Small area estimation of forest carbon using carbon maps 

and model-assisted regression in Maryland

Andrew Lister (USDA Forest Service, Forest Inventory and Analysis (FIA)

George Hurtt and Lei Ma (University of Maryland)

Ty Wilson (USDA Forest Service, Forest Inventory and Analysis)
Objectives

• Background on FIA

• Background on PS and MAR

• Results of study comparing PS and MAR

• Final thoughts, caveats, questions for YOU!

• Investigate benefits of using model-assisted regression 

(MAR) over traditional post-stratification (PS) in FIA

• Assess MAR “hyperparameters” that affect 

operationalizing MAR in FIA

Outline
What does FIA do?

Land variables

• Forest type

• Site class

• Stand size

• Physiographic 

class

• Land use

• Land cover

•Owner 

type/class

...

Tree variables

• Species

•Diameter

•Height

•Merchantability

•Damage type

• Crown ratio

•Growth

...

Other

•Invasive plant cover

•Down woody debris

•Soils

•Regeneration

....

Data Processing

• Estimation

• Modelling

• GIS + Remote 

Sensing

• Reporting

Reporting on the status of and trends in 

the Nation’s forest resources; technology 

transfer to NFS and partners; basic and 

applied research; information for policy.... 
120 degrees

4 X 1/24 acre (0.017 ha) subplots

1/6 acre (0.067 ha cluster plot)

FIA Plot Design
Sample Intensity = 1 sample location

 per hexagon (5,933 ac, 2401 ha)

Inventory Cycle Length = Between 1/5 and

1/10 of the plots will be measured each year

FIA Sample Design

>300,000 plots at full implementation!!

2004

2005

2006

2007

2008

Panels

Annual Inventory:

Any panel or cycle 

represents a full 

measurement of state.
How does FIA use remote sensing and GIS?

Plots labeled with GIS 

and Remote Sensing 

data

+

Improved estimation 

through stratification 

and other modelassisted estimation 

methods

Maps for small area 

estimation, additional 

context for tabular 

estimates 

GPS
× Area + × Area + × Area + × Area

average average average

average

Vs. 

average

× Area 

Weighted 

averaging 

procedure:

Simple Random Sampling 

– No remote sensing:

Post-stratification gives 

smaller confidence 

intervals than Simple 

Random Sampling.

Smaller confidence 

interval = better, less 

expensive estimates in 

smaller areas.

Traditional technique in FIA: Post-Stratification (PS)

Categorical

Stratum map
The mean of the attribute (y) by stratum (h) for domain d

The weighted mean of the attribute in the domain

Estimate of total: weighted average X the Total Area (AT)

Stratum Weight (Wh)

PS estimators of mean and total

Number of plots in stratum h

NN, NNh, AATTaaaaaa yyhii
Variance of the mean 

of the attribute y in 

stratum h in domain d

The variance of the 

mean, weighted with the 

stratum weight (Wh= 

Nh/N)

A penalty to address the 

fact that the number of 

plots in a stratum is not 

pre-determined, i.e., is a 

RV

PS estimator of variance of total
Newer technique in FIA: Model Assisted Regression (MAR)

Goal: a better estimate (smaller confidence interval) than 

PS or SRS

To calculate estimates 

and confidence 

intervals, need: 

1. Linear* relationship 

between map and 

ground data

2. map value for each 

plot

3. sum of all pixel 

values in the map

Continuous volume 

map

low

high Remote Sensing Map Carbon Ground Plot Carbon

*Can be used with other model types if 

df can be estimated.
MAR estimator of mean

�yy = ttttt mm vv vvvv oooo ttttt aa aaaaaa yy

= ttttt ss ss oooo ttttt rr oooo yy oo xx

XX� = ttttt mm vv vvvv oooo ttttt mm vv vvvv xx

ttttt ss

̅xx = ttttt mm vv vvvv oooo ttttt mm vv vvvv xx aa ttttt

ll oooo ttttt yy vv vvvv

aa = ttttt yy oooo ttttt rr

oooo yy oo xx

All you’re doing is adjusting the mean of the pixels using the ground plots! (or you’re 

adjusting the mean of the ground plots using the mean of the pixels....)

XX�

�yy, ̅xx, aa, , aaaaaa aa + (XX�)
Variance of the mean 

of the attribute y from 

MAR

Equivalently, the variance of the 

residuals of the simple linear 

regression ÷ n (i.e., MSE/n)

MAR estimator of variance of mean

If we know the simple random sample variance and the R2 of the regression of y on x, for large 

n, we know about how much using MAR will reduce the variance!
So what did I do? I compared vv YY� from MAR with that from PS

The plots

A (continuous) 

carbon map

The plots

A (categorical) 

stratum map
Some details: 

FIA uses subpopulations formed by 

combinations of county group, US Census 

Bureau inland water/land, and 

Public/Private ownership layer. 

The issue: So I:

Did standard FIA subpopulation-based 

estimation, and did the MAR estimates both 

with and without subpopulations

There are MANY forest carbon maps 

available. Some had predictions of C in 

nonforest areas.

Used 2 maps: a University of Maryland carbon 

map from a NASA CMS Carbon mapping 

project (Hurtt et al.) and the carbon map from 

FIA’s BigMap project (Wilson et al.). Applied a 

forest/nonforest mask to the UMD maps.

There is a spatial mismatch between an FIA 

plot (covers 4 pixels) and a single pixel (the 

xi in the equations). 

Ran the MAR in 2 ways: 1. center subplot vs. 

original map, and 2. center subplot vs. the 

original map with a 3x3 pixel low-pass filter (a 

3x3pixel mean kernel) was applied. 
RESULTS: 

Map Type slope y int

R 

square

estimate of 

total carbon 

(short tons)

sampling 

error 

relative 

efficiency

# additional plots needed by PS to 

achieve the MAR sampling error

PS subPop - - - 101928140 3.38% 1.000 0

UMD subPop 0.21 -1.63 0.46 100710240 3.24% 1.046 92

UMD state 0.22 -1.59 0.51 103356418 3.09% 1.095 194

UMD subPop - FNF mask 0.21 3.22 0.48 99862034 3.23% 1.047 95

UMD state - FNF mask 0.22 2.68 0.55 100392470 3.06% 1.107 221

UMD subPop - 3x3 filter 0.25 -4.26 0.51 100073120 3.09% 1.094 194

UMD state - 3x3 filter 0.25 -3.80 0.56 103255099 2.94% 1.151 318

Big 

Map subPop 9.00 1.63 0.56 103591553 2.91% 1.161 341

Big 

Map state 8.61 1.45 0.59 102449083 2.86% 1.184 395

Big 

Map subPop - 3x3 filter 9.20 0.75 0.56 104769239 2.83% 1.196 423

Big 

Map state - 3x3 filter 9.17 0.74 0.61 103741091 2.77% 1.222 483

estimate of 

total carbon 

(short tons)

sampling 

error 

relative 

efficiency 

(SRS vs PS)

# additional plots needed by SRS 

to achieve the PS sampling error

SRS - 

no 

map

state - - - 95026485 4.82% 0.702 1010

Lower is Better!

SS EE =

vv(YY�)

YY� ∗ 100
RESULTS: 

Map Type slope y int

R 

square

estimate of 

total carbon 

(short tons)

sampling 

error 

relative 

efficiency

# additional plots needed by PS to 

achieve the MAR sampling error

PS subPop - - - 101928140 3.38% 1.000 0

UMD subPop 0.21 -1.63 0.46 100710240 3.24% 1.046 92

UMD state 0.22 -1.59 0.51 103356418 3.09% 1.095 194

UMD subPop - FNF mask 0.21 3.22 0.48 99862034 3.23% 1.047 95

UMD state - FNF mask 0.22 2.68 0.55 100392470 3.06% 1.107 221

UMD subPop - 3x3 filter 0.25 -4.26 0.51 100073120 3.09% 1.094 194

UMD state - 3x3 filter 0.25 -3.80 0.56 103255099 2.94% 1.151 318

Big 

Map subPop 9.00 1.63 0.56 103591553 2.91% 1.161 341

Big 

Map state 8.61 1.45 0.59 102449083 2.86% 1.184 395

Big 

Map subPop - 3x3 filter 9.20 0.75 0.56 104769239 2.83% 1.196 423

Big 

Map state - 3x3 filter 9.17 0.74 0.61 103741091 2.77% 1.222 483

estimate of 

total carbon 

(short tons)

sampling 

error 

relative 

efficiency 

(SRS vs PS)

# additional plots needed by SRS 

to achieve the PS sampling error

SRS - 

no 

map

state - - - 95026485 4.82% 0.702 1010

SampErrPS / 

SampErrMAR

Higher is Better!
RESULTS: 

Map Type slope y int

R 

square

estimate of 

total carbon 

(short tons)

sampling 

error 

relative 

efficiency

# additional plots needed by PS to 

achieve the MAR sampling error

PS subPop - - - 101928140 3.38% 1.000 0

UMD subPop 0.21 -1.63 0.46 100710240 3.24% 1.046 92

UMD state 0.22 -1.59 0.51 103356418 3.09% 1.095 194

UMD subPop - FNF mask 0.21 3.22 0.48 99862034 3.23% 1.047 95

UMD state - FNF mask 0.22 2.68 0.55 100392470 3.06% 1.107 221

UMD subPop - 3x3 filter 0.25 -4.26 0.51 100073120 3.09% 1.094 194

UMD state - 3x3 filter 0.25 -3.80 0.56 103255099 2.94% 1.151 318

Big 

Map subPop 9.00 1.63 0.56 103591553 2.91% 1.161 341

Big 

Map state 8.61 1.45 0.59 102449083 2.86% 1.184 395

Big 

Map subPop - 3x3 filter 9.20 0.75 0.56 104769239 2.83% 1.196 423

Big 

Map state - 3x3 filter 9.17 0.74 0.61 103741091 2.77% 1.222 483

estimate of 

total carbon 

(short tons)

sampling 

error 

relative 

efficiency 

(SRS vs PS)

# additional plots needed by SRS 

to achieve the PS sampling error

SRS - 

no 

map

state - - - 95026485 4.82% 0.702 1010
RESULTS: 

Map Type slope y int

R 

square

estimate of 

total carbon 

(short tons)

sampling 

error 

relative 

efficiency

# additional plots needed by PS to 

achieve the MAR sampling error

PS subPop - - - 101928140 3.38% 1.000 0

UMD subPop 0.21 -1.63 0.46 100710240 3.24% 1.046 92

UMD state 0.22 -1.59 0.51 103356418 3.09% 1.095 194

UMD subPop - FNF mask 0.21 3.22 0.48 99862034 3.23% 1.047 95

UMD state - FNF mask 0.22 2.68 0.55 100392470 3.06% 1.107 221

UMD subPop - 3x3 filter 0.25 -4.26 0.51 100073120 3.09% 1.094 194

UMD state - 3x3 filter 0.25 -3.80 0.56 103255099 2.94% 1.151 318

Big 

Map subPop 9.00 1.63 0.56 103591553 2.91% 1.161 341

Big 

Map state 8.61 1.45 0.59 102449083 2.86% 1.184 395

Big 

Map subPop - 3x3 filter 9.20 0.75 0.56 104769239 2.83% 1.196 423

Big 

Map state - 3x3 filter 9.17 0.74 0.61 103741091 2.77% 1.222 483

estimate of 

total carbon 

(short tons)

sampling 

error 

relative 

efficiency 

(SRS vs PS)

# additional plots needed by SRS 

to achieve the PS sampling error

SRS - 

no 

map

state - - - 95026485 4.82% 0.702 1010
Take Home Points

MAR, compared head-to-head with 

standard FIA PS, leads to better precision.

This will help entities (like those interested in 

carbon accounting) get better information for 

smaller areas for less money.

There are many dials to turn when doing 

this, including pre-processing the default 

carbon maps (e.g., f/nf mask, 3x3 filter)

Definitional consistency between the attribute 

(carbon) and the map are important to ensure.

We have been talking about 

operationalizing this for years now.. How?

As we can see, the slope, intercept, and 

predictions are calculated with simple algebra. 

VERY compatible with our SQL-based 

estimation system! 

There’s a few big buts related to 

domain estimation and nonresponse...
Questions? Answers to my questions? Come by the USDA Forest 

Service Forest Inventory and Analysis Table and we can discuss!

https://www.fia.fs.usda.gov/

Andy Lister

andrew.lister@usda.gov
Big but 1:

The MAR estimate is computed for each cell 

of a table.. What about analysis domain 

estimation? 

We might need a new map for every cell in the 

table, and maps will likely be quite bad for 

strange attributes. 

Remote Sensing Map Carbon

Ground plot Carbon

Remote Sensing Map Carbon

# of dead red maples

Uh oh...

Question for YOU: maybe we make our MAR 

with just the nd plots to calculate a mean? 

Remote Sensing Map Carbon

# of dead red maples

nd =7
Big but 2:

There is LOTS of nonresponse in our survey. Question for YOU: Can we use MAR to improve 

upon current method (stratum mean applied 

to each missing value for the estimate, 

rrrrrrrrrrrrrrrr for the variance)? 

�yy yy

Maybe we differentially weight the imputed 

�yy values, or we use them to calculate the 

estimate but we use rrrrrrrrrrrrrrrr to calculate 

variance?