Uncertainty Analysis for a US
Inventory of Soil Organic Carbon
Stock Changes
F. Jay Breidt
Department of Statistics
Colorado State University
Stephen M. Ogle and Keith Paustian
Natural Resources Ecology Laboratory
Colorado State University
Why Inventory Soil Carbon
Stocks?



Solar energy transmitted to
earth as visible and
ultraviolet radiation
Radiation absorbed by
surface gets re-radiated as
infrared
Greenhouse Gases (GHGs)


pass visible and UV, but trap
infrared: greenhouse
effect
include (among others)
water vapor, methane,
nitrous oxide, CO2
Atmosphere
Surface
Reflected
25%
5%
Absorbed
25%
45%
Carbon Sequestration




Lithosphere: fossil fuels, limestone,
dolomite, chalk
Oceans: shells, dissolved CO2
Biosphere: organic molecules in living
and dead organisms
Soils: organic matter
Agricultural Management and
Carbon Storage


Tillage, fertilization, irrigation, etc. all
affect carbon storage
Century, a biogeophysical process
model, describes site-specific dynamics
in an agricultural system



tracks carbon, water, nutrient cycling over
long time scales (centuries to millennia)
requires inputs on soils, weather,
agricultural management
deterministic output for given inputs
cx k , z1k 
Carbon Dynamics in Century
Metherell, Harding, Cole, & Parton 1993
Inventory Goal


Estimate total carbon stock change for US
agricultural soils, 1990-2004
Report to United Nations Framework
Convention on Climate Change


pre-Kyoto agreement; nearly universal
Use Century to model carbon stock change
across US

need Century inputs on nationally-representative
set of sites in US agricultural lands
USDA National Resources
Inventory (NRI)
• Nationally-representative set of
sites in US agricultural lands
• Stratified two-stage area sample
• Fine stratification with two primary
sampling units (PSUs=quarter
sections) for every 1/3 township
• Three secondary sampling units
(points) per PSU
• Many points have
• same county, MLRA, weather
• same categorical values of
cropping history, soil, etc.
• Run Century at NRI “superpoints”
NRI Handles Sampling
Uncertainty

NRI is a nationally-representative
probability sample



straightforward and unbiased expansion of
point-level data to national total carbon
stock change
consistent design-based variance
estimation and valid confidence intervals
NRI contains many key Century inputs

site-level cropping history, soil properties
Input Uncertainty





Not all needed Century inputs are in NRI
Weather: (but treat as known from PRISM:
local interpolation of station data)
Tillage: use county-level Conservation
Technology Information Center data
Organic amendments: use county-level
USDA Manure Management Database
Fertilizer: use county-level USDA-ERS
Cropping Surveys
Tillage

Traditional Tillage:




after harvest, field contains crop residues
tillage turns over the soil to bury residues
often repeated several times prior to planting
Conservation Tillage:


Reduced-Till: limited tillage; substantial crop
residues on surface
No-Till: doesn’t use tillage; all crop residues left
on surface
Tillage Input Distribution

Conservation
Technology Information
Center (CTIC) collects
county-level information


construct discrete
distributions for Monte
Carlo (CTCT, CTRT,
CTNT, RTRT, RTNT,
etc.)
draw from these
distributions to reflect
uncertain inputs
Photo courtesy of USDA
Organic Amendments and
Fertilizer
Organic amendments and
fertilizer not included in NRI
 Use USDA Manure
Management Database

county-level data
 construct distributions for
Monte Carlo
 combine with USDA-ERS
cropping survey information to
account for negative
correlation with fertilizer

Artwork courtesy of the
Wisconsin Department of
Natural Resources
Model Uncertainty


Century is imperfect
For some long-term experimental sites, have




measured carbon stock changes
modeled carbon stock changes from Century
complete set of inputs, plus additional covariates
Adjust using regression of measured on
modeled
Measured Carbon Stock at
Long-Term Experiment Sites
Measured vs. Modeled
95
Bare-Summer
85
Fallow
Set-Aside Lands (CRP)
No-Till
75
Sqrt Measured Soil Organic C Stock (g/m2)
65
55
45
35
25
95
85
Hay/Pasture in
Annual Cropping Rotation
25
Hay/Pasture in
Annual Cropping Rotation
w/ Organic Amendments
75
65
55
45
35
25
95
85
Organic
Amendments
Other Cropping
Practices
75
65
55
45
35
25
25
35
45
55
65
75
85
95
25
35
45
55
65
75
Sqrt Modeled Soil Organic C Stock (g/m2)
85
95
35
45
55
65
75
85
95
Adjusted Century Output

Experiment sites
estimated from data
yk  f cx k , z1k , z 2 k , θ   k
measured
carbon stock

error with
dependence
from repeated
measures
known covariates
No attempt to estimate Century rate
parameters from these data (very
high dimension)
Expansion to National Total

~ 
Ideal expansion estimator

1
kNRI

k
f cx k , z1k , z 2 k , θ 
known covariates
Feasible
ˆr 

kNRI
r th replicate estimate
of national total
1
k

f cX rk , z1k , z 2 k , θˆ r

MC from modeled
distribution
MC from sampling
distribution
Complete Uncertainty Analysis
Framework
correlated
(sampling)
Cropping History
Combining Design and Monte
Carlo Uncertainties

Define






second-order inclusion prob:  kl  Prk , l  NRI
design covariance:  kl   kl   k  l
MC expectation:k  E f cXk , z1k , z 2k , θˆ
MC covariance:  kl  Cov f cXk , z1k , z 2k , θˆ , f cXl , z1l , z 2l , θˆ


 
Unconditional variance
input uncertainty

 model uncertainty

1
Var  
f cX k , z1k , z 2 k , θˆ
 kNRI  k


sampling uncertainty




    kl k l   kl kl : V
 k l
 k l
kU
kU


Variance Estimation

Combination of MC replication and design-based methods for
2
R
R
(unreplicated) sample


ˆr2   ˆr  R

 usual MC variance estimate
 r 1 
ˆ r 1
V1 

usual design-based variance estimate for MC averages
(SAS proc surveymeans or PCCARP once)
Vˆ2 

R 1
2


  f c X rk , z1k , z 2 k , θˆ r
 kl  r
R
 
RR  1 k ,lNRI  kl




R   f c Xlr ' , z1l , z 2l , θˆ r '
 r '

 k l


R

average of design-based variance estimates across MC
reps (SAS proc surveymeans or PCCARPR times)


 

r
R
ˆ r f c X r , z , z , θˆ r

f
c
X
,
z
,
z
,
θ
1
kl
k
1
k
2
k
l
1l
2l
Vˆ3 


RR  1 r 1 k ,lNRI  kl
 k l

Variance Estimation,
Continued

Unbiased estimator of V is then
Vˆ  Vˆ1  Vˆ2  Vˆ3

But note that

 
 
2
2
2






N
N
N
, E Vˆ2  O
, E Vˆ3  O

E Vˆ1  O
 n 
 n 
 Rn 

Simpler (saves R variance computations),
conservative estimator
*
ˆ
V  Vˆ1  Vˆ2
Implementation




n=123K NRI superpoints in cropland,
from almost 1M total NRI points
R=100 MC reps for each NRI superpoint
12.3M Century runs
Compute estimates and uncertainties at
national level as well as for interesting
domains
National-Scale Century Inventory Results
(Tg CO2 Eq.)
Soil Type
1990-1994
1995-2004
(71.24)
(62.52)
(69.7) to (73.0)
(60.9) to (64.2)
1.47
(2.82)
0.7 to 2.2
(2.2) to (3.3)
(8.25)
3.96
(6.2) to (10.3)
2.2 to 5.5
(12.80)
(15.99)
(12.5) to (13.2)
(15.8) to (16.1)
Mineral Soils
Croplands Remaining Croplands
95% C.I.
Lands Converted to Croplands
95% C.I.
Grasslands Remaining
Grasslands
95% C.I.
Lands Converted to Grasslands
95% C.I.
Summary


National inventory of carbon stock changes, using
variety of data sources
Combine Monte Carlo and design-based methods to
account for





sampling uncertainty
input uncertainty
model uncertainty
First phase in ongoing study
Future improvements:


Incorporate remote sensing data for estimating crop and
forage production
Account for emissions of N2O associated with agricultural
management