Jeffrey H. Gove

Moment and maximum likelihood estimators for Weibull distributions under length- and area-biased sampling

Many of the most popular sampling schemes used in forestry are probability proportional to size methods. These methods are also referred to as size-biased because sampling is actually from a weighted form of the underlying population distribution. Length- and area-biased sampling are special cases of size-biased sampling where the probability weighting comes from a lineal or areal function of the random variable of interest, respectively. Often, interest is in estimating a parametric probability density of the data. In forestry, the Weibull function has been used extensively for such purposes. Estimating equations for method of moments and maximum likelihood for two- and three-parameter Weibull distributions are presented. Fitting is illustrated with an example from an area-biased angle-gauge sample of standing trees in a woodlot. Finally, some specific points concerning the form of the size-biased densities are reported.

Multistage point relascope and randomized branch sampling for downed coarse woody debris estimation

New sampling methods have recently been introduced that allow estimation of downed coarse woody debris using an angle gauge, or relascope. The theory behind these methods is based on sampling straight pieces of downed coarse woody debris. When pieces deviate from this ideal situation, auxillary methods must be employed. We describe a two-stage procedure where the relascope is used to select pieces of downed coarse woody debris in the first stage. If the pieces so chosen on the first stage have multiple branches and detailed estimates are required for the entire piece, then a second stage sample is advocated using the randomized branch technique. Both techniques are reviewed and an example is given examining possible surrogate variables for the second stage.

Monitoring the growth of American beech affected by beech bark disease in Maine using the Kalman filter

The effect of beech bark disease on tree growth was tracked using paired resistant and susceptible American beech trees in two locations in Maine. Within each site, the paired trees were chosen in close proximity and with similar morphological characteristics (e.g. stem diameter and crown class) to minimize environment effects in subsequent analysis. A Kalman filter approach was employed to analyse the yearly time-dependent mean differences between paired susceptible and resistant tree-ring widths using simple structural time series models in state space form. On one site, under the influence of a moderate maritime climate, stand dynamics is hypothesized to account for the 34 year difference in onset of decline of trees in codominant, versus those in the intermediate crown classes. The harsher winter conditions associated with the second, more northerly site and known to limit the insect component of the disease complex, are hypothesized to be more of a factor in the close (six year) difference in decline onset between the two crown classes on this site. Some strengths and cautions in the Kalman filter approach are discussed in relation to the analysis of time-dependent trends in tree-ring series.

Critical length sampling: a method to estimate the volume of downed coarse woody debris

In this paper, critical length sampling for estimating the volume of downed coarse woody debris is presented. Using this method, the volume of downed wood in a stand can be estimated by summing the critical lengths of down logs included in a sample obtained using a relascope or wedge prism; typically, the instrument should be tilted 90° from its usual orientation in sampling live trees. The critical length of a downed tree is the length of the log section large enough in diameter to be counted from the selected sample point. A simulation study was conducted to compare critical length sampling with diameter relascope sampling and perpendicular distance sampling. For the population studied, diameter relascope sampling tended to have the smallest variance but exhibited a small design-bias. By combining critical length sampling with critical height sampling—a method previously described for estimating the volume of standing trees—the total volume of dead wood in a stand can be estimated using the same basic principle. One use of the combined method might be in rapid in-place inventories where a quick estimate of the volume of dead wood is needed for biodiversity management purposes. The combined critical length/height method provides a relatively simple means for obtaining such estimates.

A field test of point relascope sampling of down coarse woody material in managed stands in the Acadian Forest.

We field tested a new method for sampling down coarse woody material (CWM) using an angle gauge and compared it with the more traditional line intersect sampling (LIS) method. Permanent sample locations in stands managed with different silvicultural treatments within the Penobscot Experimental Forest (Maine, USA) were used as the sampling locations. Point relascope sampling (PRS) with three different angles spanning the practical range of angles for such stands was used along with 40 m of LIS sample per sample point. Compared to LIS, the three angles resulted in similar number of pieces and volume of CWM from stands with different histories of repeated partial harvests. In terms of sampling efficiency, PRS was up to 4 times more efficient than LIS. These results, while limited to only one forest type and a relatively small sample, are the first published results available on the field performance of PRS.

Estimating the Carbon in Coarse Woody Debris with Perpendicular Distance Sampling

Perpendicular distance sampling (PDS) is a design for sampling the population of pieces of coarse woody debris (logs) in a forested tract. In application, logs are selected at sample points with probability proportional to volume. Consequently, aggregate log volume per unit land area can be estimated from tallies of logs at sample points. In this chapter we provide protocols and formulae for estimating the carbon in coarse woody debris with PDS. We also provide formulae for estimating components of change in the log population between two points in time.

Simultaneous unbiased estimates of multiple downed wood attributes in perpendicular distance sampling

Perpendicular distance sampling (PDS) is a fast probability-proportional-to-size method for inventory of downed wood. However, previous development of PDS had limited the method to estimating only one variable (such as volume per hectare, or surface area per hectare) at a time. Here, we develop a general design-unbiased estimator for PDS. We then show how that estimator can be used to develop simple measurement protocols that allow simultaneous, unbiased estimation of multiple downed wood variables, including logs per hectare, length of logs per hectare, surface area or area coverage per hectare, and volume per hectare.

Optimizing the Management of Uneven-Aged Spruce-Fir Stands While Preserving Structural Diversity

The National Forest Management Act of 1976 (NFMA) final rule for the National Forest System of the United States contains the following mandate for forest managers: Management prescriptions, where appropriate and to the extent practicable, shall preserve and enhance the diversity of plant and animal communities, including endemic and desirable naturalized plant and animal species, so that it is at least as great as that which would be expected in a natural forest and the diversity of tree species similar to that existing in the planning area (NFMA, §219.27(g) p. 43051–43052).

Optimizing the horizontal structural diversity in uneven-aged northern hardwood stands

Two mathematical programming formulations are presented which allow the determination of diameter distributions that maximize the diameter class diversity in uneven-aged northern hardwood stands. Distributions generated from these models were found to be comparable from a management standpoint and could be incorporated into existing linear programming models as alternative management scenarios. The models presented here provide an initial framework for quantitatively addressing the requirements of the US National Forest Management Act in matters of diversity in the planning process.

Bimodality of the combined removal and signs-of-activities estimator for sampling closed animal populations

The possibility of a bimodal log-likelihood function arises with certain data when the combined removal and signs-of-activities estimator is used. Bimodal log-likelihoods may, in turn, yield disjoint confidence intervals for certain confidence levels. The hypothesis that bimodality is caused by the violation of the equal catchability assumption of the removal model, leading to the combination of contradictory data/models in the combined estimator is set forth. Simulations exploring the effect of the violation of removal model assumptions on estimation and inference showed that the assumption of unequal capture probability influenced the frequency of bimodal likelihoods; similarly, extreme parameter values for probability of capture influenced the number of excessively large confidence intervals produced. A sex-specific combined estimator is developed as a remedial model tailored to the problem. The simulations suggest that both the signs-of-activities estimator and the sex-specific estimator perform equally well over the range of simulations presented, though the signs-of-activities estimator is easier to implement.