Monte Carlo Simulation Tools for REDD+ Uncertainty Estimates
2024-12-19
- Introduction
- Scope of Work
- Registry Requirements
- Methods Review
- Example script
- Appendix
- Session runtime
- References
Introduction
The ART-TREES Standard V2.01 mandates specific methodologies for calculating and reporting uncertainty estimates associated with emission factors and activity data within jurisdictional and nested REDD+ projects. To strengthen compliance, the ART-TREES project team produced the following report and capacity building resources.
Scope of Work
This report focuses on the following technical areas:
- Develop Monte Carlo simulation pathways to quantify uncertainty in emission factors and activity data, ensuring consistency with ART-TREES’s emphasis on robust uncertainty analysis and corrective bias assessment.
- Use R or other software to create systems that streamline data workflows and enhance accessibility for MRV purposes. Monte Carlo Simulation for Uncertainty Estimation
- Document methodologies and provide results in formats compliant with ART-TREES reporting standards.
- Prepare technical reports that detail uncertainty estimation methods and database management workflows.
- Develop and deliver training materials to strengthen stakeholder capacity to use ART-TREES-aligned tools and methodologies.
Figure 1: Visualization of k-fold resampling (Note differences with Monte Carlo “LOGCV”)
Registry Requirements
The TREES 2.0 Standard outlines requirements for reporting uncertainty in emissions and removals, and adjusting estimates where uncertainty levels exceed the defined threshold of a half-width of a 90% confidence interval between the upper and lower bounds. Monte Carlo simulations are identified as an appropriate methodology due to their capacity to model variance and provide conservative estimates from large-scale highly-variable datasets. Specifically, “Monte Carlo simulations shall use the 90% confidence interval and a simulation n of 10,000” (p.45).
Aggregation of Uncertainty Across Crediting
Periods
The TREES Standard provides a level of flexibility in allowing
participants to aggregate uncertainty deductions across multiple
crediting periods. At the end of each crediting period, participants may
calculate a consolidated uncertainty deduction based on the summed gross
emissions reductions and removals achieved over their entire ART
participation. If prior uncertainty deductions exceeded the aggregated
deduction sum for the total period, the over-deducted credits will be
issued into the participant’s registry account.
Inclusion of Biomass Map Uncertainty
Uncertainty must be assessed and reported for emissions factors derived
from biomass maps, as these datasets directly impact the accuracy of
emission estimates. TREES participants are encouraged to adopt best
practices, such as those outlined in the CEOS LPV Biomass Protocol 2021,
to enhance calibration, validation, and reliability of spatially
explicit datasets. In this guidance document, key recommendations for
good practices include appropriate scaling, temporally & spatially
consistent reference data and remote sensing, and the use of approved
error metrics (90% CI or RMSE). In particular, three likely sources of
uncertainty in biomass estimation are highlighted separately for
consideration in assessing and calibrating predictions.
- Measurement Uncertainty in tree measurements (i.e DBH and height).
- Allometric Model Errors in statistically inferring biomass from from tree measurements
- Sampling & Spatial Uncertainty arising from autocorrelation & over-fitting
Exemption for Allometric Estimates
An exemption from requirements for Monte Carlo simulations is granted to
allometric modeled estimates. The TREES Standards V2.0 states that “such
errors are considered consistent between emissions in the crediting
level and crediting periods” which therefore do not materially influence
the net results.
Calculating Uncertainty Deductions
Cited on page 46 of the TREES Standards V2.0, calculations of
uncertainty deductions are derived using the following formulae:
\[ UNC_t = (GHG ER_t + GHG REMV_t) \times UA_t \]
| \(UNC_t\) | Uncertainty deduction for year \(t\) (\(tCO_2e\)) |
| \(GHG ER_t\) | Gross greenhouse gas emissions reductions for year \(t\) (\(tCO_2e\)) |
| \(GHG REMV_t\) | Gross greenhouse gas removals for year \(t\) (\(tCO_2e\)) |
| \(UA_t\) | The uncertainty adjustment factor for year \(t\) |
Table 1: Parameters used in Equation 10
The uncertainty adjustment factor (\(UAdj_t\)) quantifies the proportional adjustment to emissions reductions and removals based on statistical uncertainty. It is defined as:
\[ UAdj_t = 0.524417 \times \frac{HW_{90\%t}}{1.645006} \]
| \(90\%\text{ C I}_{t}\) | The half-width of 90% confidence interval as percentage of mean |
| \(1.645006\) | \(t\) value for a 90% confidence interval |
| \(0.524417\) | A scaling constant to adjust the proportion. |
Table 2: Parameters used in Equation 11
Example script
Environment setup
easypackages::packages(
"animation", "BIOMASS", "caret", "dataMaid", "DescTools", "dplyr",
"extrafont", "FawR", "ForestToolsRS", "ggplot2", "htmltools",
"janitor", "jsonlite", "lattice", "kableExtra", "kernlab",
"knitr", "Mlmetrics", "olsrr", "plotly", "psych", "RColorBrewer",
"rmarkdown", "readxl", "solarizeddox", "tibble", "tidymodels", "tidyverse",
"tinytex", "tune", "useful", "webshot", "webshot2",
prompt = F
)Monte Carlo of Emissions Factors
Import data
This section outlines the tools for importing and preparing forestry
and biomass data for analysis, a key step in building
ART-TREES-compliant MRV systems. Using the allodb package,
we load a global allometry database and a dummy dataset from the
Smithsonian Institute ForestGEO project.
library("allodb") # https://docs.ropensci.org/allodb/
set.seed(333)
#data(ufc) # spuRs::vol.m3(dataset$dbh.cm, dataset$height.m, multiplier = 0.5)
data(scbi_stem1)
dataset = scbi_stem1
head(dataset) |> tibble::as_tibble()# A tibble: 6 × 6
treeID stemID dbh genus species Family
<int> <int> <dbl> <chr> <chr> <chr>
1 2695 2695 1.41 Acer negundo Sapindaceae
2 1229 38557 1.67 Acer negundo Sapindaceae
3 1230 1230 1.42 Acer negundo Sapindaceae
4 1295 32303 1.04 Acer negundo Sapindaceae
5 1229 32273 2.47 Acer negundo Sapindaceae
6 66 31258 2.19 Acer negundo Sapindaceaepsych::describe(dataset) vars n mean sd median trimmed mad min max
treeID 1 2287 2778.66 1929.26 2525.00 2705.54 2091.95 1 6207.00
stemID 2 2287 16577.12 16197.88 5022.00 15661.27 5749.52 1 40180.00
dbh 3 2287 5.52 10.80 1.67 2.65 0.79 1 92.02
genus* 4 2287 16.37 6.52 18.00 16.71 0.00 1 31.00
species* 5 2287 13.26 9.60 8.00 11.31 0.00 1 40.00
Family* 6 2287 13.07 4.02 13.00 13.33 0.00 1 22.00
range skew kurtosis se
treeID 6206.00 0.27 -1.11 40.34
stemID 40179.00 0.40 -1.75 338.71
dbh 91.02 3.81 16.30 0.23
genus* 30.00 -0.57 0.14 0.14
species* 39.00 1.59 1.30 0.20
Family* 21.00 -0.58 1.44 0.08str(dataset)tibble [2,287 × 6] (S3: tbl_df/tbl/data.frame)
$ treeID : int [1:2287] 2695 1229 1230 1295 1229 66 2600 4936 1229 1005 ...
$ stemID : int [1:2287] 2695 38557 1230 32303 32273 31258 2600 4936 36996 1005 ...
$ dbh : num [1:2287] 1.41 1.67 1.42 1.04 2.47 ...
$ genus : chr [1:2287] "Acer" "Acer" "Acer" "Acer" ...
$ species: chr [1:2287] "negundo" "negundo" "negundo" "negundo" ...
$ Family : chr [1:2287] "Sapindaceae" "Sapindaceae" "Sapindaceae" "Sapindaceae" ...Table 3: Smithsonian Institute GEOForest dataset from
allodb package (n = 2287)
Probability density functions
Accurate selection of probability density functions (PDFs) is essential for modeling uncertainties in carbon stocks and activity data. This section describes methodologies for fitting PDFs to data, ensuring results are robust and aligned with ART-TREES best practices.
Use of statistical tests for goodness-of-fit validation.
Integration of domain expertise to refine parameter selection.
# add allometry database
data(equations)
data("equations_metadata")
show_cols = c("equation_id", "equation_taxa", "equation_allometry")
eq_tab_acer = new_equations(subset_taxa = "Acer")
head(eq_tab_acer[, show_cols])# A tibble: 6 × 3
equation_id equation_taxa equation_allometry
<chr> <chr> <chr>
1 a4e4d1 Acer saccharum exp(-2.192-0.011*dbh+2.67*(log(dbh)))
2 dfc2c7 Acer rubrum 2.02338*(dbh^2)^1.27612
3 eac63e Acer rubrum 5.2879*(dbh^2)^1.07581
4 f49bcb Acer pseudoplatanus exp(-5.644074+(2.5189*(log(pi*dbh))))
5 14bf3d Acer mandshuricum 0.0335*(dbh)^1.606+0.0026*(dbh)^3.323+0.1222*…
6 0c7cd6 Acer mono 0.0202*(dbh)^1.810+0.0111*(dbh)^2.740+0.1156*…# Compute above ground biomass
dataset$agb = allodb::get_biomass(
dbh = dataset$dbh,
genus = dataset$genus,
species = dataset$species,
coords = c(-78.2, 38.9)
)
# examine dbh ~ agb function
dbh_agb = lm(dbh ~ agb, data = dataset)
#olsrr::ols_test_breusch_pagan(lm(dbh_agb)) #<0.0000
#h = lattice::histogram(dbh ~ agb, data = dataset)
plot(
x = dataset$dbh,
y = dataset$agb,
col = factor(scbi_stem1$genus),
xlab = "DBH (cm)",
ylab = "AGB (kg)"
)
# examine univariate distributions
h1 = hist(dataset$dbh, breaks=10, col="red")
xfit<-seq(min(dataset$dbh),max(dataset$dbh),length=40)
yfit<-dnorm(xfit,mean=mean(dataset$dbh),sd=sd(dataset$dbh))
yfit <- yfit*diff(h1$mids[1:2])*length(dataset$dbh)
lines(xfit, yfit, col="blue", lwd=2)
h2 = hist(dataset$agb, breaks=10, col="red")
xfit<-seq(min(dataset$agb),max(dataset$agb),length=40)
yfit<-dnorm(xfit,mean=mean(dataset$agb),sd=sd(dataset$agb))
yfit <- yfit*diff(h2$mids[1:2])*length(dataset$agb)
lines(xfit, yfit, col="blue", lwd=2)
wilcox.test(dataset$dbh) # p<0.00001 Wilcoxon signed rank test with continuity correction
data: dataset$dbh
V = 2616328, p-value < 2.2e-16
alternative hypothesis: true location is not equal to 0wilcox.test(dataset$agb) # p<0.00001 Wilcoxon signed rank test with continuity correction
data: dataset$agb
V = 2616328, p-value < 2.2e-16
alternative hypothesis: true location is not equal to 0
Simulation Design
This section introduces the design of the Monte Carlo simulation regime, including:
Simulation parameters are defined to balance computational efficiency and statistical robustness.
Cross-validation techniques are employed to evaluate model performance and identify bias or variance.
The LGOCV acronym used in the caret package
functions below stands for “leave one group out cross validation”. We
must select the % of test data that is set out from the build upon which
the model will be repeatedly trained. Note, the following code applies
functions to full dataset without explicit training-test split.
Questions remains on whether we require cross-validation
uncertainty estimate to review internal bias, and whether we would like
to develop Monte Carlo tools for spatial uncertainty used in Activity
Data analysis. For your consideration, the consultant has
previously developed Monte Carlo tools for LULC applications, saved here
# Cross-validation split for bias detection
#samples = caret::createDataPartition(dataset_tidy$volume, p = 0.80, list = FALSE)
#train_data = dataset_tidy[samples, ]
#test_data = dataset_tidy[-samples, ]
# Simulation pattern & regime
monte_carlo = trainControl(
method = "LGOCV",
number = 10, # number of simulations
p = 0.8) # percentage resampled
# Training model fit with all covariates (".") & the simulation
lm_monte_carlo = train(
data = dataset,
agb ~ .,
na.action = na.omit,
trControl = monte_carlo)
lm_monte_carlo Random Forest
2287 samples
6 predictor
No pre-processing
Resampling: Repeated Train/Test Splits Estimated (10 reps, 80%)
Summary of sample sizes: 1832, 1832, 1832, 1832, 1832, 1832, ...
Resampling results across tuning parameters:
mtry RMSE Rsquared MAE
2 334.19769 0.6035154 114.313008
47 82.31374 0.9715600 13.758606
93 49.07356 0.9898362 8.510285
RMSE was used to select the optimal model using the smallest value.
The final value used for the model was mtry = 93.Visualize residuals
To enable access to these predictions, we need to instruct
caret to retain the resampled predictions by setting
savePredictions = "final" in our
trainControl() function. It’s important to be aware that if
you’re working with a large dataset or numerous resampling iterations,
the resulting train() object may grow significantly in
size. This happens because caret must store a record of
every row, including both the observed values and predictions, for each
resampling iteration. By visualizing the results, we can offer insights
into the performance of our model on the resampled data.
monte_carlo_viz = trainControl(
method = "LGOCV",
p = 0.8,
number = 1, # just for saving previous results
savePredictions = "final")
lm_monte_carlo_viz = train(
agb ~ .,
data = dataset,
method = "lm",
na.action = na.omit,
trControl = monte_carlo_viz)
head(lm_monte_carlo_viz$pred) # residuals intercept pred obs rowIndex Resample
1 TRUE -109.35929 0.18320423 1 Resample1
2 TRUE -63.59706 0.56378062 6 Resample1
3 TRUE -130.47898 0.18320423 10 Resample1
4 TRUE -144.19036 0.09947204 12 Resample1
5 TRUE -142.33348 0.09496032 15 Resample1
6 TRUE 1222.08038 789.41947426 18 Resample1lm_monte_carlo_viz$pred |>
ggplot(aes(x=pred,y=obs)) +
geom_point(shape=1) +
geom_abline(slope=1, colour='blue') +
coord_obs_pred()
Activity Data Uncertainty
This section showcases use of Monte Carlo simulations in reporting uncertainty of LULC classification models.
library(ForesToolboxRS)
dir.create("./data/testdata")
download.file("https://github.com/ytarazona/ft_data/raw/main/data/LC08_232066_20190727_SR.zip",destfile = "testdata/LC08_232066_20190727_SR.zip")
unzip("testdata/LC08_232066_20190727_SR.zip", exdir = "testdata") download.file("https://github.com/ytarazona/ft_data/raw/main/data/signatures.zip", destfile = "testdata/signatures.zip")
unzip("testdata/signatures.zip", exdir = "testdata")
image <- stack("./data/testdata/LC08_232066_20190727_SR.tif")
sig <- read_sf("./data/testdata/signatures.shp")
classRF <- mla(img = image, model = "randomForest", endm = sig, training_split = 80)
print(classRF)# Classification
colmap <- c("#0000FF","#228B22","#FF1493", "#00FF00")
plot(classRF$Classification, main = "RandomForest Classification", col = colmap, axes = TRUE)
##### Figure 2: LULC map classified with randomForest classifier
kernel
plot(
cal_ml$svm_mccv,
main = "Monte Carlo Cross-Validation calibration",
col = "darkmagenta",
type = "b",
ylim = c(0, 0.4),
ylab = "Error between 0 and 1",
xlab = "Number of iterations"
)
lines(cal_ml$randomForest_mccv, col = "red", type = "b")
lines(cal_ml$naiveBayes_mccv, col = "green", type = "b")
lines(cal_ml$knn_mccv, col = "blue", type = "b")
legend(
"topleft",
c(
"Support Vector Machine",
"Random Forest",
"Naive Bayes",
"K-nearest Neighbors"
),
col = c("darkmagenta", "red", "green", "blue"),
lty = 1,
cex = 0.7
)
Session runtime
devtools::session_info()─ Session info ───────────────────────────────────────────────────────────────
setting value
version R version 4.4.2 (2024-10-31)
os macOS Sequoia 15.2
system aarch64, darwin20
ui X11
language (EN)
collate en_US.UTF-8
ctype en_US.UTF-8
tz America/Vancouver
date 2024-12-24
pandoc 3.2 @ /Applications/RStudio.app/Contents/Resources/app/quarto/bin/tools/aarch64/ (via rmarkdown)
─ Packages ───────────────────────────────────────────────────────────────────
package * version date (UTC) lib source
abind 1.4-8 2024-09-12 [1] CRAN (R 4.4.1)
allodb * 0.0.1.9000 2024-12-19 [1] Github (ropensci/allodb@4207f86)
animation * 2.7 2021-10-07 [1] CRAN (R 4.4.0)
assertthat 0.2.1 2019-03-21 [1] CRAN (R 4.4.0)
backports 1.5.0 2024-05-23 [1] CRAN (R 4.4.0)
BIOMASS * 2.1.11 2023-09-29 [1] CRAN (R 4.4.0)
boot 1.3-31 2024-08-28 [1] CRAN (R 4.4.2)
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knitr * 1.49 2024-11-08 [1] CRAN (R 4.4.1)
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Matrix 1.7-1 2024-10-18 [1] CRAN (R 4.4.2)
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mnormt 2.1.1 2022-09-26 [1] CRAN (R 4.4.0)
modeldata * 1.4.0 2024-06-19 [1] CRAN (R 4.4.0)
ModelMetrics 1.2.2.2 2020-03-17 [1] CRAN (R 4.4.0)
munsell 0.5.1 2024-04-01 [1] CRAN (R 4.4.0)
mvtnorm 1.3-2 2024-11-04 [1] CRAN (R 4.4.1)
nlme 3.1-166 2024-08-14 [1] CRAN (R 4.4.2)
nnet 7.3-19 2023-05-03 [1] CRAN (R 4.4.2)
nortest 1.0-4 2015-07-30 [1] CRAN (R 4.4.0)
olsrr * 0.6.1 2024-11-06 [1] CRAN (R 4.4.1)
pander 0.6.5 2022-03-18 [1] CRAN (R 4.4.0)
parallelly 1.41.0 2024-12-18 [1] CRAN (R 4.4.1)
parsnip * 1.2.1.9004 2024-12-19 [1] Github (tidymodels/parsnip@27df158)
pillar 1.10.0 2024-12-17 [1] CRAN (R 4.4.1)
pkgbuild 1.4.5 2024-10-28 [1] CRAN (R 4.4.1)
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Literature review of current Monte Carlo methods used in REDD+ and ART-TREES projects
| Parameter | Description |
|---|---|
| Keywords | Monte Carlo simulations |
| Biomass estimation | |
| Carbon stock uncertainty | |
| REDD+ projects | |
| Forest carbon accounting | |
| Allometric uncertainty | |
| Data Sources | Scopus |
| Web of Science | |
| Google Scholar | |
| Grey Literature from REDD+ working groups (i.e. UNFCCC, IPCC) | |
| Temporal Window | 2003–2023 |
| Focus Areas | Applications of Monte Carlo simulations in biomass and carbon stock estimations. |
| Addressing uncertainty in input data (e.g., allometric equations, plot-level measurements). | |
| Integration of Monte Carlo methods in REDD+ policy frameworks and carbon accounting. | |
| Inclusion Criteria | Peer-reviewed articles and high-impact reviews |
| Case studies and empirical research involving REDD+ projects. | |
| Discussions of methodological advancements or critiques of Monte Carlo approaches. |
Table 4: Search parameters used in a review of Monte Carlo tools in REDD+
reporting.
| REDD+ scheme1 | Monte Carlo applied | Region | Key Findings | Ref |
|---|---|---|---|---|
| ADD | Uncertainty of SAAB estimate | Rondônia, Brazil | Estimated ± 20% measurement error in SAAB using Monte Carlo simulations; emphasized large trees’ role in biomass. | 3 |
| ADD | AGB Uncertainty | Kenya, Mozambique | Assessed mixed-effects models in estimating mangrove biomass. | 4 |
| ADD | Blanket uncertainty propagation | Ghana | AGB prediction error >20%; addressed error propagation from trees to pixels in remote sensing. | 5 |
| ADD | Plot-based uncertainty | New Zealand | Cross-plot variance greatest magnitude of uncertainty | 6 |
| JNR | Multi-scale AGB uncertainty modeling | Minnesota, USA | Cross-scale tests showing effects of spatial resolution on AGB uncertainty. | 7 |
| NA | Allometric uncertainty modeling | Panama | Allometric models identified as largest source of biomass estimation error. | 8 |
| ADD | Sampling and allometric uncertainty | Tapajos Nat Forest, Brazil | Significance of allometric models on uncertainty of root biomass, 95% CI, 21 plots. | 9 |
| ADD | Uncertainty of volume estimates | Santa Catarina, Brazil | Negligible effects of residual uncertainty on large-area estimates | 10 |
| NA | Uncertainty metrics in model selection | Oregon, USA | Uncertainty estimates call for local validation or new local model development | 11 |
| ADD | AGB model uncertainty | French Guiana | AGB sub-model errors dominate uncertainty; height and wood-specific gravity errors are minor but can cause bias. | 12 |
| IFM | Emission factor uncertainty | Central Africa | Model selection is the largest error source (40%); weighting models reduces uncertainty in emission factors. | 13 |
| NA | Uncertainty in ecosystem nutrient estimate | New Hampshire, USA | Identified 8% uncertainty in nitrogen budgets, mainly from plot variability (6%) and allometric errors (5%). | 14 |
Table 5: Results of a review of literature on Monte Carlo methodologies in REDD+ projects
References
ART, S. The REDD+ Environmental Excellence Standard; 2021. https://www.artredd.org/wp-content/uploads/2021/12/TREES-2.0-August-2021-Clean.pdf
Duncanson, L.; Disney, M.; Armston, J.; Nickeson, J.; Minor, D.; Camacho, F. Aboveground Woody Biomass Product Validation Good Practices Protocol. 2021. https://doi.org/10.5067/DOC/CEOSWGCV/LPV/AGB.001
Brown, I. F.; Foster Brown, I.; Martinelli, L. A.; Wayt Thomas, W.; Moreira, M. Z.; Cid Ferreira, C. A.; Victoria, R. A. Uncertainty in the Biomass of Amazonian Forests: An Example from Rondônia, Brazil. Forest Ecology and Management 1995, 75 (1–3), 175–189. https://doi.org/10.1016/0378-1127(94)03512-u
Cohen, R.; Kaino, J.; Okello, J. A.; Bosire, J. O.; Kairo, J. G.; Huxham, M.; Mencuccini, M. Uncertainty to Estimates of Above-Ground Biomass for Kenyan Mangroves: A Scaling Procedure from Tree to Landscape Level. In Forest ecology and management; 2013; Vol. 310, pp 968–982. https://doi.org/10.1016/j.foreco.2013.09.047
(5)] Chen, Q.; Laurin, G. V.; Valentini, R. Uncertainty of Remotely Sensed Aboveground Biomass over an African Tropical Forest: Propagating Errors from Trees to Plots to Pixels. Remote Sensing of Environment 2015, 160, 134–143. https://doi.org/10.1016/j.rse.2015.01.009
Holdaway, R. J.; McNeill, S. J.; Mason, N. W. H.; Carswell, F. E. Propagating Uncertainty in Plot-Based Estimates of Forest Carbon Stock and Carbon Stock Change. Ecosystems 2014, 17, 627–640. https://doi.org/10.1007/s10021-014-9749-5
Chen, Q.; McRoberts, R. E.; Wang, C.; Radtke, P. J. Forest Aboveground Biomass Mapping and Estimation Across Multiple Spatial Scales Using Model-Based Inference. Remote Sensing of Environment 2016, 184, 350–360. https://doi.org/10.1016/j.rse.2016.07.023
Chave, J.; Condit, R.; Aguilar, S.; Hernandez, A.; Lao, S.; Perez, R. Error Propagation and Scaling for Tropical Forest Biomass Estimates. Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences 2004, 359 (1443), 409–420
Keller, M.; Palace, M.; Hurtt, G. Biomass Estimation in the Tapajos National Forest, Brazil. Forest Ecology and Management 2001, 154, 371–382
McRoberts, R. E.; Moser, P.; Oliveira, L. Z.; Vibrans, A. C. A General Method for Assessing the Effects of Uncertainty in Individual-Tree Volume Model Predictions on Large-Area Volume Estimates 222 with a Subtropical Forest Illustration. Canadian Journal of Forest Research 2015, 45
Melson, S. L.; Harmon, M. E.; Fried, J. S.; Domingo, J. B. Estimates of Live-Tree Carbon Stores in the Pacific Northwest Are Sensitive to Model Selection. Carbon Balance and Management 2011
Molto, Q.; Rossi, V.; Blanc, L. Error Propagation in Biomass Estimation in Tropical Forests. Methods in Ecology and Evolution 2013, 4, 175–183. https://doi.org/10.1111/j.2041-210x.2012.00266.x
Picard, N.; Bosela, F. B.; Rossi, V. Reducing the Error in Biomass Estimates Strongly Depends on Model Selection. Annals of Forest Science 2015, 72 (6), 811–823. https://doi.org/10.1007/s13595-014-0434-9
Yanai, R. D.; Battles, J. J.; Richardson, A. D.; Blodgett, C. A.; Wood, D. M.; Rastetter, E. B. Estimating Uncertainty in Ecosystem Budget Calculations. Ecosystems 2010, 13, 239–248. https://doi.org/10.1007/s10021-010-9315-8
ADD: Avoided Deforestation and Degradation, JNR: Jurisdictional & Nested REDD+, IFM: Improved Forest Management↩︎