profiles of heights that can be used to predict biophysical parameters (Maclean & Krabill, 1986; Nelson et al., 1988).

With the introduction of small-footprint scanning lidar systems, also known as airborne laser scanning (ALS), for forest research in approximately 1995, lidar-based height information became available as three-dimensional, wall-to-wall data. Such data can be used either to segment and characterize individual tree crowns (Hyyppä & Inkinen, 1999; Persson et al., 2002) or to estimate areal means for tree height, tree stem volume, total biomass, tree size distribution and other biophysical parameters (Næsset, 1997a, 1997b; Gobakken & Næsset, 2004). With the areal approach, field sample plot data and the lidar data are combined to construct models that are then used to predict biophysical parameters of interest for individual cells of a grid that covers the entire area of interest (Næsset, 2002). Cell sizes are equal to the field plot sizes, and cell predictions can be aggregated to produce estimates for stands or areal units of interest. Whereas the single-tree approach requires lidar measurements at densities greater than 2–5 pulses m⁻², the areal approach can provide accurate estimates with pulse densities of only approximately 0.5–1.0 pulses m⁻² (Gobakken & Næsset, 2008; Magnussen et al., 2010a). Non-parametric methods such as the kNN technique have been used successfully for both tree-level (Maltamo et al., 2009) and areal applications (Maltamo et al., 2006; Packalén & Maltamo, 2007).

To date, most successful lidar-based applications have been for management inventories, for which they have been shown to be more cost efficient than conventional methods that use a combination of aerial photography and/or field data (Eid et al., 2004). Fully operational or semi-operational management inventories using mainly low-pulse lidar data have been conducted in the Nordic countries, North America, southern Europe and Australia in support of commercial thinning and management (e.g. Næsset, 2004, 2007).

For operational use in NFIs, airborne lidar data may play different roles including partial replacement of tree-level plot measurements or as ancillary data for estimation of parameters such as total biomass over larger areas. Full integration of lidar into NFIs currently faces multiple challenges: (1) tree heights are always underestimated, although corrections can be made; (2) errors are introduced as the result of failure to detect suppressed trees or prediction of non-existing trees; (3) tree species can be difficult to identify; and (4) estimates must be provided by timber assortments such as saw timber and pulp wood, not just total tree stem volume. For the Nordic countries, where many of the lidar studies have been conducted, progress has been made on species identification. Holmgren et al. (2008) showed that fusion of aerial imagery and lidar data can improve species classification for individual trees, and Packalén and Maltamo (2007) showed that the same fusion can be used to estimate species specific parameters in areal estimation. Further, dense lidar data without supporting aerial imagery can produce accurate species predictions for individual trees (Vauhkonen et al., 2009). Breidenbach et al. (2010) proposed methods for statistically correcting errors occurring with individual tree crown segmentation algorithms. However, additional progress is necessary for regions that have more species than in the Nordic countries and that have extensive mixed species and uneven-aged stands. For management inventories, timber assortments have been better estimated with lidar than with conventional methods using field assessments (Peuhkurinen et al., 2007; Packalén & Maltamo, 2008), but positive results must be demonstrated for NFI applications.

Finally, wall-to-wall airborne lidar coverage is costly if estimates are sought only for large regions. Thus, applications using samples of lidar data collected along strips, rather than for entire areas, are under development (Andersen et al., 2009; Gregoire et al., 2010; Ståhl et al., 2010). However, efficient methods for producing valid statistical areal inferences for inventory population parameters based on lidar data require further investigation. A more extended discussion of the use of lidar for forestry applications is presented in McRoberts et al. (2010).

Summary

NFIs have had a long history, although their major features such as statistical sampling and estimation date only to the early years of the twentieth century. In addition, NFIs have been at least partially reshaped in recent years as a result of concern for issues such as acid deposition, biodiversity and forest sustainability, increased demand for and use of forest data, and international reporting requirements. Technological advances, particularly the widespread availability of remotely sensed data, have led to greater inventory efficiencies, map construction and accurate estimation for small areas. Other technological advances such as lidar that are now on the horizon have the potential to lead to even greater efficiencies and even more useful spatial products and estimates. Finally, the current concern for global climate change and recognition of the ecological necessity of preserving tropical forests has added a new impetus for initiating NFIs in developing, southern hemisphere countries. Unique challenges will emerge; in particular, although general inventory principles will remain unchanged, the sampling designs, plot configurations and measurement protocols that have become familiar to inventory experts in northern hemisphere countries with mostly boreal and temperate forests will require modification before they can be applied in countries with tropical forests.

Acknowledgements

The authors acknowledge the valuable comments on the lidar section received from Johannes Breidenbach and Hans Ole Ørka of the Norwegian University of Life Sciences and Matti Maltamo of the University of Joensuu. No external funding was received.