Background

Research over the past decades has clearly established the vital role of downed woody debris (DWD) in forest ecosystems, where it influences biodiversity, nutrient cycling, soil development, and wildfire behavior (Harmon et al. 1986; Schoennagel et al. 2004; Stokland et al. 2012). Recent attention has shifted to carbon stores and fluxes in DWD given the growing interest in carbon dynamics in the context of climate change (Russell et al. 2014; Woodall et al. 2015). Methods of reliably estimating DWD abundance in forest systems are thus critical for carbon estimates (e.g. Russell et al. 2015), as well as a broader range of applications.

As with live-tree sampling, a complete census could be conducted of all DWD pieces above a minimum diameter threshold within a location of interest; however, this may be impractical for inventories at stand or larger scales. As a consequence, sampling methods such as line-intersect sampling (LIS) are employed to inventory DWD abundance (Woodall et al. 2009). LIS produces an area-based estimate of DWD volume derived from a sample of the logs present. In the field, the intersect method proceeds by measuring diameters of pieces (above a given size threshold) at the point where an inventory transect crosses the pieces’ central longitudinal axes. The resulting data (tallied diameters and total transect length) are converted to area-based estimates of DWD volume, typically using equations provided by Van Wagner (1968) and Brown (1974). The theory underlying LIS is well developed (de Vries 1979; Kaiser 1983). No assumption of piece shape is required, as LIS leads naturally to an unbiased Monte Carlo estimate of volume based on the cross-section of pieces where they are intersected by the transect line. Moreover, despite some derivations that assume pieces are randomly distributed (e.g. Warren and Olsen 1964), from a design-based perspective, the positions of pieces are regarded as fixed even if they originated from a random process. Because it is time-efficient and design-based unbiased, LIS is widely used in ecological studies and fuel inventories (e.g. Brown 1974), as well as in many national-scale forest inventories (Woodall et al. 2009). Kershaw et al. (2016, ch. 12) provide a practical and theoretical review of LIS along with alternative DWD inventory approaches.

Despite its widespread use in large-scale forest inventories, the appropriate LIS transect lengths needed under local forest settings has received relatively little attention (but see Woldendorp et al. 2004; Sikkink and Keane 2008). Considering that many ecological studies intend to characterize DWD attributes at particular locations, where forest stands or small reserves are themselves the populations of interest (Jönsson et al. 2011; Fraver and Palik 2012), surprisingly few studies have evaluated DWD sampling methods at this local scale. Stokland et al. (2004) caution that a DWD sampling design that is efficient for assessing average ecological values over large scales may be poorly matched to the task of assessing those same values at local scales.

Previous empirical work as well as sampling theory demonstrates that for a given site the precision of the LIS estimate consistently improves (lower variance) with increasing cumulative transect length (Pickford and Hazard 1978; Woldendorp et al. 2004; Miehs et al. 2010) and that sites with lower DWD volume generally require longer transects to achieve a desired level of precision (Brown 1974; Pickford and Hazard 1978; Woldendorp et al. 2004). The objective of the present study was two-fold: 1) to better quantify the effect of increasing transect length and DWD volumes on the precision of LIS volume estimates, and 2) to provide practical guidance for selecting transect lengths that achieve a desired level of precision regarding LIS volume estimates. We addressed these objectives through simulations constrained by empirical LIS data from 33 locations representing eight diverse forest types in the northeastern USA. These locations provide a range of forest conditions and DWD volumes, making them ideal for addressing these objectives. Additionally, we present a novel application of copula models to account for within-transect spatial autocorrelation of DWD volumes; without accounting for such autocorrelation, simulations would underestimate the variance within transects, resulting in spurious assessments of appropriate transect lengths.

Methods

Study locations and field data collection

We tested the effect of increasing transect length (up to ca. 340 m) and DWD volume on LIS precision using 33 locations representing eight mature or old-growth forest types in northeastern USA (Table 1). These forest types and locations were utilized opportunistically, given that we needed DWD inventories for these locations as part of on-going research (e.g., Fraver and Palik 2012; Fraver et al. 2017). We thus opted to intensify sampling (i.e., long transects) to provide data for the modelling efforts presented here. These locations represent diverse forest compositions and structures, and they conveniently capture a large range of DWD volumes necessary for modelling. Forest types include mature red-spruce (Picea rubens), mature spruce–hemlock (P. rubens, Tsuga canadensis), mature hemlock–pine (T. canadensis, Pinus strobus), mature mixed conifer–hardwood (T. canadensis, Acer saccharum, Fagus grandifolia, Betula alleghaniensis), mature northern hardwoods (A. saccharum, F. grandifolia, B. alleghaniensis, with P. rubens), old-growth northern white-cedar (Thuja occidentalis), old-growth red pine (Pinus resinosa), and old-growth black ash (Fraxinus nigra). A minimum of three locations was established in each forest type. Additional site information is provided in Table 1.

Table 1 Characteristics of the eight forest types from which empirical data were collected for evaluation of LIS precision
Full size table

At each location we conducted LIS sampling using a cluster of four transects that intersected at their midpoints as follows: two diagonal 100-m transects (arranged southwest to northeast and northwest to southeast) and two 70-m transects (arranged in cardinal directions). This configuration formed a spoke-like ‘union jack’ with equal angles between transects and a total transect length of 340 m. At four of the eight sites, the centers of the union jacks were placed on centers of existing long-term continuous forest inventory plots. At the remaining four sites, where previous plots did not exist, centers were placed randomly within sampled stands. This union jack configuration was initially employed at the red-pine locations (the first to be sampled; Fraver and Palik 2012), as an exploratory comparison of co-located LIS and fixed area inventories on 0.5-ha (70.7 m × 70.7 m) plots (unpublished). For sampling consistency, we then used this same configuration at all remaining locations. A multi-directional, spoke-like arrangement is often used to reduce potential sample variance when DWD pieces exhibit fairly uniform fall directions (Van Wagner 1968). The most common estimating equations used with LIS assume the angle of intersection between individual pieces and the sample line is random; when the sample line orientation itself is random, these equations provide unbiased estimates, but strong correlation in the pieces’ fall directions can inflate sample variance (Kaiser 1983; Kershaw et al. 2016, ch. 12). Spoke-like transect designs have been the subject of some debate in large-area inventories (Affleck et al. 2005; but see Woodall and Monleon 2008); however, their use represents common field practice (e.g., Dunn and Bailey 2015), and we feel the design is adequate to evaluate the effect of varying transect lengths, the focus of our study. For each DWD piece intersected by the sampling transect with a diameter ≥ 10 cm at the intersection, we recorded diameter at intersection, distance from center, species, and decay class (five-class system as per Sollins 1982). By convention, if a piece intersected two transects, or if it crossed the same transect twice, it was recorded twice (Van Wagner 1968, Brown 1974).

We calculated DWD volume for each site as follows,

$$ V=\left({\uppi}^2\sum \frac{d^2\ }{8L}\right)\times 10\ 000 $$

where V is the area-based volume (m3∙ha− 1), d is the DWD piece diameter (m) at point of intersection, and L is the total transect length at the site (m) (from Van Wagner 1968). Similar calculations were used to provide volume estimates for short 5-m segments of each transect, to be used in the sampling simulations described below.

Simulation of sampling variability

To assess how transect length and population characteristics influence sampling variability, we employed a model-based approach using simulation based on a zero-inflated, autoregressive copula. To motivate this approach, let us first consider a naive approach that would be simple to implement, and would have some desirable features, but could lead to inaccurate results. Given the line intersect data from a site, with the positions of all the intersections recorded, one could subdivide transects into shorter segments (e.g., 5 m) that would allow simulation of the sampling variance for different transect lengths, with a resolution on length that would be meaningful for operational inventory design. Then, to simulate transects of different total length, one would draw segments at random with replacement from the data for that site, concatenating them to form transects of the desired length. For example, to form a 20-m transect, one would choose four segments of 5 m each with replacement, and compute any desired values (such as DWD volume) for the new, simulated transect. By repeating this process many times, and varying the transect length, one would develop a Monte Carlo estimate of the sampling distribution of DWD volume estimates for LIS at that site and its dependence on transect length, with a richer potential inference beyond second-order statistics such as variance (which, under the assumptions implicit in this approach, could also be calculated using the usual formulae for simple random sampling; e.g., Thompson 2012). Essentially, this approach uses the same rationale as the bootstrap (Efron and Tibshirani 1993), substituting the empirical distribution of DWD volume on short segments for the unknown population distribution of possible segments, to arrive at the sample distribution through resampling.

Unfortunately, the bootstrap approach makes a strong independence assumption that is inappropriate for DWD data under either a design- or model-based approach (for details on the distinction, see Gregoire 1998). Within a design-based framework (Kaiser 1983; Gregoire and Valentine 2008), DWD pieces are fixed but sample selection is random. The entire transect is a single randomly selected sample unit, and treating segments within a sample unit as if they were drawn independently is anathema. Without substantial additional information (such as the joint inclusion probabilities of the individual pieces of DWD) it is difficult to make progress from this perspective, beyond inference based on the design that was actually employed (except for changes in the number, rather than the length, of sample transects). While a design-based approach has substantial benefits, both in terms of formulating a design for a specific inventory, and computing defensible estimates once the data are in hand, to understand how different designs (such as varying transect lengths) would impact estimates and their uncertainty, a model is invaluable. However, from a model-based perspective, the independence of short, contiguous segments is also suspect. One might expect positive autocorrelation of the values from adjacent segments, not only because of the potential for spatially clumped inputs from tree mortality, but also because local terrain might cause DWD pieces that are close to each other also to share similar orientation. In the presence of positive autocorrelation, the variance of estimates from transects of a given length would be higher than that implied by independent segments, and the transect length needed to reach a given level of accuracy would be longer. To summarize: from a design-based perspective, successive segments within a randomly selected transect are likely to display autocorrelation because of the fixed characteristics of the downed woody debris population. From a model-based perspective, the independence of successive short segments is also suspect, because of likely clumping in the random process that generates the population. Whichever perspective is taken in inference, it would appear important to incorporate autocorrelation within any simulations.

To address the challenge of autocorrelation among successive segments, while remaining faithful to the observed marginal distribution of estimates from those segments, we employ a copula modeling approach. Within the forestry literature, we believe this is the first application to employ an autoregressive model (to account for correlation between successive segments), and also to address a zero-inflated distribution (many segments are empty, with no DWD intersections). Copula models are relatively new in the forestry literature; they have been used to describe relationships between continuous variables such as tree diameter and height (Wang et al. 2008, 2010; MacPhee et al. 2017), to inform spatially-explicit stand simulations (Kershaw et al. 2010), and for imputation-based growth and yield models (Kershaw et al. 2017). Eskelson et al. (2011) and Fortin et al. (2013) extended copula models in forestry to account for spatial autocorrelation. More recently, copula methods have been used to simulate spatial populations for remote sensing-assisted inventory (Ene et al. 2012, 2013a, 2013b; Grafström et al. 2014). For a somewhat theoretical overview of copulas, see Genest and Mackay (1986) and Nelsen (2006).

Strictly defined, a bivariate copula describes the relationship between two random variables x and y in terms of two random deviates that are uniformly distributed on [0, 1]. Specifically, if U1 and U2 ~ Uniform (0, 1), then the copula C is defined as

$$ C\left({u}_1,{u}_2\right)=\Pr \left\{{U}_1\le {u}_1,{U}_2\le {u}_2\right\} $$

If FX(x) and FY(y) are the cumulative distribution functions of x and y, then

$$ {F}_{XY}\left(x,y\right)=C\left({F}_X(x),{F}_Y(y)\right) $$

In our application, x and y are the volume estimates (m3∙ha− 1) associated with successive pairs of short LIS segments within the same dead wood population; thus, they share the same cumulative distribution function F(x) = FX(x) = FY(y). A variety of formulations have been proposed to model the copula function C. For simulation purposes, one of the simplest is the Gaussian or normal copula (Wang 1998). Let Z1 and Z2 have a standard normal marginal distribution, i.e., Z1 and Z2 ~ N (0, 1), with correlation coefficient ρ. Let Φ(∙) be the cumulative distribution function of the standard normal distribution. Then Φ(Z1) and Φ(Z2) will be bivariate uniform, with Kendall’s τ = (2/π)acrsin(ρ). More generally, if we simulate a sequence of Zi, i = 1,…, m, the sequence will be a first-order autoregressive [AR(1)] time series, and each pair Φ(Z1), Φ(Zi + 1) will be described by the same copula function C. To complete a simulation of a single transect composed of m segments, each having marginal distribution F(x), we employ the following algorithm:

  1. 1)

    Generate a standard normal AR(1) time series of length m, with correlation coefficient ρ. This can easily be done through the sequential generation of independent random deviates, with appropriate summation to generate successive entries in the time series.

  2. 2)

    Convert the resulting series to an autocorrelated series of uniform deviates ui = Φ(Z1).

  3. 3)

    Apply the inverse of the empirical cumulative distribution function to translate the uniform series to the marginal distribution of downed wood volume estimates, xi = F− 1(ui).

The only slight complication in the simulation phase of our application arises due to the zero-inflated nature of DWD volume data. Specifically, let p0 be the probability of obtaining xi = 0 in a given population. Then, in step 3 above, xi = 0 whenever ui < p0.

The zero-inflated nature of the data does cause a more important problem in the estimation stage that must precede simulation, however. Specifically, it complicates the estimation of the autocorrelation parameter ρ. In the absence of zero-inflation, the Zi, ui, and xi would share a common value of Kendall’s τ (or Spearman’s rank-correlation coefficient), both of which share a one-to-one relationship with ρ. Thus, it would be possible to calculate a reasonable value of ρ based on the value of τ calculated from the sample pairs xi, xi + 1 in the transects for a given population. Alternatively, one could reverse the process used in simulation: calculate ui = F(xi) to translate the sample series to an autocorrelated uniform series, then Zi = Φ− 1(ui) to obtain a standard normal AR(1) series, and finally estimate ρ from the serial correlation of the Zi. However, this approach fails for zero-inflated data, because it is unclear what value to assign to ui (and hence Zi) when xi = 0.

To overcome this challenge, we adopted a maximum-likelihood approach based on the estimation of correlation coefficients in the presence of censoring, as originally developed by Lyles et al. (2001). In fact, our situation is somewhat simpler than that described by Lyles et al. (2001), because the Zi are defined to have standard normal distribution; we do not need to concern ourselves with the mean and variance as nuisance parameters. We reverse-engineer the Zi values as described above, but record the values as censored whenever xi = 0; that is, we only know that Zi ≤ Φ− 1(p0). Then, we may recognize four distinct categories of outcomes for Zi, Zi + 1 pairs:

  • Category 1: In this category, neither Zi nor Zi + 1 are censored. Pairs of this type contribute lnf(Zi, Zi + 1) = ln f(Zi + 1| Zi) + ln f(Zi) to the log-likelihood. Following Lyles et al. (2001), we denote this contribution as ln(ti1):

$$ \ln \left({t}_{i1}\right)=-\frac{1}{2\left(1-{\rho}^2\right)}\left({Z}_i^2+{Z}_{i+1}^2-2\rho {Z}_i{Z}_{i+1}\right)-\ln \left[2\pi \left(1-{\rho}^2\right)\right] $$

  • Category 2: In this category, Zi is known but Zi + 1 is censored. Let Z' = Φ−1(p0). The contribution of these pairs to the log-likelihood is

$$ \ln \left({t}_{i2}\right)=\ln \left[\Pr \left({Z}_{i+1}\le {Z}^{\hbox{'}}|{Z}_i\right)\right]+\ln f\left({Z}_i\right) $$

which equals

$$ \ln \left({t}_{i2}\right)=\frac{Z_i^2}{2}-\ln \left(2\pi \right)+\ln \Phi \left[\frac{Z^{\hbox{'}}-{Z}_i}{\sqrt{1-{\rho}^2}}\right] $$

  • Category 3: In this category, Zi + 1 is known but Zi is censored; by the same rationale as Category 2, the contribution to the log-likelihood is

$$ \ln \left({t}_{i3}\right)=\frac{Z_{i+1}^2}{2}-\ln \left(2\pi \right)+\ln \Phi \left[\frac{Z^{\hbox{'}}-{Z}_{i+1}}{\sqrt{1-{\rho}^2}}\right] $$

  • Category 4: In this category, both Zi and Zi + 1 are censored. Then the contribution to the log-likelihood is ln{Pr[(Zi ≤ Z') ∩ (Zi + 1 ≤ Z')]}, i.e.

$$ \ln \left({t}_{i4}\right)=\ln \left\{\frac{1}{\sqrt{2\pi }}{\int}_{-\infty}^{Z\hbox{'}}\Phi \left[\frac{Z^{\hbox{'}}-Z}{\sqrt{1-{\rho}^2}}\right]\exp \left(\frac{Z^2}{2}\right)\mathrm{d}Z\right\} $$

Unfortunately, the integral has no convenient, closed-form expression, but it yields readily to numerical integration. By summing the contributions to the log-likelihood across all unique adjacent pairs of segments within a site, one obtains the log-likelihood function; maximization with respect to ρ yields the maximum likelihood estimate, and the inverse of the negative of its second derivative provides the asymptotic variance of the estimate, from which the standard error and confidence limits may be computed.

To summarize: For each location, we used all unique pairs of adjacent 5-m segments, and estimated a site-specific ρ and its standard error by maximum likelihood. Then, we simulated artificial transects of varying length using a zero-inflated autogressive copula, based on the estimated value of ρ and the empirical distribution of volume estimates for 5-m segments from that location. We simulated transect lengths from 20 m to 340 m by 20-m intervals, with 1000 simulated replications at each length for each location. For comparison purposes, we performed identical simulations with ρ = 0, i.e., no autocorrelation, for each location. In this case, the simulation is mathematically identical to the independent bootstrap approach. We summarized the distributional characteristics of the 1000 replicates and used the results as response variables for further modeling to characterize the dependence of sampling variability on transect length and DWD volume.

Modeling of simulation results

Using output from our simulation set, we tallied mean DWD volume estimates and standard deviations for use in the modeling described below. That is, each transect length from each of the 33 locations yielded a mean and a standard deviation, calculated from the 1000 runs unique to each transect length–location combination. We used linear and non-linear mixed-effects models to evaluate the influence of transect length and total DWD volume (predictor variables, unique to each location) on the coefficient of variation (CV) of these estimates. Total DWD volume refers to that calculated from the full 340-m transect length for each location. We first tested the predictors in individual models and then combined them to determine the best-fit two-term model. Forest type was included as a random factor in all models, given that we are not making inferences about particular forest types, for which our sample size (3–7 locations per type) is inadequate. Logarithmic and square-root transformations of response and predictor variables were explored for stabilizing variance and improving residual-versus-predicted diagnostics. The fact that our candidate models included transformed and non-transformed variables limited our reliance on Akaike’s information criterion (Burnham and Anderson 2002) for evaluating model performance. Instead, we relied on pseudo R2 (see below) and root mean square error (RMSE, back transformed as needed) values, as well as graphs of observed-versus-predicted and residual-versus-predicted values to determine which models were best supported by the data. For the one-term models, we tested linear, negative exponential, and power function model forms; for the two-term model, we tested linear, negative exponential, power, quadratic, and partial quadratic forms. Analyses were conducted in the nlme package (Pinheiro et al., 2016) in R (version 3.0.3; Core Team, 2016), using the weighting option to compensate for non-homogenous variance. Goodness-of-fit (pseudo R2) for the models was expressed as the correlation between observed and predicted CVs (cf. Canham et al. 2004).

Results

Simulations of various transect lengths revealed that, as expected, the precision (here, CV) of the LIS estimate consistently improved with increasing cumulative transect length. Model comparisons revealed that the relationship between CV and transect length was best described by a power function (P < 0.0001, R2 = 0.801, RMSE = 0.127) as shown in Fig. 1. Importantly, our analysis of DWD values along transects indicated the presence of spatial autocorrelation, with correlation coefficients (one for each of 33 locations) forming a bimodal distribution; that is, some locations showed positive and others negative autocorrelation (Fig. 2).

Fig. 1
figure 1

Modelled relationship (black line, accounting for spatial auto-correlation) between line intersect sampling precision (expressed as coefficient of variation, CV) and transect length, showing that precision increases (lower CV) with increasing transect length. Ignoring spatial autocorrelation of DWD values (grey dots) falsely suggests greater precision can be obtained for a given transect length

Full size image
Fig. 2
figure 2

Frequency distribution of ρ coefficients for the 33 locations, suggesting tendencies toward both negative and positive spatial autocorrelation of DWD values within sampling transects

Full size image

Simulations also revealed that precision of the LIS estimate improved with increasing DWD volume present, meaning that for a given transect length, greater precision could be achieved where higher DWD volumes are encountered. The relationship between CV and DWD volume was best described by a linear function, as shown in Fig. 3. Although this model was statistically significant (P < 0.01), the relationship is quite weak (R2 = 0.228, RMSE = 0.206).

Fig. 3
figure 3

Relationship between line intersect sampling precision (expressed as coefficient of variation) and downed woody debris (DWD) volume. For a given transect length, sites with greater volumes generally produce more precise estimates

Full size image

When the CV was modeled as a function of both transect length and DWD volume, model comparisons revealed that the exponential relationship was best supported by the data, expressed as CV = a × eb × volume × ec × sqrt_length, where sqrt_length is the square root of transect length (Fig. 4). However, diagnostics for this model were poorer than those for the length-only model (here, R2 = 0.795, RMSE = 0.128), and the volume term became non-significant. For this reason, the length-only model was used for inferences regarding appropriate transect lengths.

Fig. 4
figure 4

Modelled relationship between line intersect sampling precision (expressed as coefficient of variation), transect length, and downed woody debris (DWD) volume, using simulations constrained by empirical field data from 33 locations within eight forest types of northeastern USA. Top model form: CV = 1.30268 × e–0.0002277 × volume × e–0.1059549 × sqrt_length, where sqrt_length is the square root of transect length. Diagnostics for this model (R2 = 0.795, RMSE = 0.128) were poorer than those for the length-only model (Fig. 1), and the volume term was non-significant

Full size image

Discussion

Our study was motivated by the need to provide guidance in selecting a LIS transect length appropriate for estimating DWD volumes for particular locations of interest. We note that a sampling design efficient for assessing average ecological values of DWD over large scales (as in national forest inventories) may not be well suited for assessing those same values at local scales, such as small forest reserves (Stokland et al. 2004), where the reserve itself may be the population of interest.

Our finding that the precision of LIS estimates consistently improved with increasing transect length corroborates a number of previous studies from quite different forest types (e.g., Van Wagner 1982; Pickford and Hazard 1978; Woldendorp et al. 2004; Miehs et al. 2010; Keane and Gray 2013). Similarly, our finding that precision of LIS estimates improved with increasing DWD volume present has been reported from a range of distinct forest systems (Brown 1974; Pickford and Hazard 1978; Woldendorp et al. 2004). However, the best-fit model that included both transect length and DWD volume as predictors showed poorer diagnostics than that of the length-only model described above, suggesting that the length-only model (shown in Fig. 1) could be used for inferences regarding appropriate transect lengths.

If DWD pieces were distributed at random, basic sampling theory (e.g. Thompson 2012) predicts that the variance of the total DWD intersected by a transect would scale linearly with transect length; thus, the CV would scale with the − 0.5 power of transect length for a given DWD density. However, clumping or regularity in DWD pieces would produce a departure from this scaling relationship. Indeed, our analysis of DWD values along transects revealed the presence of spatial autocorrelation, with some locations tending toward positive and others toward negative autocorrelation (Fig. 2). We found no discernable pattern among locations regarding forest type or DWD volume that might explain the bimodal distribution. We minimized the confounding effects of autocorrelation through simulations based on a zero-inflated autoregressive copula model, which accounted for correlations between adjacent 5-m transect segments unique to each location. Without accounting for such autocorrelation, our simulations would have underestimated variance (Fig. 1), leading to spurious assessments of appropriate transect lengths. To the best of our knowledge, this is the first attempt to employ an autoregressive model to account for spatial autocorrelation, while also addressing a zero-inflated distribution (segments may contain no DWD).

A frequent question in the design of research or monitoring protocols is how long transects should be to provide a reliable DWD volume estimate for a given location. A perusal of literature where LIS was used to estimate DWD volume for particular locations reveals substantial variation in the transect length employed: total transect lengths can range from < 40 m (Bradford and Kastendick 2010; Buma et al. 2014) to 200 m or more (Ekbom et al. 2006; Lõhmus and Kraut 2010). For the forest types and conditions sampled here, and considering the particular copula model framework employed, transect lengths of ca. 120 m provide what we feel is a reasonable level of precision, ranging from 18% to 60% CV from the observed simulations, and 37% from the fitted model results, across the range of DWD volumes encountered (Fig. 1). Practitioners working in these or similar forest types and requiring greater precision could chose a transect length based on the curve or equation presented in Fig. 1. For our own inventories, we find it convenient to arrange three 40-m transects radiating equi-angularly from a center point. This total length is somewhat comparable to the 100 m recommended by Woldendorp et al. (2004) for eucalypt forests of Australia to remain below a 100% CV threshold. Regarding the national DWD inventory of the US (Woodall and Monleon 2008), the total transect length sampled in a fully forested plot is 88 m, which would place the estimated CV below 43% for the locations sampled here. This level of uncertainty may be acceptable, as the strategic-scale inventory is designed to produce reliable DWD estimates (e.g., CV’s below 10%) at the scale of entire forest types or regions (Woodall et al. 2013), not for particular locations of interest, the focus of our study.

LIS provides an attractive method for estimating DWD volume for a given localized area of interest, particularly when transects of reasonable length are employed. LIS sampling has been repeatedly shown to be much more time efficient than the other common sampling alternative, namely censuses of fixed area plots (Bailey 1970; Jordan et al. 2004). Nevertheless, a complete census of large plots, including mapping of DWD pieces, may remain the method of choice in studies where spatial information is needed, or where a particular plot size is needed for comparability to preexisting benchmarks (see Rouvinen and Kouki 2002; Jönsson et al. 2008). It should not be assumed, however, that a complete census of a large plot yields a precise estimate of DWD volume, in part because of the chance of overlooking hard-to-detect DWD pieces (Jordan et al. 2004; Kershaw et al. 2016), and more importantly because of the bias inherent in an assumed piece shape (Fraver et al. 2007; Ducey and Fraver 2018), both problems against which LIS provides natural safeguards. That is, LIS requires tallying only those pieces that cross a specific, well-defined transect, thus avoiding the need to search a large area for hard-to-detect pieces, and as above, no assumption of piece shape is required for LIS sampling.

Conclusions

As interest in forest carbon accounting, woody fuels assessments, and forest structure inventories increases, so does the need for practical DWD sampling guidelines based upon metrics of uncertainty, as we have provided in this study. Our simulations, based on empirical field data, allowed us to test the influence of transect length and DWD volume on the precision of LIS volume estimates, using a range of forest conditions. Our study is among the first to provide practical guidelines for appropriate LIS transect lengths for particular locations, where forest stands or small reserves may themselves represent the entire populations of interest. To the best of our knowledge, our study is the first to apply copula models to account for within-transect spatial autocorrelation of DWD volumes during our simulations, while also addressing a zero-inflated distribution (transect segments may contain no DWD), thereby properly addressing variance estimates. As expected, precision of DWD volume estimates improved dramatically with increasing cumulative transect length; precision improved only slightly with increasing DWD volumes. For the forest types sampled here, as well as the particular copula model framework employed, transect lengths of ca. 120 m provide a reasonable level of precision, ranging from 18% to 60% coefficients of variation. Finally, we note that although we have focused our analyses on DWD volume, these same results would apply to estimates of DWD biomass or carbon density, as these are typically calculated directly from volume data.