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Historical forest conditions are often used to inform contemporary management goals because historical forests are considered to be resilient to ecological disturbances. The General Land Office (GLO) surveys of the late 19th and early 20th centuries provide regionally quasi-contiguous data sets of historical forests across much of the Western United States. Multiple methods exist for estimating tree density from point-based sampling such as the GLO surveys, including distance-based and area-based approaches. Area-based approaches have been applied in California mixed-conifer forests but their estimates have not been validated. To assess the accuracy and precision of plotless density estimators with potential for application to GLO data in this region, we imposed a GLO sampling scheme on six mapped forest stands of known densities (159–784 trees/ha) in the Sierra Nevada in California, USA, and Baja California Norte, Mexico. We compared three distance-based plotless density estimators (Cottam, Pollard, and Morisita) as well as two Voronoi area (VA) estimators, the Delincé and mean harmonic Voronoi density (MHVD), to the true densities. We simulated sampling schemes of increasing intensity to assess sampling error. The relative error (RE) of density estimates for the GLO sampling scheme ranged from 0.36 to 4.78. The least biased estimate of tree density in every stand was obtained with the Morisita estimator and the most biased was obtained with the MHVD estimator. The MHVD estimates of tree density were 1.2–3.8 times larger than the true densities and performed best in stands subject to fire exclusion for 100 yr. The Delincé approach obtained accurate estimates of density, implying that the Voronoi approach is theoretically sound but that its application in the MHVD was flawed. The misapplication was attributed to two causes: (1) the use of a crown scaling factor that does not correct for the number of trees sampled and (2) the persistent underestimate of the true VA due to a weak relationship between tree size and VA. The magnitude of differences between true densities and MHVD estimates suggest caution in using results based on the MHVD to inform management and restoration practices in the conifer forests of the American West.

Accurate forest density estimates based on United States Public Land Surveys have long been questioned because of doubts about randomness of both the surveyors' selection of witness trees and the underlying tree dispersion. This study analyzes the surveyor sampling of witness trees in six Midwestern states in the mid‐1800s. It develops universal methods for identification, quantification, and correction of bias, and then calculation of unbiased density. Applying these techniques produces unbiased site‐specific densities before Euro‐American settlement, which are the essential baseline for determining historic changes in forest structure. Previous analysts used untested assumptions, inaccurate estimators, unknown or unrealistic sampling designs, and omitted or poorly corrected surveyor bias, resulting in hundreds of unreliable density estimates. The surveyors' recording of the empirical distance, bearing, and diameter of witness trees documented the exact sampling design. The intended design and deviation from it are investigated with a combination of descriptive statistics, probability theory, computer simulations, analogue geometric models, and modern stand conditions. Herein, analyst bias is eliminated using the robust Morisita II density estimator matching the predominant sampling design of two trees in opposing semicircles. Six widespread surveyor biases deviating from the nearest tree to the corner are evaluated. Quadrant bias and diameter bias for medium‐sized trees are subsumed under newly framed design and small tree biases. Two novel surveyor positional biases (pair angle and near‐post) are introduced here. Previously recognized azimuthal and species biases are analyzed with new techniques. Widely postulated surveyor bias for certain species was found to be minimal. Bias correction and density estimation are applied in detail over 68 townships in northern Wisconsin. The estimated historical forest density in northern Wisconsin, corrected for bias and small tree truncation, averaged 323 trees/ha ≥20 cm. Over 80 Midwestern subregions, surveyors bypassed an estimated 17% of the nearest trees due to their position, resulting in an average bias correction of +24% over the base density. If censored trees below a 20‐cm “veil‐line” are considered, the surveyors bypassed 48% of the nearest trees >12.7 cm in diameter. This study resolves a 70‐year‐old conundrum of surveyor and analyst biases in historical density estimation.

Sampling point‐to‐tree distances is a simple plotless technique for estimating forest density that is readily applied in modern stands and retroactively with historical surveys. Although plotless density estimators (PDEs) have been applied in over 1000 ecological publications, the accuracy and precision of the techniques remain poorly understood and depend on the statistical estimator used, the underlying spatial pattern of the forest sampled, and the tree survey methodology. The four most commonly applied PDEs are related formulations: Cottam, Pollard, Morisita, and Shanks, a family of equations that differ in the order of mathematical operations. Since the 1950s, the Cottam IV PDE has found common use as the “point‐quarter method.” The Pollard PDE prevails in the statistical literature. Both Cottam and Pollard PDEs are theoretically rigorous for trees distributed according to a complete spatial randomness (CSR) spatial point process. The Morisita PDE was developed in a 1957 publication, with four‐tree (Morisita IV) and two‐tree (Morisita II) variants, and is the basis for higher distance rank g‐tree estimators. The Shanks PDE is formally described here for the first time. We review and evaluate the performance of these four PDEs on CSR and a variety of non‐CSR forests using spatial patterns simulated from known spatial point processes, 14 mapped modern stands, and historical public land surveys (PLSs). We found that the Cottam and Pollard PDEs lacked accuracy for non‐CSR patterns. The Morisita PDEs are appropriate for non‐CSR forests, but the Morisita IV has sensitivity to local dispersion. The Morisita II PDE has high accuracy even under non‐CSR distributions yielding density estimates within 10% of the true value for a variety of non‐CSR patterns, but has considerable variability at small sample sizes. In conjunction with the Morisita II, the potentially biased Cottam and Pollard PDEs can be indicators of the type of non‐CSR pattern. No plotless estimator is efficacious for use with small sample sizes such as found in a single stand. Morisita II PDE is recommended as a robust choice for sampling for large and non‐CSR data sets such as the PLS witness tree database.

The accuracy of methods for reconstructing parameters (e.g., tree density) of historical forest structure from General Land Office (GLO) survey data has not been thoroughly assessed. Past simulation and statistical assessments of plotless density estimators have focused on minimizing estimation error, but not congruent with the specific data available in the GLO surveys. Most GLO reconstruction studies do not reconstruct absolute measures of density, basal area, or diameter-class distributions, key measures used for forest restoration. We tested the accuracy of a suite of plotless density estimators and other survey methods to accurately reconstruct forest attributes using both a field-based modern calibration and a cross-validation with tree-ring reconstructions. In addition to the common distance estimators, we developed several Voronoi-based plotless density estimators that can be used with GLO data. Estimators were assessed using modern survey and plot data collected in the same location and spatial arrangement as the original survey locations in three geographically distinct areas. Results showed that Voronoi-based density estimators were superior to distance-based estimators. Data need to be pooled across locations. Voronoi estimators yielded more accurate measures of density and basal area and can be used at smaller pooling levels without sacrificing much accuracy. At spatial extents of 260 and 520 ha (3- and 6-corner pools), relative mean absolute error (RMAE) averaged 29%% and 22%%, respectively, for density estimates in all three study areas. To estimate basal area as accurately (i.e., 23%%), data must be pooled to 780 ha (9-corner pool). Composition and diameter-class distributions also required larger pooling areas to achieve accurate results. In the cross-validation, accuracy of density and basal area were both superior to accuracy in the modern calibration, and RMAE for density and basal area at all pooling levels averaged 16.6%% and 15.7%%, respectively. Composition and diameter-class distribution estimates were lower in accuracy. Voronoi-based methods can accurately estimate historical forest parameters across large landscapes and are accurate at finer scales (e.g., 260 ha, 3-corner pool) than previously thought possible. GLO reconstructions complement tree-ring reconstructions but can provide more spatially comprehensive estimates of the historical range of forest variability, facilitating landscape-level restoration.

The variable area transect (VAT) is a plotless density estimator that has received little attention in the ecological literature despite having potentially robust estimation properties. VAT allows for density estimations without the lengthy search times associated with other plotless density estimators. In spite of this, little has been written about the effect of varying transect widths on its density estimation properties or on the practical implementation of the VAT in field settings. An artificial population sampler was used to examine the effect of transect width on density estimates obtained using the VAT. Three transect widths were chosen corresponding to the mean object size, the largest object size, and twice the size of the largest object. Transect width had a marked effect on the quality of the density estimation, with the largest transect width resulting in significant negative biases in estimation. For the narrowest width, most estimates were within 10% of the true value for a nonrandomly distributed population. The practical considerations of choosing a VAT transect width are enumerated.

Population density is the most basic ecological parameter for understanding population dynamics and biological conservation. Distance-based methods (or plotless methods) are considered as a more efficient but less robust approach than quadrat-based counting methods in estimating plant population density. The low robustness of distance-based methods mainly arises from the oversimplistic assumption of completely spatially random (CSR) distribution of a population in the conventional distance-based methods for estimating density of non-CSR populations in natural communities. In this study we derived two methods to improve on density estimation for plant populations of non-CSR distribution. The first method modified an existing composite estimator to correct for the long-recognized bias associated with that estimator. The second method was derived from the negative binomial distribution (NBD) that directly deals with aggregation in the distribution of a species. The performance of these estimators was tested and compared against various distance-based estimators by both simulation and empirical data of three large-scale stem-mapped forests. Results showed that the NBD point-to-tree distance estimator has the best and most consistent performance across populations with vastly different spatial distributions. This estimator offers a simple, efficient and robust method for estimating density for empirical populations of plant species

Estimation of tree density from point-tree distances is an attractive option for quick inventory of new sites, but estimators that are unbiased in clustered and dispersed situations have not been found. Noting that bias of an estimator derived from distances to the kth nearest neighbor from a random point tends to decrease with increasing k, a method is proposed for estimating the limit of an asymptotic function through a set of ordered distance estimators. A standard asymptotic model is derived from the limiting case of a clustered distribution. The proposed estimator is evaluated against 13 types of simulated generating processes, including random, clustered, dispersed, and mixed. Performance is compared with ordered distance estimation of the same rank and with fixed-area sampling with the same number of trees tallied. The proposed estimator consistently performs better than ordered distance estimation and nearly as well as fixed-area sampling in all but the most clustered situations. The estimator also provides information regarding the degree of clustering or dispersion.

An extensive simulation study was conducted to optimize the number, r, of population members to be encountered from each random starting point in variable area transect (VAT) sampling. The quality of estimation provided by the original calculation formula presented by K. R. Parker in 1979 was compared to another formula that was a Morisita analog intended to reduce bias when sampling aggregated populations. Monte Carlo simulations covered 64 combinations of four spatial patterns, four sample sizes, and four densities. Values of r from 3 through 10 were considered in each case. Relative root mean squared error was used as the primary assessment criterion. Superior estimation properties were found for r > 3, but diminishing returns, relative to the potential for increased effort in the field, were found for r > 6. The original estimation formula consistently provided results that were superior to the Morisita analog, with the difference most pronounced in the aggregate patterns for which the Morisita analog was intended. As long as the sampled populations displayed randomness in location of individuals, rather than systematic patterns that are uncommon in nature, the variance formula associated with the original estimation formula performed well. Additional simulations were conducted to examine four confidence interval methods for potential use in association with the Parker original estimation method. These simulations considered only the sample sizes for which the best estimation was achieved in the earlier simulations. The confidence interval method developed by Parker worked well for populations with random spatial patterns, but it rarely achieved 80% (generally much less) of target coverage for populations displaying aggregation. A nonparametric confidence interval method presented here, or a combination of it with the Parker method, is recommended for general use.

One of the most fundamental decisions that must be made concerning a sampling program is the choice of sampling units, which is the topic of this chapter. A wide variety of sampling units can be used for intertidal sampling. The most common units include line transects and plots or quadrats (fig. 5.1). In addition, plotless designs are sometimes used. The choice of sampling unit depends on the goals of the sampling program, especially the species to be sampled. Chapter 2 addresses how to decide where to sample on the scale of study sites. Chapter 4 considers issues of sampling design,