Propagating uncertainty through individual tree volume model predictions to large-area volume estimates

Published Date
September 2016, Volume 73, Issue 3, pp 625–633

Title

Propagating uncertainty through individual tree volume model predictions to large-area volume estimates

• Author

• Ronald E. McRoberts, 
• James A. Westfall

• Introduction 

• 
  

• 2. Materials and Methods

• 2.1 Study area

  The study area was Minnesota Survey Unit 1 of the Forest Inventory and Analysis (FIA) program of the Northern Research Station, US Forest Service (Fig. 1). The study area includes approximately 33,353 km2 (12,877 mi2) and consists of forest land dominated by aspen-birch and spruce-fir associations, agricultural land, wetlands, and water.
  

  

  Fig. 1
  

  Minnesota Survey Unit 1 (black), area of Northern Research Station inventory responsibility (gray), and geographic source of model calibration data (Minnesota, Wisconsin, Michigan)

  2.2 Data

  The FIA program conducts the National Forest Inventory (NFI) of the United States of America (USA) and has established field plot centers in permanent locations using a quasi-systematic sampling design that is regarded as producing an equal probability sample (McRoberts et al. 2010). Field crews observe species and measure diameter at breast height (dbh) (1.37 m, 4.5 ft) and height (ht) for all trees with dbh of at least 12.7 cm (5 in.). Volumes (V) for individual trees are predicted using statistical models, aggregated at plot level, expressed as volume per unit area, and typically considered to be observations without error. For this study, data were used for 2178 FIA plots on forest land with 50,176 trees representing 38 species. McRoberts and Westfall (2014, Table 1) describe these data in greater detail. For future reference, these data are characterized as the estimation dataset.
  

  Table 1
  

  Effects on large-area volume estimates of model prediction uncertainty due to uncertainty from underlying sources

  Source of uncertainty	Simple random sampling (SRS) estimators		Stratified (STR) estimators (number of strata)
  					2		4		8
  	Mean	SE	Mean	SE	Mean	SE	Mean	SE
  None	90.17	1.69	90.17	1.11	90.17	0.76	90.17	0.54
  Diameter measurement error	90.17	1.69	90.17	1.11	90.17	0.76	90.17	0.54
  Height measurement errora	90.17	1.69	90.17	1.11	90.17	0.77	90.17	0.54
  Height measurement errorb	90.17	1.69	90.17	1.11	90.17	0.77	90.17	0.55
  Diameter and height measurement errorsa	90.17	1.69	90.17	1.11	90.17	0.77	90.17	0.54
  Diameter and height measurement errorsb	90.17	1.69	90.17	1.11	90.17	0.77	90.17	0.55
  Parameter uncertainty and residual variance	90.17	1.71	90.17	1.14	90.17	0.82	90.17	0.61
  Alla	90.17	1.71	90.17	1.14	90.17	0.82	90.20	0.61
  Allb	90.17	1.71	90.17	1.15	90.17	0.83	90.17	0.62

  aWithin-plot height measurement error correlation: ρ = 0.00

  bWithin-plot height measurement error correlation: ρ = 0.25

  The data used to calibrate the allometric volume model were originally acquired for a taper model study (Westfall and Scott 2010) encompassing 24 northeastern states of the USA (Fig. 1). For the current study, the geographic source of the calibration data was restricted to the States of Michigan, Minnesota, and Wisconsin which span the ecological province that includes the study area. For the 2398 individual trees in the dataset, diameter measurements were obtained using a Barr and Stroud dendrometer at heights of 0.3, 0.6, 0.9, 1.4, and 1.8 m and at approximately 2.5-cm diameter taper intervals up to total tree height. Volumes of sections between height measurements were calculated using Smalian’s formula (Avery and Burkhart 2002, p. 101) as the product of mean cross-sectional area and section length, and total stem volumes for individual trees were calculated by adding volumes for all sections. McRoberts and Westfall (2014) describe the sampling procedure and protocols for individual tree measurements in detail. For future reference, these data are characterized as the calibration dataset.

  2.3 Volume models

  An allometric model of the relationship between V as the response variable and dbh and ht as the predictor variables was formulated as
  

  
  

  Vi=β0×dbhβ1i×htβ2i+εi,$$V_i{\beta }_0{\mathrm{d}\mathrm{b}\mathrm{h}}_i^{{\beta }_1}{\mathrm{h}\mathrm{t}}_i^{{\beta }_2}{\epsilon }_i$$

  (1)

  where i indexes individual trees, ε i is a random residual, and the βs are parameters to be estimated. McRoberts and Westfall (2014) documented the global popularity of this model form. They also demonstrated that a set of species-specific models and a non-specific model were nearly indistinguishable with respect to estimates of large-area volume means and their standard errors (SE); McRoberts et al. (2014b) confirmed this result for a sub-tropical Brazilian dataset. Further, the boundaries for Minnesota Survey Unit 1, which are also the boundaries of the study area, were specifically selected to ensure a large degree of within-unit homogeneity with respect to climate, topography, and forest types. Therefore, only a single, non-specific model was considered for this study.

  Before model fitting, natural logarithmic (ln) transformations of the response and predictor variables were calculated, and the model was reformulated as
  

  
  

  ln(Vi)=α0+α1×ln(dbhi)+α2×ln(hti)+εi,$$\ln V_i{\alpha }_0{\alpha }_1\ln {\mathrm{d}\mathrm{b}\mathrm{h}}_i{\alpha }_2\ln {\mathrm{h}\mathrm{t}}_i{\epsilon }_i$$

  (2)

  where the αs are parameters to be estimated, and the ε i of Eq. (2) are not necessarily the same ε i as for Eq. (1). The advantages of the transformations were that the model could be expressed in linear form which facilitates estimation of the parameters, and heteroskedasticity was removed, thereby eliminating the necessity of weighted regressions. On the original scale, predictions were calculated as
  

  
  

  Vˆi=exp[αˆ0+αˆ1×ln(dbhi)+αˆ2×ln(hi)+σˆε2],$${\stackrel{}{V}}_i\exp {\stackrel{}{\alpha }}_0{\stackrel{}{\alpha }}_1\ln {\mathrm{d}\mathrm{b}\mathrm{h}}_i{\stackrel{}{\alpha }}_2\ln h_i\frac{{\stackrel{}{\sigma }}_{\epsilon }}{2}$$

  (3)

  where σˆε${\stackrel{}{\sigma }}_{\epsilon }$ is the residual standard deviation on the ln-ln scale, and the term σˆε2$\frac{{\stackrel{}{\sigma }}_{\epsilon }}{2}$ compensates for bias that accrues when transforming from the ln-ln scale back to the original scale (Baskerville 1972).

  The quality of model fit on both the ln-ln and original scales was assessed in terms of the proportion of the variability explained by the model. On the ln-ln scale, the proportion was calculated and denoted R 2. The predictions were also transformed from the ln-ln scale back to the original scale using Eq. (3), and the proportion of variability explained by the model was denoted pseudo-R 2 because the assumptions underlying R 2 are not completely satisfied when using non-linear models (Anderson-Sprecher 1994).

  2.4 Estimators

  The simplest approach for estimating large-area parameters is to use the familiar simple random sampling (SRS) estimators,
  

  
  

  μˆSRS=1n∑j=1nyj$${\stackrel{}{\mu }}_{\mathrm{S}\mathrm{R}\mathrm{S}}\frac{1}{n}{\displaystyle \underset{j1}{\overset{n}{}}{y}_j}$$

  (4a)

  and
  

  
  

  Vaˆr(μˆSRS)=∑j=1n(yj-μˆSRS)2n(n-1),$$\mathrm{V}\stackrel{}{\mathrm{a}}\mathrm{r}{\stackrel{}{\mu }}_{\mathrm{S}\mathrm{R}\mathrm{S}}\frac{{\displaystyle \underset{j1}{\overset{n}{}}{y_j\text{-}{\stackrel{}{\mu }}_{\mathrm{S}\mathrm{R}\mathrm{S}}}^2}}{nn\text{-}1}$$

  (4b)

  where n is the total sample size and y j is the observation for the jth plot selected for the sample. The primary advantages of the SRS estimators are that they are intuitive, simple, and unbiased when used with an SRS design; the disadvantage is that variances are frequently large, particularly for highly variable populations and/or small sample sizes. Although Vaˆr(μˆSRS)$\mathrm{V}\stackrel{}{\mathrm{a}}\mathrm{r}{\stackrel{}{\mu }}_{\mathrm{S}\mathrm{R}\mathrm{S}}$ from Eq. (4b) may be biased when used with systematic sampling, it is usually conservative in the sense that it overestimates the variance (Särndal et al. 1992, p. 83). For this study, the finite population correction factor was ignored because of the small sampling intensity of approximately one 670 m2 plot per 1200 ha of study area.

  Because the uncertainty in volume model predictions is independent of the particular large-area estimators, the relative effects of model prediction uncertainty will be greater for estimators that reduce the effects of population variability than for the SRS estimators. Multiple regional FIA programs use post-stratified estimators to reduce variances of estimates with strata based on satellite spectral data. McRoberts et al. (2012) showed that stratifications derived from lidar-based maps of growing stock volume reduced the variance of mean volume per unit area by factors as great as 3.5 relative to the SRS estimators. Therefore, for this study, the effects of volume model uncertainty on large-area estimates of mean volume per unit area were also assessed using post-stratified estimation.

  Post-stratified (STR) estimates of means and variances are calculated using estimators provided by Cochran (1977, pp. 134–135),
  

  
  

  μˆSTR=∑h=1Hwhμˆh$${\stackrel{}{\mu }}_{\mathrm{S}\mathrm{T}\mathrm{R}}{\displaystyle \underset{h1}{\overset{H}{}}{w}_h{\stackrel{}{\mu }}_h}$$

  (5a)

  and
  

  
  

  Vaˆr(μˆSTR)=∑h=1H[wh⋅σˆ2hn+(1−wh)σˆ2hn2],$$\mathrm{V}\stackrel{}{\mathrm{a}}\mathrm{r}{\stackrel{}{\mu }}_{\mathrm{S}\mathrm{T}\mathrm{R}}{\displaystyle \underset{h1}{\overset{H}{}}{w}_h\frac{{\stackrel{}{\sigma }}_h^2}{n}1w_h\frac{{\stackrel{}{\sigma }}_h^2}{n^2}}$$

  (5b)

  where
  

  
  

  μˆh=1nh∑j=1nhyhj,σˆ2h=1nh−1∑j=1nh(yhj-μˆh)2,$$\begin{array}{c}{\stackrel{}{\mu }}_h\frac{1}{n_h}{\displaystyle \underset{j1}{\overset{n_h}{}}{y}_{hj}}\\ {\stackrel{}{\sigma }}_h^2\frac{1}{n_{h}1}{{\displaystyle \underset{j1}{\overset{n_h}{}}{y}_{hj}\text{-}{\stackrel{}{\mu }}_h}}^2\end{array}$$

  n is the total sample size; h = 1,…, H denotes strata; and for the hth stratum, y hj is the jth sample observation, w h is the weight calculated as the proportional area of the stratum, n h is the sample size, and μˆh${\stackrel{}{\mu }}_h$ and σˆ2h${\stackrel{}{\sigma }}_h^2$ are the sample estimates of the mean and variance, respectively.

  Because lidar data were not available for the entire study area, stratifications were simulated by ordering the predicted plot volumes from smallest to largest and dividing the range into intervals with approximately the same number of plots per interval. These intervals simulated strata for which strata weights were estimated as the proportions of plots in the strata. The consequences on estimates of using these simulated strata rather than strata based on an actual lidar-based volume map were twofold. First, when using the same data, the SRS and STR estimates of the study area mean will be exactly the same, regardless of the number of strata used. Operationally, however, differences between stratum weights and proportions of plots per stratum cause SRS and STR estimates of means to differ. Second, the simulated approach assumes that each plot is assigned to the correct stratum, whereas operationally map prediction and geo-location errors cause some plots to be assigned to incorrect strata. These incorrect assignments do not induce bias into the estimators, but they increase the within-stratum variances and the overall variance. The overall result is that the relative effects of volume model prediction uncertainty for this study will be slightly overestimated for the simulated stratifications.

  Cochran (1977, pp. 132–134) suggests that more than six strata are usually not useful; McRoberts et al. (2012) reported that little was gained when using more than six lidar-based strata; and the FIA program uses five spectral-based strata in the study area. For this study, stratifications based on 2, 4, and 8 strata were considered as representative of the range of possibilities.

  2.5 Simulating uncertainty

  The study focused on the effects on estimates of large-area mean volume per unit area of model prediction uncertainty arising from four sources: dbh measurement error, height measurement error, parameter uncertainty, and model residual variance. The tolerance for dbh measurement errors specified by the FIA protocols is that 95 % of measurements are to be within 0.5 % of the true dbh (U.S. Forest Service 2012). Assuming that dbh measurement errors follow a Gaussian distribution with mean 0, the standard deviation of the distribution is σdbhε=0.005×dbh1.96≈0.00255×dbh${\sigma }_{\epsilon }^{\mathrm{d}\mathrm{b}\mathrm{h}}\frac{0.005\mathrm{d}\mathrm{b}\mathrm{h}}{1.96}0.00255\mathrm{d}\mathrm{b}\mathrm{h}$. The tolerance for height measurement errors specified by the FIA protocols is that 90 % of measurements are to be within 10 % of the true height (U.S. Forest Service 2012). Assuming that the height measurement errors follow a Gaussian distribution with mean 0, the standard deviation of the distribution is σhtε=0.10×ht1.645≈0.06079×ht${\sigma }_{\epsilon }^{\mathrm{h}\mathrm{t}}\frac{0.10\mathrm{h}\mathrm{t}}{1.645}0.06079\mathrm{h}\mathrm{t}$.

  Uncertainty in the linear model parameter estimates on the ln-ln scale was assessed using a 3-step Monte Carlo approach: (i) the transformed calibration dataset was aggregated into 10 dbh size classes, each with approximately the same number of observations; (ii) each dbh size class was resampled with replacement until the original class sample size was achieved; and (iii) the model was fit to the resampled data and the parameters were estimated. Steps (i)–(iii) were then replicated until the means and standard deviations of the distributions of parameter estimates stabilized. The resulting multiparameter distribution of parameter estimates represented the uncertainty in estimates of the linear model parameters on the ln-ln scale.

  Residual uncertainty was assessed on the original scale where the models were applied using a 4-step procedure that accommodated heteroskedasticity: (i) the pairs (Vi,Vˆi)$V_i{\stackrel{}{V}}_i$ were ordered with respect to the model prediction, Vˆi${\stackrel{}{V}}_i$; (ii) the pairs were aggregated into groups of size 25; (iii) within each group, g, the mean of the observations V¯¯¯¯g${\stackrel{}{V}}_g$, the mean of the predictions Vˆ¯¯¯¯g${\stackrel{}{\stackrel{}{V}}}_g$, and the standard deviation σˆg${\stackrel{}{\sigma }}_g$ of the residuals εi=Vi-Vˆi${\epsilon }_i{V}_i\text{-}{\stackrel{}{V}}_i$ were calculated; and (iv) the relationship between the group standard deviations, σˆg${\stackrel{}{\sigma }}_g$, and the group prediction means, Vˆ¯¯¯¯g${\stackrel{}{\stackrel{}{V}}}_g$, was represented using the model,
  

  
  

  σˆg=γ1×Vˆ¯¯¯¯gγ2+εg,$${\stackrel{}{\sigma }}_g{\gamma }_1{{\stackrel{}{\stackrel{}{V}}}_g}^{{\gamma }_{{}_2}}{\epsilon }_g$$

  (6)

  where the γs are parameters to be estimated.

  2.6 Uncertainty in large-area volume estimates

  A 6-step Monte Carlo simulation procedure was used to estimate the effects of model prediction uncertainty on the uncertainty of large-area estimates of mean volume per unit area.
  

  1. Step 1.
     For the kth replication, a set of model parameter estimates, βˆk${\stackrel{}{\beta }}^k$, was randomly selected from the distribution constructed in “Section 2.5”.
  2. Step 2.
     For the ith tree on the jth plot in the estimation dataset, a random number, ε, was drawn from a Gaussian (0,1) distribution; if |ε| > 2.5, ε was redrawn. A dbh observation was then simulated as
     

     
     

     dbhij=dbh0ij+ε×σdbhε,$${\mathrm{d}\mathrm{b}\mathrm{h}}_{ij}{\mathrm{d}\mathrm{b}\mathrm{h}}_{ij}^0\epsilon {\sigma }_{\epsilon }^{\mathrm{d}\mathrm{b}\mathrm{h}}$$

     where dbh ij 0 was the observation from the estimation dataset, and σ ε dbh was as described in “Section 2.5.”
  3. Step 3.
     For the ith tree on the jth plot in the estimation dataset, a random number, ε, was drawn from a Gaussian (0,1) distribution; if |ε| > 2.5, ε was redrawn. A height observation was then simulated as
     

     
     

     htij=ht0ij+ε×σhtε,$${\mathrm{h}\mathrm{t}}_{ij}{\mathrm{h}\mathrm{t}}_{ij}^0\epsilon {\sigma }_{\epsilon }^{\mathrm{h}\mathrm{t}}$$

     where ht ij 0 was the observation from the estimation dataset, and σ ε ht was as described in “Section 2.5.”
  4. Step 4.
     For the ith tree on the jth plot in the estimation dataset, an initial volume observation was calculated using the parameter values from step (1) and the simulated dbh and height observations from steps (2) and (3) as
     

     
     

     Vk,0ij=βˆk1×dbhijβˆk2×htijβˆk3.$$V_{ij}^{k0}{\stackrel{}{\beta }}_1^k{{\mathrm{d}\mathrm{b}\mathrm{h}}_{ij}}^{{\stackrel{}{\beta }}_2^k}{{\mathrm{h}\mathrm{t}}_{ij}}^{{\stackrel{}{\beta }}_3^k}$$

     A random number, ε, was drawn from a Gaussian (0,1) distribution; if |ε| > 2.5, ε was redrawn. The residual standard deviation, σˆij${\stackrel{}{\sigma }}_{ij}$, was then calculated using Eq. (6) with V ij k,0 as the value of the predictor variable. The individual tree volume was then simulated as
     

     
     

     Vkij=Vk,0ij+ε×σˆij.$$V_{ij}^k{V}_{ij}^{k0}\epsilon {\stackrel{}{\sigma }}_{ij}$$
  5. Step 5.
     The total volume for the jth plot in the estimation dataset was calculated as Vkj=∑i=1njVkij$V_j^k{\displaystyle \underset{i1}{\overset{n_j}{}}{V}_{ij}^k}$ where n j is the number of trees on the plot.
  6. Step 6.
     The overall study area mean, V¯¯¯¯k${\stackrel{}{V}}^k$, and variance of the mean, Vaˆr(V¯¯¯¯k)$\mathrm{V}\stackrel{}{\mathrm{a}}\mathrm{r}{\stackrel{}{V}}^k$, for the kth replication were estimated using both the SRS and STR estimators as described in “Section 2.4.”

  Steps (1)–(6) were replicated, and the mean and variance over replications were estimated as per Rubin (1987, pp.76–77),
  

  
  

  μˆ=1nrep∑k=1nrepV¯¯¯¯k,$$\stackrel{}{\mu }\frac{1}{n_{\mathrm{r}\mathrm{e}\mathrm{p}}}{\displaystyle \underset{k1}{\overset{n_{\mathrm{r}\mathrm{e}\mathrm{p}}}{}}{\stackrel{}{V}}^k}$$

  (7)

  and
  

  
  

  Vaˆr(μˆ)=(1+1nrep)×W1+W2,$$\mathrm{V}\stackrel{}{\mathrm{a}}\mathrm{r}\stackrel{}{\mu }1\frac{1}{n_{\mathrm{r}\mathrm{e}\mathrm{p}}}{W}_1{W}_2$$

  (8)

  where W1=1nrep-1∑k=1nrep(V¯¯¯¯k-μˆ)2$W_1\frac{1}{n_{\mathrm{r}\mathrm{e}\mathrm{p}}\text{-}1}{\displaystyle \underset{k1}{\overset{n_{\mathrm{r}\mathrm{e}\mathrm{p}}}{}}{{\stackrel{}{V}}^k\text{-}\stackrel{}{\mu }}^2}$ is the among-replications variance, W2=1nrep∑k=1nrepVaˆr(V¯¯¯¯k)$W_2\frac{1}{n_{\mathrm{r}\mathrm{e}\mathrm{p}}}{\displaystyle \underset{k1}{\overset{n_{\mathrm{r}\mathrm{e}\mathrm{p}}}{}}V\stackrel{}{\mathrm{a}}\mathrm{r}{\stackrel{}{V}}^k}$ is the mean within-replication variance, and n rep is the number of replications. The replications continued until μˆ$\stackrel{}{\mu }$ and SE(μˆ)=Vaˆr(μˆ)−−−−−−√$\mathrm{S}\mathrm{E}\stackrel{}{\mu }\sqrt{V\stackrel{}{\mathrm{a}}\mathrm{r}\stackrel{}{\mu }}$ stabilized.

  In Steps (2) and (3), dbh and height measurement errors for the same tree were assumed to be independent, as were dbh errors for trees on the same plot. However, because of the greater difficulty in accurately measuring height and because plot canopy conditions tend to affect measurements of the heights of all trees on the same plot in a similar manner, height measurement errors for trees on the same plot may not be independent. Therefore, simulations were conducted separately for height correlations of ρ = 0.00 and ρ = 0.25. In Step (4), spatial correlations among residuals for trees on the same plot were ignored based on Berger et al. (2014), Breidenbach et al. (2014), and McRoberts and Westfall (2014) who all reported that the effects were negligible.

  Parameter uncertainty and residual variance are highly correlated because each necessarily affects the other. This phenomenon is easily understood by considering the parametric form of the parameter covariance matrix for a linear model; in particular,
  

  
  

  Var(βˆ)=σ2×(X′X)-1,$$\mathrm{V}\mathrm{a}\mathrm{r}\stackrel{}{\text{β}}{\sigma }^2{\mathbf{X}\mathrm{\prime }\mathbf{X}}^{\text{-}1}$$

  where X is the matrix of values of the independent variables and σ 2 is the residual variance (Bates and Watts 1988, p. 5). Thus, if σ 2  = 0, the covariances of the parameter estimates are necessarily 0; conversely, the only way the covariances can all simultaneously be 0 is if σ 2  = 0. Therefore, for this study, neither parameter uncertainty nor residual variance was incorporated into the simulation procedure apart from the other.

• 3. Results 

• The fit of the model to the data on the ln-ln scale produced R 2  = 0.98 with corresponding pseudo-R 2 = 0.97 on the original scale (Figs. 2 and 3). These large values justify the initial decision not to consider model misspecification for this study (“Section 1”) and also suggest that other model forms would likely not have produced more accurate predictions. However, as with most similar datasets, the number of observations for large trees is smaller than for other trees. Although this phenomenon could contribute to serious lack of fit of the model for large trees, such was not the case for this study (Figs. 2 and 3).
  

  

  Fig. 2
  

  Group observation means versus group prediction means on ln-ln scale

  

  Fig. 3
  

  Group observation means versus group prediction means on original scale

  On the ln-ln scale, the distribution of simulated parameter estimates exhibited an ellipsoidal pattern as expected for linear models (Fig. 4). Parameter uncertainty was simulated by random draws from this distribution. The approach to estimating the relationship between heteroskedastic residual standard deviations and volume model predictions as described in “Section 2.5” was somewhat arbitrary, but the relationship was well estimated (Fig. 5).
  

  

  Fig. 4
  

  Distribution of parameter estimates used to simulate parameter uncertainty

  

  Fig. 5
  

  Observed versus predicted heteroskedastic residual standard deviations

  For all combinations of dbh measurement error, height measurement error, and parameter uncertainty and residual variance, 5000 replications of the simulation procedure were sufficient for estimates of both means and SEs to stabilize (Fig. 6). Further, no prediction for any of the more than 50,000 trees in the estimation dataset over the 5000 replications was proportionally less than 0.87 or greater than 1.20 than the prediction with the original parameter estimates. For large trees, the proportions were 0.95 and 1.05.
  

  

  Fig. 6
  

  Simulated mean biomass per unit area (Mg/ha) and standard error versus replications for four strata and incorporating all sources of uncertainty

  Means of tree-level dbh measurement errors ranged from approximately −0.12 to 0.12 cm with nearly 98 % between −0.05 and 0.05 cm, and means of tree-level height measurement errors ranged from approximately −2.4 to 2.1 m with nearly 98 % between −1.0 and 1.0 m. These relatively small tree-level errors have minimal effects at the population level.
  For the STR estimators, the effects of uncertainty from the four sources on SEs increased as the number of strata increased, although nearly all the increase is attributed to the combined effects of parameter uncertainty and residual variance. For two strata, the proportional increase in SE was 0.036; for four strata, the proportional increase was 0.092; and for eight strata, the proportional increase was 0.148. Although the increase for two strata would likely be considered negligible, such may not be the case for four and eight strata. As previously noted, the simulated stratifications likely reduce SEs more than would be realized with actual stratifications with the result that the actual increases in SE may be slightly less than those reported in Table 1.

• 4. Discussion 

• The simulated stratifications accomplished the intended objective by grouping plots into strata with greater homogeneity than the population as a whole and thereby reducing the variance of the estimate of the population mean relative to the variance of the SRS mean (Tables 1 and 2). As previously noted, the lack of differences among the SRS and STR estimates of the means is attributed to using proportions of plots per stratum as stratum weights rather than proportions of the population.
  

  Table 2
  

  Stratified estimates

  Stratum	Weight	Sample size	Mean	SE
  1	0.25	545	12.42	0.35
  2	0.25	544	49.13	0.54
  3	0.25	545	100.14	0.71
  4	0.25	544	199.14	2.90
  Total	1.00	2178	90.17	0.76

  No uncertainty incorporated

  When using the SRS estimators, no source of uncertainty individually or in combination had a meaningful effect on SEs (Table 1). This result is consistent with the results reported by McRoberts et al. (2014b) for a sub-tropical dataset. For both the SRS and STR estimators, the effects of dbh and height measurement errors, including correlated height measurement errors, were negligible. This result can be attributed to the fairly large number of trees per plot (mean 23, maximum 134) which resulted in relatively small mean dbh and height measurement errors at the plot level. Berger et al. (2014) and Qi et al. (2015) reported similar results. Overall, these results suggest that as long as height measurements satisfy FIA protocols, these sources of uncertainty produce no meaningful adverse consequences. However, experience suggests that height measurements may fail to satisfy the protocols. Nevertheless, results obtained using a 20 % rather than 10 % tolerance and an 80 % rather than 90 % satisfaction rate were essentially unchanged.

  The combined effects of parameter uncertainty and residual variance were greater than the combined effects of dbh and height measurement error. This result can be at least partially attributed to how parameter uncertainty affects plot-level estimates. Whereas measurement errors and prediction residuals are incorporated separately for individual trees and compensate for each other, parameter uncertainty is realized at the population level and therefore should be expected to have a greater population-level effect.

  The important finding is that as the effects on SEs of population variability are reduced by using the STR estimators rather than the SRS estimators, the relative effects of underlying sources of model prediction uncertainty increase. Model-assisted estimators, which are receiving increasing attention for inventory applications, often reduce the effects of population variability even more than do stratified estimators (McRoberts et al. 2013, 2014a). Thus, the proportional adverse effects of measurement error, parameter uncertainty, and residual variance on the uncertainty of the large-area volume estimates may be even greater when the effects of sampling variable populations are further reduced.

• 5. Conclusions

Three conclusions may be drawn from the study. First, when using the simple random sampling estimators, the effects on large-area volume estimates of uncertainty in individual tree volume model predictions due to diameter and height measurement error, parameter uncertainty, and residual variance were negligible. Second, however, when the effects of variability in the population on uncertainty were reduced via stratified estimation, the effects of model prediction uncertainty on the large-area volume estimates increased as the number of strata increased. For four and eight strata, the proportional increases in the stratified SEs were as great as 0.092 and 0.148, respectively, which may not be negligible. Third, nearly all the effects of model prediction uncertainty can be attributed to parameter uncertainty and residual variance. Finally, all results for this study are contingent on the calibration dataset sample size and the quality of fit of the model to the data, both of which directly affect parameter uncertainty and residual variance.

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