United States Department of Agriculture / Forest Service

Rocky Mountain Research Station

General Technical Report RMRS-GTR-239

July 2010

Complex Sample Survey 

Estimation in Static State-Space

Raymond L. Czaplewski

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Czaplewski, Raymond L. 2010. Complex sample survey estimation in static 

state-space. Gen. Tech. Rep. RMRS-GTR-239. Fort Collins, CO: U.S. 

Department of Agriculture, Forest Service, Rocky Mountain Research Station. 

124 p.

Abstract

Increased use of remotely sensed data is a key strategy adopted by the Forest 

Inventory and Analysis Program. However, multiple sensor technologies require 

complex sampling units and sampling designs. The Recursive Restriction 

Estimator (RRE) accommodates this complexity. It is a design-consistent Empirical 

Best Linear Unbiased Prediction for the state-vector, which contains all suffi cient 

statistics for the sampled population. RRE reduces a complex estimator into a 

sequence of simpler estimators. Also included are model-based pseudo-estimators 

and multivariate Taylor series approximations for covariance matrices. Together, 

these provide a unifi ed approach to detailed estimation in large, complex sample 

surveys.

Keywords: FIA, sampling, recursive, Pythagorean regression, EBLUP, remote 

sensing

Author

Raymond L. Czaplewski is a Research Mathematical Statistician with the USDA 

Forest Service, Rocky Mountain Research Station (RMRS), in Fort Collins, CO. 

He is a member of the RMRS Inventory, Monitoring, and Analysis Program and 

the Interior West Forest Inventory and Analysis (FIA) program, with headquarters 

in Ogden, UT. He was an RMRS Project Leader with the national FIA program, 

a Program Analyst with the Bighorn National Forest, and a Wildlife Planner with 

the Wyoming Game and Fish Department. He received his Ph.D. from Colorado 

State University in quantitative range science, his M.S. degree from University of 

Wyoming in systems ecology, and his B.A. degree from Northwestern University in 

biology. His professional interests include statistical estimators for sample surveys 

that integrate time-series of remotely sensed observations with FIA fi eld data.

Acknowledgments

Detailed technical reviews by Jean Opsomer, Ph.D., Jianqiang Wang, Ph.D., 

and David Turner, Ph.D., improved the original draft of this manuscript. Steen 

Magnussen, Ph.D., also provided valuable suggestions. Any remaining errors or 

omissions remain the sole responsibility of the author.

The use of trade or fi rm names in this publication is for reader 

information and does not imply endorsement by the U.S. 

Department of Agriculture of any product or service.
Contents

Introduction ...........................................................................................................................1

Outline .........................................................................................................................3

Stratifi cation in FIA................................................................................................................6

Recursive Restriction Estimator (RRE) ................................................................................8

State-Space .................................................................................................................8

Recursive Estimation ...................................................................................................9

The Basic Idea ...........................................................................................................10

The Univariate Composite Estimator .........................................................................12

The Restriction Estimator (RE) Object .......................................................................14

Relationship to an Optimal Sample Survey Estimator ...............................................16

RRE for Complex Sampling Designs: The Basic Idea Revisited ...............................19

Two-Phase Sampling for Continuous Variables with a Sample at Phase 1 ........................20

Phase 1 ......................................................................................................................22

Phase 2 ......................................................................................................................22

Recursive Restriction Estimator (RRE) ......................................................................23

Two-Phase Sampling for Continuous Variables with a Census at Phase 1 ........................27

Numerically Stable Algorithm .....................................................................................30

RRE as the Optimal Linear Calibration Estimator ......................................................32

Three-Phase Sampling for Continuous Variables ...............................................................36

A Potentially More Robust but Less Effi cient Estimator .............................................40

Single-Phase Sampling for Continuous Variables with Cluster Plots .................................43

Two-Phase Sampling for Continuous Variables with Cluster Plots .....................................46

Categorical Variables ..........................................................................................................53

Robust Numerics ................................................................................................................55

Optimal PSU Expansion Values for Each FIA Plot .............................................................57

Expansion Values for FIA Population Estimates ........................................................59

Expansion Values for FIA Variance Estimates ...........................................................60

Further Research .......................................................................................................61

Multivariate Vector of FIA Study Variables ..........................................................................62

Multivariate Vector of Auxiliary Variables ............................................................................63

Digital Processing of Multispectral Landsat Data ......................................................64

Geopolitical Domains of Study ...................................................................................65

Photo-Interpretation and Photogrammetry ................................................................66

LiDAR Laser Technology ...........................................................................................67

Synthetic Aperture Radar Technology (SAR) .............................................................67

Linear, Small-Area, Synthetic Estimator with Census Predictors .......................................68

RRE as a Synthetic Small-Area Estimator .................................................................75

Non-Linear Pseudo-Estimators ..........................................................................................76

Product Estimator: Missing Data Imputation ..............................................................77

Ratio Pseudo-Estimator: Volume per Estimated Unit Area of Forest Cover ..............82

Recursive Object-Oriented Simplifi cation of Taylor-Series Approximations ...............83

A Model-Based Estimator for Small-Domain Predictions ...........................................87

Second-Order Taylor-Series Approximation to Assess Bias ......................................94

Accuracy and Registration of Remotely Sensed Data........................................................99

Discussion ........................................................................................................................101

Seminal Literature ....................................................................................................105

Future Research and Development..................................................................................106

Conclusions ......................................................................................................................109

References ....................................................................................................................... 111
USDA Forest Service RMRS-GTR-239. 2010. 1

Introduction

The USDA Forest Service, Research and Development branch, through its 

Forest Inventory and Analysis (FIA) program, is committed to delivering current, consistent, and credible information about the status, condition, and trends 

of America’s forests (Gillespie 1999; Van Deusen and others 1999; Smith 2002; 

Bechtold and Scott 2005; McRoberts 2005; Patterson and Reams 2005; Reams 

and others 2005; Scott and others 2005). The FIA strategic plan for 2007 to 2011 

(USDA Forest Service 2007) charts the course for continuous improvement in 

the near term. The FIA plan details plans to utilize the latest technologies to acquire data through remote sensing, fi eld activities, primary forest product studies, 

utilization studies, and landowner surveys. FIA plans to elevate its emphasis on 

remote sensing and fi eld data to analyze the location, status, and trends in land 

use, land cover, and fragmentation.

The 2007 FIA strategic plan identifi es opportunities to expand its service to 

the nation after full implementation of the base FIA program, which is described 

in its 1998 strategic plan (USDA Forest Service 1998). Development and implementation of these opportunities are not included in the current or target funding 

for FIA. However, FIA can provide the leadership necessary to pursue these opportunities through research and development, innovation, strategic partnerships, 

and cost sharing with other programs.

One key opportunity is to increase the role of remote sensing to improve cost 

effectiveness and data quality. One of many options might include a shift in FIA 

investments toward observations with remotely sensed data. However, verifi cation of remotely sensed data with suffi cient fi eld data remains essential. A shift 

toward more remotely sensed observations is most compelling in the Interior 

West, interior Alaska, and other regions with forests dominated by open canopies, lower levels of stocking, and lower productivity. FIA leadership envisions 

using high-resolution, remotely sensed imagery and light detection and ranging (LiDAR) technology. Other sensors include Moderate Resolution Imaging 

Spectro-radiometer (MODIS) with 250-m pixels (Justice and others 2002), 

Landsat (30-m pixels), and IKONOS satellites (1-m pixels). Lower-resolution 

satellite data, such as MODIS and Landsat, are suitable for synoptic, wall-towall, full-coverage, thematic mapping. Sampling is needed for higher resolution, 

remotely sensed data such as IKONOS, small-footprint LiDAR, and large-scale 

aerial photography.

Another key opportunity is to enhance support to the National Forest System 

(NFS) within the U.S. Forest Service. FIA could extend its sampling grid of fi eld 

plots to all NFS lands, not just to those that meet the defi nition of forest. This 

expansion would also require an inventory compilation package be developed 

for NFS business needs, including mid-level vegetation map products. This is 

consistent with the FIA guiding principle to develop inventory methods and new 

techniques that will assist NFS in meeting the requirements of the National Forest 

Management Act. Remote sensing might reduce the cost of this ambitious strategy.

A remote sensing module might support other opportunities for FIA service. A 

rapid assessment capacity could be developed within FIA to provide emergency 

surveys of resource impacts within days or weeks following a major environmental disturbance such as a wildfi re, storm, hurricane, or sudden insect or disease 

outbreak (Van Deusen and others 1999).

Urban trees and forests affect quality of life of urban populations, which 

constitute approximately 80 percent of the U.S. population. Urban forestry is especially important to states with increasing populations. However, the FIA system 

does not include urban forests. A systematic approach to collecting and reporting 

data on status and trends of trees and forests in urban settings is another strategic opportunity. The FIA sample frame could be expanded to urban settings. The 

Federal role would best remain at the strategic scale, but an FIA system could 
2 USDA Forest Service RMRS-GTR-239. 2010.

serve as a platform for institutions to intensify to the individual city level. In principle, remote sensing technologies could contribute to a cost effective solution in 

urban settings.

FIA has the opportunity to help build a national rangeland inventory. The 2002 

House Interior Appropriations Committee report (USDA Forest Service 2007) 

included language directing the Secretaries of Agriculture and the Interior to collaborate in implementing a rangeland monitoring system. There is an ongoing 

effort to explore this opportunity in partnership with the USDA Forest Service’s 

National Forest System, the USDA Natural Resources Conservation Service’s 

National Resources Inventory, and the USDI Bureau of Land Management. 

Remote sensing technologies offer a potential contribution to cost-effective implementation. The feasibility of this major undertaking might be improved with 

remotely sensed data. This too is an FIA strategic priority (USDA Forest Service 

2007).

FIA also has the strategic opportunity to monitor wildlife habitat to mitigate 

concerns about management actions on species viability. However, standard FIA 

indicators, plot size, and measurements protocols can miss important habitat features. The geographic scale of a habitat feature for many wildlife species might 

not match the scale of an FIA fi eld plot. Enhancements or embellishments might 

be necessary to the traditional FIA design and processing procedures. Synoptic 

remote sensing methods might produce geospatial data on certain habitat features with suffi cient accuracy. Other habitat features require higher resolution 

observations that are feasible with a sample of large-scale aerial imagery or smallfootprint LiDAR data that cover sampling units 5 acres to 2500 acres in size. 

Holthausen and others (2005) provide a detailed assessment of strategic monitoring of wildlife populations that could be linked to the FIA program.

There is another strategic opportunity to conduct strategic-level monitoring of 

status and trends on all treed land. Indicators could be developed for ecosystem 

health, biodiversity, carbon sequestration, wildlife corridors and habitat, narrow 

riparian features, windrows, and agroforestry stands. Some indicators might be 

reliably measured with high-resolution remote sensing technologies.

Integrating new technologies is critical to the effi cient implementation of the 

FIA program (Van Deusen and others 1999). One focus of the 2007 FIA strategic 

plan is to build technology partnerships with other national programs that produce data related to FIA objectives. These national programs include: National 

Resources Inventory conducted by the USDA Natural Resources Conservation 

Service; the Multi-Resolution Land Characterization Consortium and its National 

Land Cover Data products, which are coordinated by the USGS Earth Resources 

Observation and Science Center; the GAP Analysis Program by the USGS 

Biological Resources Division; NASA; and NOAA (USDA Forest Service 2007). 

Most opportunities for improved technologies involve sharing costs and logistical burdens to use remotely sensed data from satellites and low-altitude aircraft 

more effi ciently and effectively in order to facilitate broad-scale applications of 

Geographical Information Systems.

FIA plans to increase capacity to conduct research and development on techniques that improve how data are collected, analyzed, and disseminated (USDA 

Forest Service 2007). Potential research areas include improved sampling and 

estimation procedures, better linkages to small-area or tactical assessments, development of improved geospatial analytical tools, and better mechanisms for 

making data and results available to the public. The highest priorities are to improve effi ciency of FIA, expand the scope of FIA, and develop new product lines 

that demonstrate value added to FIA. An emphasis on national techniques will 

increase FIA involvement in internally directed extramural research that will increase the knowledge and uses of FIA data.

Accomplishing these strategic FIA objectives will likely require statistical 

estimators that combine annual time-series of FIA fi eld data with separate timeseries of remotely sensed data from different sensors with different resolutions 
USDA Forest Service RMRS-GTR-239. 2010. 3

in order to estimate status, trends, and location by class for land use/land cover/

fragmentation change analysis (de Gruijter and others 2006). This report is one 

attempt to advance these programmatic objectives. It describes a suite of statistical estimation methods that accommodate complexities inherent to using diverse 

remote sensing technologies. It is a unifi ed and cohesive approach to estimation. 

These methods rely upon the extensive literature on Kalman fi ltering, and they 

build upon recent developments by Knottnerus (2003) that are closely related to 

the static Kalman fi lter.

Outline

This paper is organized as follows. The section “Stratifi cation in FIA” includes 

the limitations of FIA’s post-stratifi cation estimator. The next section, “Recursive 

Restriction Estimator (RRE)”, introduces the core estimation concepts used 

throughout this report. This section illustrates the algebraic identity between the 

Generalized Least Squares survey estimator and the Kalman fi lter. These concepts 

are applied with Recursive Restriction Estimation in complex sampling designs 

using the following examples.

The fi rst example is “Two-Phase Sampling for Continuous Variables with a 

Sample at Phase 1”. It considers the case of a large sample of points that are measured with high-resolution remote sensing at Phase 1 (P1) and a small sub-sample 

of these points that are measured by a fi eld crew. It is similar to FIA methods used 

in the 1960s to 1990s. This example does not include pre-stratifi cation, but it does 

apply to the case of double-sampling for stratifi cation with equal inclusion probabilities among strata. This example illustrates estimation in state-space with the 

static Kalman fi lter, although it requires only a single recursion of the RE object, 

which is a special case of RRE.

The second example is “Two-Phase Sampling for Continuous Variables with a 

Census at Phase 1”. Rather than a sample of points at Phase 1, this example uses 

one or more censuses of remotely sensed pixels at Phase 1. A special case of this 

design is used by FIA throughout the United States. However, this FIA estimator, which uses temporally indifferent post-stratifi cation, captures only a portion 

of the available auxiliary pixel data (see “Stratifi cation In FIA,”). RRE is a more 

capable alternative. Computation of optimal RRE weights requires inversion of a 

covariance matrix, which is the partition of the sample covariance matrix for the 

auxiliary variables. However, whenever the auxiliary data at the population level 

are known constants, this covariance matrix is singular or ill-conditioned, and the 

inversion is infeasible. This example introduces one form of RRE that solves this 

numerical problem. The vector of census constraints, which contains the exact 

population totals for all auxiliary variables, is processed recursively, one element 

at a time. The RRE sequentially applies the scalar inverse of the sample variance 

for each auxiliary variable rather than attempt an infeasible matrix inversion. This 

section also demonstrates RRE is equivalent to optimal calibration.

The third example is “Three-Phase Sampling for Continuous Variables”. It 

starts with the previous example and then adds a third phase, which is an equalprobability sub-sample of the Phase 2 (P2) sample. This example is directly 

relevant to the FIA design that includes the Phase 3 (P3) Forest Health Monitoring 

plots. It uses all remotely sensed data and the Phase 2 fi eld measurements to improve population estimates for the Phase 3 variables. Sequential RRE is a simple 

solution to this otherwise complex estimation problem. Simplicity is achieved by 

fi rst using RRE with the remotely sensed data to improve population estimates of 

variables that are measured with FIA Phase 2 fi eld protocol. In the next step, the 

RRE object combines the results of the fi rst step with the Phase 3 sub-sample. 

This two-step process is an example of a different type of recursion. The primary purpose is to simplify the estimator rather than solve numerical problems. 

However, the dimensions of the multivariate vectors can become large, posing 

numerical risks and other problems. This example includes a sub-optimal RRE 

alternative that uses smaller matrix dimensions to be more numerically robust in.
4 USDA Forest Service RMRS-GTR-239. 2010.

The fourth example is “Single-Phase Sampling for Continuous Variables 

with Cluster Plots”. This section introduces an RRE for experimental designs 

such as the FIA Nevada Photo-based Inventory Pilot (NPIP) study. NPIP uses a 

50-acre photo-plot, which is a cluster of 49 points that are measured with a sample 

of high-resolution aerial photography. One of these 49 points is the center of a 

standard FIA Phase 2 fi eld plot. RRE uses the sample of photo-interpreted measurements to improve precision of population estimates from the sample of Phase 

2 fi eld measurements. The previous examples consider remotely sensed data from 

independent samples at the different phases. That type of independence does not 

exist when remotely sensed and fi eld measurements are made within the same 

cluster plot in this two-stage design.

The fi fth example is “Two-Phase Sampling for Continuous Variables with 

Cluster Plots”. This is based on the previous NPIP example with the addition of 

a large, independent sample of 50-acre cluster plots. This large Phase 1 sample 

is measured with photo-interpretation of the same type of high-resolution aerial 

photography; there are no fi eld measurements made within this Phase 1 sample of 

cluster plots. Furthermore, this example includes wall-to-wall, full-coverage data 

from spaceborne sensors. RRE is a simple solution to this complex design.

All of the examples assume the remotely sensed data are continuous variables. 

Each continuous variable is represented by a single element in a multivariate 

vector. The section “Categorical Variables” expands the scope of RRE. A polychotomous variable with C categories may be readily transformed into a vector 

of C dichotomous variables and then used directly with the estimators in previous 

sections. Rather than a single element for a continuous variable, there will be C

elements for a categorical variable. If there are two polychotomous categorical 

variables, then a C1

-byC2 contingency table of cross-classifi cations is possible. 

Each of the C1×C2 dichotomous cross-classifi cations would require a separate 

vector element. This can produce vectors and covariance matrices with large dimensions, which often lead to numerical problems. Sub-optimal alternatives are 

discussed.

The section “Robust Numerics” briefl y covers the subject of numerical roundoff error and the need for robust algorithms. U-D factorization, which uses the 

modifi ed Gram-Schmidt orthogonalization, is an example. Optimal sample 

survey estimation is hindered by numerical problems that have been solved by 

engineering statisticians. The Kalman fi ltering literature offers an abundance of 

reliable solutions

FIA relies on “expansion factors” to simplify post-stratifi ed estimation in FIA’s 

large database. FIA expansion factors are analogous to calibration weights, which, 

likewise, are widely used in large government survey programs to conveniently 

improve statistical effi ciency with auxiliary information. FIA expansion factors 

and calibration weights modify the design weights for each Primary Sampling 

Unit (PSU) to incorporate the auxiliary data, such as the population census of predictions for remotely sensed pixels. The section “Optimal PSU Expansion Values 

for Each FIA Plot” develops analogous methods for the RRE. This method assigns a vector of expansion values to each PSU. The RRE expansion vector for 

each PSU uses the auxiliary data specifi cally observed for that PSU. The sum of 

RRE expansion values for each PSU in the sampled population will exactly equal 

the estimated vector of population totals from the RRE. The RRE covariance matrix is similarly expressed as the sum of expansion value vector cross-products at 

the PSU level. This solves problems with the FIA database that are identifi ed by 

Scott and others (2005). Further development is needed for FIA plot expansion 

factors from RRE algorithms.

The section “Multivariate Vector of FIA Study Variables” describes the core 

variables being estimated by FIA. In principle, all these variables can be simultaneously estimated with RRE. Multivariate FIA response variables enable 

pseudo-estimators for other FIA statistics in later sections. However, it is always 

possible to estimate one study variable at a time, which is the situation in FIA.
USDA Forest Service RMRS-GTR-239. 2010. 5

The section “Multivariate Vector of Auxiliary Variables” briefl y covers the 

different types of remotely sensed and geospatial variables that might improve 

precision of FIA statistics. RRE can simply accommodate the complex sampling 

designs required to utilize auxiliary information from these diverse technologies.

Next, a series of examples illustrate various pseudo-estimators. These are 

transformations of vector estimate from a design-based estimator. They depend 

upon multivariate estimation and are not generally feasible with the usual univariate estimators. The section “Linear, Small-Area, Synthetic Estimator with 

Census Predictors” is an example of a multivariate transformation for small-area 

estimation with inverse calibration (Tenenbein 1972; Brown 1982). The example 

uses a complex version of the NPIP design (discussed in “Single-Phase Sampling 

for Continuous Variables with Cluster Plots”), which uses a small sample of 50-

acre photo-plots that surround a 1-acre FIA fi eld plot, a large sample of 50-acre 

photo-plots that do not include a fi eld plot, and wall-to-wall remotely sensed data. 

However, the same pseudo-estimation methods apply to any RRE estimate and 

its covariance matrix. The small areas in this example are individual 6000-acre 

FIA hexagons, which FIA uses for national-scale geospatial presentations. These 

pseudo-estimators are consistent in that the sum of small-area estimates over all 

small areas equals the RRE estimates for the total population. This would support 

a geospatial presentation database based on FIA hexagons.

The section “Non-Linear Pseudo-Estimators” covers more examples that are 

relevant to FIA. Computation of these non-linear transformations is straightforward. However, the estimator for the associated covariance matrix requires 

multivariate Taylor-series transformations. The sub-section “Product Estimator: 

Missing Data Imputation” is a simple example in which the fi rst-order Taylorseries produces an exact model-based estimator for the covariance matrix. The 

example uses model-based imputation of FIA biomass accretion estimates for 

lands that are too dangerous to measure by a fi eld crew and for private lands for 

which the landowner has denied legal trespass to an FIA fi eld crew. The example 

assumes classifi cations of wall-to-wall pixels from a spaceborne sensor are available for all lands. These remotely sensed data serve as predictors of the missing 

FIA fi eld measurements. The model assumes that the expected biomass accretion 

for each remotely sensed category is independent from the missing data status. 

Given these model assumptions, it readily produces variance estimates for population totals that include both the sampling and estimation errors for the measured 

sampling units and prediction error from the model for missing data.

The sub-section “Ratio Pseudo-Estimator: Volume per Estimated Unit Area of 

Forest Cover” considers the ratio operator, which is a more complex non-linear 

transformation. The example is estimation of wood volume per forested acre. The 

pseudo-estimator is the population estimate of total wood volume divided by the 

estimated total area of forest. The covariance matrix for this pseudo-estimator 

uses the fi rst-order Taylor-series approximation. The sub-section “Recursive 

Object-Oriented Simplifi cation of Taylor-Series Approximations” presents 

several techniques that simplify derivation of multivariate Taylor-series approximations for covariance matrices. The sub-section “A Model-Based Estimator for 

Small-Domain Predictions” is a more complex example of multivariate pseudoestimators that combines both linear and non-linear transformations. It uses the 

Taylor-series simplifi cations given in the previous sub-sections. The example uses 

unbiased estimates of tree volume with expensive measurements of upper-stem 

diameters on a small sub-sample of FIA Phase 2 fi eld plots. It assumes inexpensive volume estimates for Phase 2 plots are biased by volume equations that are fi t 

to historical data, which no longer represent the current sampled population. This 

is a form of calibration in which an expensive, unbiased estimate from a small 

sample is empirically used to correct for measurement bias in a less-expensive 

estimate from a large sample. The calibration is performed for many small-domains. The sum of estimates for all domains is consistent with the estimate for the 

population total. The sub-section “Second-Order Taylor-Series Approximation to 
6 USDA Forest Service RMRS-GTR-239. 2010.

Assess Bias” considers the error in fi rst-order Taylor-series approximations by 

computing the second-order approximation. Recursive methods presented in previous sub-sections simplify derivation of the second-order approximation.

The fi nal sections depart from mathematical statistics and concentrate on application topics. The section “Accuracy and Registration of Remotely Sensed 

Data” explores the degree of association required between the study variables and 

the remotely sensed auxiliary variables. The “Discussion” covers the following 

topics: the advantages of RRE compared to post-stratifi cation, Risk management 

with complex sampling designs, and seminal literature. “Future Research and 

Development” briefl y discusses the next steps needed to implement RRE confi -

dently with the FIA database. “Conclusions” follows.

Stratifi cation in FIA

FIA is a pioneer in the use of remotely sensed data for stratifi cation to improve effi ciency of its statistical products (e.g., Bickford and others 1963). Today, 

this is exclusively accomplished with the post-stratifi cation statistical estimator 

(Scott and others 2005). FIA applies post-stratifi cation with full-coverage, remotely sensed thematic maps of forest versus non-forest cover produced with 

wall-to-wall Landsat data (e.g., Hoppus and Lister 2002; Nelson and others 2007). 

Hansen and Wendt (2000) and McRoberts and others (2002) further stratify based 

on edge conditions between the following remotely sensed classifi cations of forest and non-forest: forest, forest edge, non-forest edge, and non-forest. McRoberts 

(2006) uses a related approach with four strata that are based on predictions of 

the proportion forest area in each Landsat pixel from a logistic regression model. 

These stratifi ers separate heterogeneous edge conditions from more homogeneous 

conditions within the interior of stands. Most stratifi cation schemes help isolate 

many of the most serious classifi cation errors that are caused by misregistration 

between FIA plots and their corresponding pixels. A relatively small frequency of 

misclassifi cation errors can degrade the statistical effi ciencies offered by remote 

sensing (Czaplewski and Patterson 2001, 2003).

Certain FIA areal summary statistics are constrained to match administrative 

records for geopolitical entities. These include Census Bureau statistics for total 

area of each county, regardless of terrestrial land cover, and the area of signifi cant 

water bodies by county. Areal constraints used by FIA also include major land 

ownership categories. FIA cross-classifi es remotely sensed pixel data and these 

administrative areal statistics, and this merged cross-classifi cation is applied with 

the post-stratifi cation estimator to impose areal control. The result is numerous 

strata, many of which have very small samples of FIA sample plots. Stratifi cation 

is best used to reduce variance in sample estimates. However, FIA also uses stratifi cation to impose areal control for administrative statistics. Areal control does 

not necessarily produce variance reduction. Stratifi cation for areal control and 

remotely sensed data produces numerous small strata that contain few FIA plots.

Prior to 1999, FIA would survey all fi eld plots in a state, produce statistical and 

analytical reports, and then move to the next state in the regional survey sequence. 

Each state was re-surveyed every 7 to 15 years (Gillespie 2000). FIA documents 

cite this as a periodic survey. These periodic surveys served the states well shortly 

after the full survey was completed. However, confi dence in the data decreased 

as the data aged more than fi ve years (American Forest Council 1992:1-15). The 

Agricultural Research Extension and Education Reform Act of 1998 directed all 

FIA units to change these statewide periodic systems into a nationally consistent 

annual survey system (Gillespie 1999). Beginning in 1999, FIA responded to this 

Congressional direction by systematically subdividing its sampling frame into 5 

to 10 separate panels. Each panel includes 10 to 20 percent systematic sample of 

all FIA fi eld plots in each state. Ideally, a single panel is fully measured during a 

single year (Van Deusen 2002). However, this subsampling frequency introduces 
USDA Forest Service RMRS-GTR-239. 2010. 7

an unintended consequence—it further aggravates problems that are inherent with 

detailed stratifi cation and very small, within-stratum sample sizes.

One solution to the relatively small sample size in each annual FIA panel is 

to merge fi eld data from multiple years. A time-series of panel measurements 

is treated as though it was measured during a single year. Post-stratifi cation 

uses a single set of remotely sensed images (Bechtold and Scott 2005) rather 

than an annual time-series of remotely sensed data. However, changes in plot 

conditions between acquisition of the remotely sensed data and the most recent 

fi eld measurements are not distinguishable from prediction errors with remotely 

sensed data. This type of disagreement between fi eld data and remotely sensed 

classifi cations will reduce the statistical effi ciency from stratifi cation. The loss in 

effi ciency will be rapid as the agreement deteriorates (Czaplewski and Patterson 

2001, 2003), regardless of the cause (i.e., classifi cation error versus change in 

land cover). This means that the value of remotely sensed data slowly diminishes 

in dynamic landscapes, which are exactly the types of domains that often have the 

greatest need for current monitoring data. These same areas could benefi t the most 

from remotely sensed data if the limits from post-stratifi cation can be mitigated.

Inaccurate variance estimates can be common for strata that have rare or common attributes. For example, if all plots sampled in a forest stratum are truly 

forest, then the estimated variance for forest cover will be zero in that stratum. 

Even though the stratum has some non-forest cover, the stratum estimate for nonforest area would be zero with zero variance. Likewise, if a particular condition 

exists in a stratum but not in the sample for that stratum, then the sample survey 

estimate will be zero with zero estimated sampling error. This is known as a sampling zero in the analysis of contingency tables. The frequency of this type of error 

in realized variance estimates increases as the number of strata increases because 

within-stratum sample sizes become small (e.g., 4≤nh

≤10), as recommended by 

Scott and others (2005). Seminal references recommend larger sample sizes (e.g., 

20≤nh; Särndal and others 1992:251, 267; de Gruijter and others 2006:118).

Stratifi cation on remotely sensed data for both variance reduction and areal 

control has another undesirable consequence. FIA fi eld plots frequently straddle 

stratum boundaries, which compromises the assumed independence of sampling 

errors among strata in the post-stratifi cation estimator (Hahn and others 1995; 

Bechtold and Scott 2005). FIA treats a single plot that straddles multiple strata 

as if it is partially included in each stratum. The plot expansion factors are used 

to account for “mixed-condition” plots (Scott and others 2005). This means that 

the variance estimators FIA uses assume that this departure from underlying statistical assumptions of stratifi cation does not cause substantial bias in variance 

estimators. Van Deusen (2005) found that these “variance estimates will often be 

optimistic when weighted estimators are applied to arbitrary population subsets.” 

Scott and others (2005) reported that “FIA is evaluating the frequency of plots 

that straddle NFS boundaries and may switch to the ratio-of-means estimators 

described by Zarnoch and Bechtold (2000), if necessary.”

The following example attempts to clarify the current situation. Auxiliary 

variables are available from different sources of remotely sensed and geospatial 

data. The post-stratifi cation estimator used by FIA (Scott and others 2005) requires cross-classifi cation of multiple sources of categorical auxiliary data (Zhang 

2000). For example, consider two such sources: (1) a detailed thematic map of 

fi ve forest types that uses remotely sensed data; and (2) a map of fi ve ecoregions, 

each of which is associated with different components of potential natural vegetation (Bailey 2004). Cross-classifi cation produces 25 potential strata. Further, 

assume there are 1000 FIA fi eld plots (i.e., a typical FIA inventory unit), of which 

100 are measured each year in a single panel. If only one of the potential sources 

of auxiliary data is used for post-stratifi cation, which has fi ve categories, then 

there would be an average of (100/5) = 20 FIA plots in each stratum. If both 

sources of auxiliary data are used in post-stratifi cation, there would be an average 

of [(100/5)/5] = 4 FIA plots in each cross-classifi ed stratum. Twenty FIA plots 
8 USDA Forest Service RMRS-GTR-239. 2010.

might be suffi cient for estimating conditions in a single stratum, but fi ve FIA 

plots might not be suffi cient for reliably estimating conditions in a single crossclassifi ed stratum. Because the number of strata is limited by the FIA sample 

size, the number of categories created through cross-classifi cation is also limited. 

Inevitably, different cross-classifi ed categories must be merged, or some thematic 

information must be ignored, in order to create strata with suffi cient sample sizes. 

Furthermore, stratifi cation reduces the information in continuous remotely sensed 

data into categorical values (Mandallaz 2008:89). For example, remotely sensed 

data can predict continuous variables that are correlated with relevant variables, 

such as total biomass, but some of that remotely sensed information is lost when 

categorized into an ordinal variable.

Recursive Restriction Estimator (RRE)

The Recursive Restriction Esti mator (RRE) refers to the sequential application 

of the Restriction Estimator (RE), in which the RE output from one recursion is 

used as input to the next recursion. RRE is a special case of the Kalman fi lter, 

namely the static Kalman fi lter. The Kalman fi lter was originally developed as 

a multivariate time-series estimator for dynamic stochastic systems. In contrast, 

the static Kalman fi lter is not used for time-series applications. Rather, it is used 

here to estimate more precisely a vector of population totals at one point in time, 

which is considered the static “state” of the system. The Kalman fi lter, either in a 

time-series or static application, is a sequential recursive estimator in multivariate 

state-space.

Pollock (2003) reviews the history of recursive estimation. Gauss originally 

published the concept of recursive least squares during 1821 to 1826. Gauss refers 

to augmentation with new observations, which is essentially recursive leastsquares estimation. Recursive estimation was rediscovered by Kalman (1960) and 

Kalman and Bucy (1961), which soon spawned the vast amount of literature on 

the Kalman fi lter. Kalman and Bucy cast recursive estimation into the state-space 

model with time-varying parameters, which complicates the mathematics but expands the applications.

Knottnerus (2003) provides a detailed treatise on estimation for complex 

sample surveys. His unifying “standard sampling model” uses Pythagorean regression, and he demonstrates the intimate connection between Pythagorean 

regression and the Kalman fi lter. Knottnerus further illuminates the relationships 

among regression theory, systems theory, and sampling theory. Similar connections in more applied settings are made by Czaplewski and others (1988), 

Czaplewski (1989, 1990, 1996, 1999, 2000, 2001, in prep.[b]), and Czaplewski 

and Thompson (2009).

State-Space

State-space is a mathematical model of a stochastic process. The state-space 

concept of the Kalman fi lter originated in electrical engineering for applications 

in astronautical navigation systems (Jazwinski 1970). It has been extended to a 

large number of different stochastic processes, including those that evolve over 

both time and space (Wikle and Cressie 1999). Harvey and Todd (1983) equate 

state-space to “structural” or “unobserved components” time-series models. In 

the context of probability sampling, the stochastic structure is defi ned by the 

probability distributions of the various random sampling and measurement errors (Särndal and others 1992:21, 515, 538, 606; Knottnerus 2003:11). The U.S. 

Bureau of Labor Statistics has used the state-space perspective for two decades in 

longitudinal panel surveys (Tiller 1992; Pfeffermann and Tiller 2006).

Kalman (1960) used the state-space paradigm to describe a dynamic stochastic 

system: “The fundamental concept is the notion of the state. By this is meant, 
USDA Forest Service RMRS-GTR-239. 2010. 9

intuitively, some quantitative information (a set of numbers, a function, etc.) 

which is the least amount of data one has to know about the past behavior of the 

system in order to predict its future behavior.” Jazwinski (1970) defi nes the statespace as a family of random variables or vectors. The state-vector is described 

by Maybeck (1979:26, 112) as “the set of n variables, the values of which are 

suffi cient to describe the system ... completely.” State-space is “memoryless” 

in the sense that the state at time t fully contains all suffi cient information about 

the past (Kalman and others 1969); it is not necessary to store past information. 

Otherwise, the dimensions of the estimation problem would be larger, as in most 

other time-series models.

In the context of the static Kalman fi lter, the state-space of the multivariate 

time-invariant system contains all relevant quantitative information, namely the 

population parameters, about a sampled population. In a simple design, the statespace, or the vector of population parameters, includes the primary study variables. 

In a complex sampling design, the state-space is enlarged to include complex 

auxiliary parameters at the population or domain levels. Auxiliary variables are 

generally inexpensive, correlated to some degree with the study variables, and, 

therefore, available to improve estimates of the study variable (Särndal and others 

1992:220, 304, 397).

In the methods that follow, the state-space does not explicitly include individual population elements, sampling units, or inclusion probabilities, although 

those essential concepts in sample surveys are used for unbiased, design-based 

estimation of a state-vector and its covariance matrix during execution of the RE 

(Sugden and Smith 2002; Knottnerus 2003:357). The sequential processing of 

time-series information in the dynamic Kalman fi lter is replaced in the static

Kalman fi lter with sequential processing of information from different components of a complex sample survey, such as different stages or phases. Because the 

Kalman fi lter is memoryless, the static Kalman fi lter need not store all auxiliary 

information as it sequentially processes individual components of a complex sample survey. This reduces the dimensions of the sequential multivariate estimator.

Recursive Estimation

Meinhold and Singpurwalla (1983) describe the recursive process as “the 

evolution of a series of regression functions, each having a potentially different 

intercept and regression coeffi cient; the evolution stems from a learning process 

involving all the data.” The dynamic Kalman fi lter is a recursive estimator that 

uses two prior vector estimates: (1) a model-based estimate of the current statevector based on the best estimate at the previous time step; and (2) a current 

direct measurement (e.g., design-based sample estimate of population totals) for 

a linear transformation of the same state-vector. The Kalman fi lter combines both 

estimates into a more precise estimate of the current state-vector using a multivariate composite estimator. Assuming unbiased measurements of the current 

state-vector, unbiased estimators for the state-vector at the previous time period, 

and unbiased model predictions of the current state based on the previous state, 

the Kalman fi lter removes all information available from the current time and 

all previous time steps. Given these assumptions, the remaining random residual 

estimation errors are “white noise” and are independent of all previous estimation errors (Knottnerus 2003:311). The recursion process is repeated at the next 

time step as the state of the stochastic system changes and new measurements 

of the system become available. There is no need to process any past information beyond the previous time step because the Kalman fi lter estimate contains 

all previous information. In other words, the Kalman fi lter is memoryless in that 

it propagates all suffi cient statistics for the problem conditional on the realized 

independent observations.

The Kalman fi lter is seldom applied in sample surveys. Contributions by 

Knottnerus (2003), Knottnerus and van Duin (2006), Sõstra (2007), and Sõstra 

and Traat (2009) are among the few exceptions. More often, the application of 
10 USDA Forest Service RMRS-GTR-239. 2010.

the Kalman fi lter is in the context of time-series estimation with longitudinal panel surveys (e.g., Harvey 1978; Tam 1987; Tiller 1992; Jones 1993; Frees 2004; 

Oud 2004; de Gruijter and others 2006; Pfeffermann and Tiller 2006). These authors use models at the level of individual population elements or sampling units. 

However, state-space perspective (e.g., Rosenbrock and MacFarlane 1972) is 

equally valid. For example, Czaplewski and Thompson (2009) estimate the state 

of a large forest with the state-vector defi ned as the number of trees in several 

different condition categories within the entire sampled population. Individual 

sampling units in a probability survey are used in the measurement vector as estimates of these population parameters, but otherwise, they do not explicitly appear 

in the time-series model within the Kalman fi lter.

The Basic Idea

The Merriam-Webster Dictionary defi nes a recursion process as the “successi o n of elements (as numbers or functions) by operation on one or more preceding 

elements according to a rule or formula involving a fi nite number of steps.” In 

the context of RRE, the recursion starts with exactly two elements, called objects 

here, of a complex survey design. For example, consider FIA research on the 

Nevada Photo-based Inventory Pilot Study (Frescino and others 2009a), in which 

a sample of high-resolution aerial photography is added to the existing design of 

FIA Phase 2 fi eld samples and the wall-to-wall, full-coverage census of Landsat 

satellite pixels.

• The Phase 1 Object is a relatively large probability sample of PSUs. Each PSU 

is measured with two protocols: (1) remotely sensed vector measurements from 

a spaceborne, earth-observing satellite sensor and (2) vector measurements 

based on photo-interpretation of a sample of high-resolution aerial photography. 

The spaceborne sensor protocol is inexpensive, but it contains considerable 

measurement error relative to photo-interpretation. However, measurements 

from the spaceborne sensor are correlated to some degree with more accurate 

photo-interpreted measurements. Standard, design-based methods produce 

sample survey estimates for the vector of population totals and its covariance 

matrix. This is used as the state vector in the Kalman fi lter, which is discussed 

in more detail below. The state-vector includes a partition for the variables 

measured with the spaceborne sensor protocol. The remaining partition 

contains the variables measured with the photo-interpretation protocol. The 

covariance matrix includes a diagonal partition for the covariances among 

variables measured with the spaceborne protocol and another diagonal 

partition for variables measured with photo-interpretation. The remaining offdiagonal partition contains the estimated sample cross-covariances among the 

population estimates for the spaceborne and photo-interpreted variables.

• The “Census Object” is based on 30- by 30-m pixels, each of which is 

measured with the same spaceborne sensor protocol as in Phase 1. The photointerpretation protocol in Phase 1 is not used in the census. This is a true census 

because the set of remotely sensed pixels fully covers the entire sampled 

population. Enumeration of the spaceborne measurements for each and every 

pixel in the sampled population produces a vector constant of population totals 

for variables measured with the spaceborne protocol. The covariance matrix 

for the population totals exactly equals zero because there are no sampling 

errors. This vector constant contains the auxiliary information, and it is called 

the measurement vector in the Kalman fi lter.

• The RE combines the measurement vector from the Census Object with the 

state-vector from the Phase 1 Object. Depending on the magnitude of the 

off-diagonal cross-covariances among the spaceborne and photo-interpreted 

variables from the Phase 1 sample, the RE population estimate for the photointerpreted variables will be more precise than the corresponding design-based 

estimates from the Phase 1 Object alone. This is a design-based calibration 
USDA Forest Service RMRS-GTR-239. 2010. 11

estimator, which is discussed in the section “Relationship to an Optimal 

Sample Survey Estimator.”

• The RE vector estimate contains a partition that corresponds to the 

spaceborne variables. That partition will exactly equal the measurement 

vector from the remotely sensed census. Its corresponding partition in 

the RE covariance matrix is exactly zero. The remotely sensed census 

statistics essentially constrain the RE estimator. (See Knottnerus 

2003:Chapter 12 on the General Restriction Estimator) The RE estimate 

for the photo-interpreted variables contains all information contributed 

by the correlation with the spaceborne variables. Therefore, the partitions 

for the spaceborne variables may be deleted from further consideration. 

The remaining partition of improved population estimates for the photointerpreted variables is redefi ned as the measurement vector in the 

second recursive process, which follows.

So ends the fi rst recursion. Now, an example of the second recursive 

process, in which the RE results from the fi rst recursion are combined with 

another independent design object:

• The Phase 2 Object is a probability sample of PSUs with a relatively small 

sample size. Each Phase 2 PSU is measured with two protocols: (1) the 

same photo-interpretation of high-resolution aerial photography used in 

Phase 1 (above) and (2) expensive, on-the-ground vector measurements 

made by FIA fi eld crews. Measurements from the photo interpretation 

are reasonably well correlated with certain fi eld measurements. FIA uses 

these fi eld measurements as the minimum acceptable protocol for data 

quality of its statistical tables. As with the Phase 1 object above, standard 

design-based methods produce a sample survey estimate for the vector 

of population totals and its covariance matrix from Phase 2. This is used 

as the new defi nition of the state-vector during this second recursion. 

The off-diagonal partition of its covariance matrix contains the estimated 

sample cross-covariances among the population estimates for photointerpreted and FIA fi eld variables from the small Phase 2 sample.

• The Composite Object is identical to the partition of RRE vector estimate 

from the fi rst recursion that corresponds to the photo-interpreted variables. 

Its covariance matrix is the corresponding partition from the outcome 

of the fi rst recursion. Because the Composite Object fully incorporates 

all relevant information from the census in the fi rst recursion, the size 

of this covariance matrix is smaller than the corresponding partition in 

the design-based Phase 1 estimate. The Composite vector estimate is 

redefi ned as the measurement vector for this second recursion. It does 

not contain any elements for variables measured with the FIA fi eld 

protocol in Phase 2.

• The RE combines the measurement vector from the Composite Object 

with the state-vector from the Phase 2 Object. Because the off-diagonal 

cross-covariances among the photo-interpreted and fi eld variables from 

the Phase 2 sample are non-zero, RRE population estimates for the 

FIA fi eld variables will be more precise than the corresponding designbased estimates from the Phase 2 Object alone. The RRE partition that 

corresponds to the photo-interpreted variables is less precise in the fi rst 

recursion relative to the outcome of the second recursion because the 

latter gains strength from the Phase 2 sample. The remaining partition 

that corresponds to the FIA fi eld variables contains all information 

available from the Census, Phase 1, and Phase 2 objects. Unless needed 

for other purposes, the partition for the photo-interpreted variables may 

be discarded because all relevant information has been fi ltered out and 

incorporated into RRE for the partitions corresponding to the FIA fi eld 

variables.
12 USDA Forest Service RMRS-GTR-239. 2010.

Therefore, the design-consistent Empirical Best Linear Unbiased Estimator 

(EBLUE) of population estimates for the FIA fi eld variables is achieved through 

two recursive steps. The census of remotely sensed spaceborne measurements 

improves the population estimates for the photo-interpreted variables in the fi rst 

recursion. The second recursion uses these improved estimates results to further 

improve population estimates for the FIA fi eld variables. The recursive method 

separates the complexity of the sampling design into simpler objects that are 

sequentially combined with RRE. As briefl y discussed in the following section 

“Relationship to an Optimal Sample Survey Estimator,” RRE is also a multivariate recursive calibration estimator in state-space.

Now, the focus shifts to the details of the composite estimator. The univariate 

composite estimator is a conditionally unbiased statistical method that combines 

a weighted sum of two separate, previously calculated statistical estimates into a 

single, more effi cient estimate (e.g., Cochran 1977:346, 355). The more precise 

of the two previously calculated estimates is weighted more heavily. The scalar 

weights are computed to produce an approximately minimum variance estimator. 

The less familiar Restriction Estimator (RE) combines two previously calculated

vector estimates, which need not have equal dimensions, with a matrix of optimal 

weights computed with the estimated covariance matrix for each vector estimate. 

In the context of multilevel sampling, the composite estimator is used to combine estimates sequentially from the various sampling phases or stages. Sampling 

units are segregated into independent samples for each phase or stage (Magnussen 

2003, including references to Card 1982; Cochran 1977; Tenenbein 1972), and 

the multivariate composite estimator is used to combine pairs of stages, phases, 

censuses, or prior composite estimates into a more effi cient estimate of the variables of interest. But fi rst, the univariate composite estimator will be used as an 

introduction to the more complex multivariate composite estimator.

The Univariate Composite Estimator

The univariate composite estimator is well known in sample survey applications, such as sampling over time, small-area estimation, and adaptive sampling 

(e.g., Särndal and others 1992:371; Schreuder and others 1993:321; Binder and 

Hidiroglou 1988:190; Valliant and others 2000:400; De Gruijter and others 

2006:226; Gregoire and Valentine 2007:388). Czaplewski and Thompson (2009) 

provide a tutorial description of the univariate Kalman fi lter in the context of 

time-series data that assists intuitive understanding. The univariate composite estimator is presented here with notation that is a slight modifi cation of Särndal and 

others (1992:371).

Assume t

X and tZ

 are two unbiased estimators of the same univariate population 

total t for the sampled population (e.g., total woody biomass or total forest area). 

As an example, consider two independent estimates from simple random sample 

1 (s = 1) and an independent simple random sample 2 (s = 2):

( )

( ) ( )

( )

( )

( ) ( )

( )

¦

¦

¦

¦

∈ =

∈ =

∈ =

∈ =

−

 ̧

 ̧

1

·

 ̈

 ̈

©

§

− =

=

=

−

 ̧

 ̧

1

·

 ̈

 ̈

©

§

− =

=

=

2

2

2

2

1

2

2

1

1

1 ˆ ˆ

ˆ

1

1 ˆ ˆ

ˆ

i s Z

i Z

Z

Z

Z Z

i s Z

i

Z

i s X

i X

X

X

X X

i s X

i

X

n

y y

n

V t A

t Ay

n

y

y

n

y y

n

V t A

t Ay

n

y

y

 

 (1)

Let the linear combination of the two estimators be the weighted (i.e., constant 

“k,” 0<k<1) sum o f the two prior estimates. From Särndal and others (1992:Eq. 

9.9.3), the composite estimator is defi ned as t

RE is:
USDA Forest Service RMRS-GTR-239. 2010. 13

( )

( )

[ ]

RE

ˆˆ ˆ 1

ˆ ˆˆ

ˆ 1 ˆ

X Z

Z XZ

X

Z

t kt k t

t kt t

t

k k

t

= +−

=+ −

a o = − « » « » ¬ 1⁄4

. (2)

The variance estimator for the composite estimate (Särndal and others 1992:Eq. 

9.9.4) in Eq. 2 is:

( ) () ( ) ( ) ( ) ( )

() () ( )

[ ] () ( )

( ) ()

2 2

RE ˆˆ ˆ ˆ ˆ ˆ ˆˆ ˆ 1 21 ,

ˆ ˆ ˆ ˆ ˆˆ ˆ ,

ˆ ˆ ˆˆ ˆ ,

1 ˆ ˆˆ ˆ , ˆ 1

X Z XZ

Z Z XZ

X ZX

XZ Z

V t kV t k V t k k C t t

Vt kVt Ct t

Vt Ct t k

k k

Ct t Vt k

= +− + −

=− + a o ¬ 1⁄4

a o a o = − « » « » − ¬ 1⁄4 ¬ 1⁄4

 (3)

where C is the scalar cross-covariance among the random errors from the two 

estimators. Because the tw o samples are independent in this example, C = 0, and:

( ) () ( ) ( )

[ ] ( )

( )

2 2

RE ˆ ˆ ˆˆ ˆ 1

ˆ ˆ 0

1

0 ˆ ˆ 1

X Z

X

Z

V t kV t k V t

V t k

k k

V t k

= +−

a o a o = − « » « » « » − ¬ 1⁄4 ¬ 1⁄4

The unbiased minimum variance univariate composite estimator is achieved with 

the following k weighting constant (Bierman 1977):

  

    

       ^ `      1

ˆ ˆˆ,

ˆ ˆ ˆˆ 2 ,

ˆ ˆˆ ˆ ˆ ˆˆ , 2, 

  

  

Z XZ

X Z XZ

Z XZ X Z XZ

Vt Ct t

k

Vt Vt Ct t

Vt Ct t Vt Vt Ct t

 (4)

If the two estimates of t

X and tZ

 are independent, then C = 0 and Eq. 4 simplifi es to:

 

  

ˆ

ˆ ˆ 

Z

X Z

V t

k

Vt Vt

 (5)

(See Cochr an (1977:346), Maybeck (1979:12-14), and De Gruijter and others (2006:226).) Relatively large deviations from the optimal scalar weight (k) 

in Eq. 5 do not substantially degrade the effi ciency of the composite estimator 

(Schaible 1978). Therefore, a sub-optimal weight computed from sample estimates of the variances is expected to perform well in most univariate cases:
14 USDA Forest Service RMRS-GTR-239. 2010.

( )

() () X Z

Z

V t V t

V t k ˆ ˆ ˆ ˆ

ˆ ˆ

+ = (6)

The theoretical estimator in Eq. 5 is the Best Linear Unbiased Estimator (BLUE). 

The applied estimator in Eq. 6 is a sub-optimal Empirical Best Linear Unbiased 

Estimator (EBLUE) (Militino and others 2008). Because both estimators t

X and tZ

are design-consistent, their weighted sum is also design-consistent.

As seen from Eq. 6, the univariate composite estimator weights each independent estimate inversely proportional to its relative variances. The most precise 

estimate (least variance) receives the greatest weight, although the less precise 

estimate does receive a positive weight.

If the fi rst estimator is twice as precise as the second (i.e., Vt Vt ( ) ( )/2 = z x

t t ) 

and each estimator is mutually independent, then the fi rst estimate will be weighted more heavily (Eq. 6) by a factor of k = (2/3), while the second estimate, which 

is less precise, will receive the lesser weight of (1 - k) = (1/3):

( ) ( ) [ ] ( )

( ) [ ] ( )

( )

( ) Z

X

X X

X Z

V t

V t

V t V t

V t V t V t

ˆ ˆ 3

1

ˆ ˆ 3

2

2 ˆ ˆ 9

1 ˆ ˆ 94

0

3

2 1

3

2 ˆ 2 ˆ

3

2 ˆ 1 ˆ

3

2 ˆ ˆ

2 2

MvCE

 ̧

1

·  ̈

©

§ =

 ̧

1

·  ̈

©

§ =

 ̧

1

·  ̈

©

§  ̧ + 1·  ̈

©

§ =

»

1⁄4

o «

¬

a  ̧

1

·  ̈

©

§

 ̧ − 1

·  ̈

©

§  ̧ + 1

·  ̈

©

§  ̧ + − 1·  ̈

©

§ =

 (7)

The estimated variance for the resulting composite estimate (Eqs. 3 and 6) would 

be less than the variance of either independent estimate by itself (Eq. 7).

As another example, if both estimators are independent and equally precise 

(i.e., Vt Vt () () t t

x z = ), then each estimate receives an equal weight of k = 0.5. The 

composite estimate would have an estimated variance that is one-half of either 

independent estimate by itself (Eqs. 3 and 6).

The Restriction Estimator (RE) Object

The Restriction Estimator (RE) is a generalization of the univariate estimator 

in Eqs. 2 to 6. The RE is also a special case of the static Kalman fi lter (Maybeck 

1979:114; Gregoire and Walters 1988; Czaplewski 2000). RE is a “static linear 

model” in the terminology of Maybeck (1977:114). It is a special case of the 

Kalman fi lter in Eqs. 9 through 12. Like the Kalman fi lter, RRE is: (1) a recursive 

formulation of the Bayes estimator (e.g., Jazwinski 1970:145; Maybeck 1979:205), 

(2) the minimum mean square error predictor (e.g., Jazwinski 1970:149; Maybeck 

1977:232), and (3) a maximum likelihood estimator when joint densities are all 

Gaussian (e.g., Jazwinski 1970:207; Maybeck 1977:234). Duncan and Horn 

(1972), Diderrich (1985), Gregoire and Walters (1988), and Piepho and Ogutu 

(2007) demonstrate the close relationship between the Kalman fi lter and Theil and 

Goldberger’s (1961) mixed estimator. As a special case of the Kalman fi lter, RRE 

is an iterative least-squares approach to estimation (Jazwinski 1970:276; Bierman 

1977). Based on a matrix inversion lemma employed by earlier authors, Diderrich 

(1985) and Czaplewski (2001) illustrate the algebraic equivalency between the 

generalized least squares estimator and the composite estimator in this context 

(Householder 1964; Maybeck 1979:213). Knottnerus (2003:315, 326) uses the 
USDA Forest Service RMRS-GTR-239. 2010. 15

perspective of Pythagorean regression and the matrix inversion lemma to make 

the strong connection between the Kalman fi lter and estimators for complex sample surveys. This includes constraints on the estimands. De Gruijter and others 

(2006) give a more contemporaneous exposition on the Kalman fi lter in the context of sampling in time and space for monitoring of natural resources. Because 

the sample survey applications considered here are temporally static, the terms 

“RRE” and “static Kalman fi lter” are used synonymously.

The following illustration uses the seminal work by Maybeck (1979) on 

the Kalman fi lter as an introduction to RE. Blending the notation of Maybeck 

(1979:246-247) with that of Särndal and others (1992:Appendix A), the stochastic 

model for two multivariate design-based sample matrix estimates as:

() ( )

[ ]

[ ] ( ) ( ) () ()

() ()

[ ]

[ ] ( ) ( ) ( )

[ ] () () E ( ) ˆ ,ˆ ( ) 1

E ˆ 1

E

ˆ

E ˆ 1 1

E

ˆ

1 1

1

1 1

1

1 0

İ İ C t t C

İ İ V t R

İ 0

t H t İ

İ İ V t P Q

İ 0

t t İ

′ = =

′ = =

=

= +

′ = = =

=

= +

W V XY Z

V V Z

V

Z XY V

W W XY

W

XY X W

 (8)

where tXY(1) is the (qX + qY)-by-1 state-vector of population totals for the qX auxiliary variables (X) and qY

 s tudy variables (Y), and tZ(1) is the qZ-by-1 measurement 

vector of population totals from a linear transformation (matrix constant H) of 

population parameters for all measurement protocols for variables X and Y. The 

state-vector tXY(1) and measurement vector tZ(1) (Eq. 8) may have different dimensions (i.e., qXY ≠ qZ

) or the same dimensions (i.e., qXY = qZ). The RE (below) will 

combine vector estimates of t

XY(1) and tZ(1) into a more precise, minimum-variance 

estimate of the full (qX + qY)-by-1 state-vector . This structure follows that used 

by Knottnerus (2003:335) for the regression estimator as a partitioned restriction 

estimator.

The notation in Eq. 8 follows: εW is the qXY-by-1 vector of random estimation 

errors with qXY-by-qXY covariance matrix ; H is the qZ-by-qXY matrix of known 

constants that identify the linear transformation of tXY(1) that produces tZ(1); εZ is the 

qZ-by-1 vector of random estimation errors for the sample estimate of the auxiliary variables with qZ

-by-qZ covariance matrix ; and is qXY-by-qZ cross-covariance 

matrix between εW and εZ.

Translation of remaining notation used by Maybeck (1979:246-247) in Eq. 

8 follows: the time-series transition matrices Φ = G = I (i.e., qXY-by-qXY identity 

matrix); the qXY-by-qXY covariance matrices P(0) = 0 (i.e., the zero matrix) and

[P(1) = ΦP(0)Φ́ + GQ(1)Ǵ = Q(1)]; and the input control vector B×u = 0 (i.e., 

qXY-by-1 zero vector). Given the abbreviated stochastic model in Eq. 8, the minimum variance composite estimator follows equations 5-115 through 5-117 for 

the Kalman fi lter in Maybeck (1979:247). Notation starts with that of Maybeck 

(1979) and is followed with equivalent notation that is a facsimile of Särndal and 

others (1992).

The qXY-by-qZ gain matrix K, which is a matrix extension of the univariate 

weighting constant k in Eq. 4 (Maybeck 1979:Eq. 5-115), is:

[ ] () () ([ ] () () () () )

( )

( )

( ) ()

( )( )  ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

1⁄4

o

«

«

«

¬

a

′ ′ + +

′ +

»

»

1⁄4

o

«

«

¬

a

+ ′

′

=

= ′ + ′ + + + ′ ′

−

−

1

1

ˆ, ˆ ˆ ˆ , ˆ ˆ

ˆ ˆ ˆ ˆ

ˆ, ˆ ˆ

ˆ ˆ

1 1 1 1 1 1

H C t t C t t H

H V t H V t

C t t

V t H

K P H C HP H R HC C H

XY Z XY Z

XY Z

XY Z

X XY (9)
16 USDA Forest Service RMRS-GTR-239. 2010.

Use of estimated covariances in Eq. 9 does not affect the asymptotic variance 

(Knottnerus 2003:336).

Using the gain matrix K in Eq. 9, the RE (Maybeck 1979:Eq. 5-116) is:

RE ( ) ˆˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

XY Z XY

X X

Z

Y Y

=+ −

ao ao § ·

=+ − «» «»  ̈  ̧

¬1⁄4 ¬1⁄4 © 1

t t K t Ht

t t

Kt H

t t

, (10)

The gain matrix K is the matrix of approximately optimal weights on the 

residual difference between the Phase 1 and Phase 2 population estimates for 

the auxiliary variables. See Knottnerus (2003:335-357) for a comparison of the 

Kalman fi lter with the minimum variance regression estimator.

Eq. 10 is the multivariate extension of the univariate composite estimator in 

Eq. 2. The corresponding covariance matrix (Maybeck 1979:Eq. 5-117) is:

( ) () () ()

( ) () ( )

RE

ˆ 1 11

ˆ ˆ ˆ ˆ ˆˆ ˆ , XY XY XY Z

=− + a o ¬ 1⁄4

=− + a o ¬ 1⁄4

V t P K HP C

V t K HV t C t t

 (11)

which is the multivariate extension of the univariate composite estimator in Eq. 

3. (The covariance matrix in Eq. 11 is expressed in a more familiar form in the 

next section.)

Special cases of the RE are algebraically identical to optimal sample survey 

estimators, as shown in the next section for two-phase sampling. More general 

expressions of the RE may be applied to more complex designs, such as two-stage 

sampling. Examples include “Single-Phase Sampling for Continuous Variables 

with Cluster Plots”. The sequential application of the RRE addresses even more 

complex designs. One example is “Two-Phase Sampling for Continuous Variables 

with Cluster Plots”, which uses photo-interpretation and sub-sampling for fi eld 

measurements for a small sample of cluster plots, photo-interpretation (without 

fi eld measurements) for a large independent sample of cluster plots, and a census 

of pixels from a spaceborne sensor. Furthermore, because RRE is a special case of 

the Kalman fi lter, the methods that follow are compatible with time-series estimators for longitudinal surveys.

Relationship to an Optimal Sample Survey Estimator

While the structure of the covariance matrix estimator in Eq. 11 is familiar 

in the Kalman fi ltering literature, it is much less familiar in the sample survey 

literature. The major exception is Knottnerus (2003). He simplifi es estimators 

for complex sample survey designs through the Pythagorean regression form of 

Generalized Least Squares regression. Chapter 2.8 in Knottnerus’ book demonstrates that Pythagorean regression is algebraically equivalent to the static Kalman 

fi lter.

A special case of the RE Eq. 11 is algebraically identical to the minimum 

variance Generalized Least Squares regression estimator in the sample survey 

literature. Let the vector tZ be estimated from a Phase 1 sample or census. Let vector tXY be estimated from an independent Phase 2 sample. Assume the fi rst qX-by-1 

partition of the state vector tXY represents the population totals for the auxiliary 

variables (t

X) and the remaining qY

-by-1 partition represents the population totals 

for the study variables (t

Y). Furthermore, Let the vectors tZ and tX represent the 

same remotely sensed auxiliary variables (i.e., tZ = tX), the only difference being 

that t

Z is estimated from the Phase 1 sample and tX is estimated from the Phase 2 

sample; therefore, H = [I | 0] in Eqs. 9 through 11. Furthermore, C(1) = 0 in Eqs. 9 
USDA Forest Service RMRS-GTR-239. 2010. 17

through 11 because the Phase 1 and Phase 2 samples are independent. Thus, Eqs. 

9 through 11 simplify to:

() ( ) ( )

( ) ( ) ( )

RE

RE

ˆˆ ˆ ˆ

ˆˆ ˆ ˆˆ ˆ

ˆ ˆ

XY Z XY

XY XY

XY

=+ − a o ¬ 1⁄4

= − a o ¬ 1⁄4

′ =− − a o ¬ 1⁄4

t t K t Ht

V t V t KH V t

I KH V t I KH

 (12)

where

( ) () ( )

( ) () »

»

1⁄4

o

«

«

¬

a

′ =

X Y Y

X X Y

XY

C t t V t

V t C t t

V t ˆ ˆ ˆ , ˆ ˆ

ˆ, ˆ ˆ ˆ ˆ ˆ ˆ (13)

After combining Eqs. 9 and 13, the qXY-by-qZ gain matrix of optimal weights 

equals:

() ( )

( ) () [ ] ( ) ()

( )

( ) ( [ ] () () )

() () () ( [ ] )

( ) () () ( [ ] ) »

»

»

1⁄4

o

«

«

«

¬

a

′ +

′ + =

» +

1⁄4

o «

¬

a =

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

+»

1⁄4

o

«

¬

a »

1⁄4

o

«

¬

a

»

»

1⁄4

o

«

«

¬

a

′ =

−

−

−

−

1

1

1

1

ˆ ˆ ˆ ˆ ˆ , ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ, ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ , ˆ ˆ

ˆ, ˆ ˆ ˆ ˆ

Y X X Z

X X Z

X Z

Y X

X

XY Z

X Y Y

X X Y

C t t V t V t

V t V t V t

V t V t

C t t

V t

V t

0

I

I 0 V t

0

I

C t t V t

V t C t t

K

 (14)

The equations for the matrix K in Eq. 14 a nd the covariance matrix estimator in 

Eq. 12 may be combined as follows:

() () ()

( ) ( ) ()

[ ]

1

RE

1

ˆ ˆˆ ˆ ˆˆ

ˆˆ ˆ ˆ

ˆ ˆˆ ˆ ˆ ˆ ˆ ,

X XZ

XY Z XY

YX X Z

−

−

a o ′ § · « »  ̈  ̧ a o + ¬ 1⁄4 © 1

=+ − a o ′ ¬ 1⁄4 § ·  ̈  ̧ a o + ¬ 1⁄4 © 1 ¬ 1⁄4

Vt Vt Vt

t t t I 0t

Ct t Vt Vt

 (15)

Because th e objective is to improve estimates of the study variables with the 

auxiliary variables, our ultimate interest is limited t o the partition of the vector 

estimate in Eq. 15 that corresponds to the study variables (tY).
18 USDA Forest Service RMRS-GTR-239. 2010.

RE RE

1

1

1

ˆ ˆ

ˆ ˆˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ,

ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ,

Y

X XZ

X

Z X

Y

YX X Z

Y YX X Z Z X

t 0 It

Vt Vt Vt

t 0I 0I t t

t

Ct t Vt Vt

t Ct t Vt Vt t t

 (16)

Like Eq. 16, the equations for the weighting matrix K and the covariance matrix estimator in Eq. 12 may be combined.

() ()

() () ()

( ) () ()

[ ]

() ( )

( ) ()

() ( )

( ) ()

() () ()

1

RE

1

1

ˆ ˆˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ,

ˆ ˆ ˆ ˆ

ˆ ˆˆ ˆ ˆ , ˆ ˆˆ ˆ ˆ ˆ ˆ ,

ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆˆ ,

ˆ ˆˆ ˆ ˆ ,

X XZ X XY

XY

XY Y

YX X Z

X XY X XZ

XY Y

−

−

−

a o ′ § · « »  ̈  ̧ a o + a o ¬ 1⁄4 « » © 1 « » = − ′ ′ § ·  ̈  ̧ a o + ¬ 1⁄4 « » ¬ 1⁄4 © 1 ¬ 1⁄4

§ · a o  ̈  ̧ a o + ¬ 1⁄4 « » © 1 = − ′

¬ 1⁄4

Vt Vt Vt Vt Ct t

Vt Vt I 0

Ct t Vt Ct t Vt Vt

Vt Ct t Vt Vt Vt

Ct t Vt ( ) () ()

() ( )

( )

( ) ( )

( ) ( )

( )

( ) ( )

( ) ( )

( )

( ) ( )

1

1 1

ˆ ˆ ˆˆ ˆ ,

ˆ ˆˆ ˆ ˆ ˆ ˆ ,

ˆ ˆ ˆ ˆ ˆ,

ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ,

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ,

ˆ ˆ

ˆ ˆ ˆ,

X XY

YX X Z

X X Y

X X

X XY X X

Z Z

X Y

X

Y X

−

− −

a o ′

« »

a o ′ ¬ 1⁄4 § ·  ̈  ̧ a o + ¬ 1⁄4 © 1 ¬ 1⁄4

′ § · § · a o a o  ̈  ̧  ̈  ̧ « » − « » −  ̈  ̧  ̈  ̧ « » « »  ̈  ̧ +  ̈  ̧ « » + « » ¬ 1⁄4 ¬ 1⁄4 © 1 © 1 = ′

−

+

Vt Ct t

Ct t Vt Vt

V t Ct t

V t V t

Vt Vt Vt Ct t

V t V t

Ct t

V t

Ct t

( ) ( )

( )

( ) ( )

( ) ( )

1 1

ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ , ,

ˆ ˆ ˆ ˆ

Y

X

YX XY X

Z Z

− −

a o

« »

§ · § · a o a o  ̈  ̧  ̈  ̧ « » « » −  ̈  ̧ « »  ̈  ̧ « » « »  ̈  ̧  ̈  ̧ « » + « » « » ¬ 1⁄4 ¬ 1⁄4 © 1 « » © 1 ¬ 1⁄4

V t

V t

V t Ct t Ct t

V t V t

 

 (17)
USDA Forest Service RMRS-GTR-239. 2010. 19

The partition of Eq. 17 that corresponds to the study variables is:

( ) [ ] ( )

() ( ) ( ) () ( )

RE RE

1

ˆ ˆ

ˆ ˆˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ , ,

Y

Y YX X Z XY

−

a o = « » ¬ 1⁄4

§ · § · =− + a o  ̈  ̧  ̈  ̧ ¬ 1⁄4 © 1 © 1

0

Vt 0 IVt

I

Vt Ct t Vt Vt Ct t

 (18)

The expression of the RE covariance matrix for the study va r iables in Eq. 18, 

which is a special case of RE, is algebraically identica l to the optimal calibration 

estimator. Because both estimators tXY and tZ in Eq. 8 are design-consistent, the 

weighted sum of those estimators with RRE in Eq. 10 is also design-consistent 

(Fuller 2009:117).

RRE for Complex Sampling Designs: The Basic Idea Revisited

The presentation above demonstrates the relative simplicity of RE as an optimal multivariate estimator. The next example is a Census Object produced with a 

spaceborne remote sensing protocol, a Phase 1 Object that features photo-interpretation of high-resolution sample imagery for a large sample, and a smaller Phase 

2 sample of FIA fi eld measurements. This example uses sequential application of 

the RE, which is termed the recursive RE (RRE). The following exposition is intended to explain this application further in simple, heuristic terms. Mathematical 

details follow in later sections.

Complex surveys can include two or more multiphase and/or multistage modules, among others (Magnussen 2003). An example of these modules in the forest 

health monitoring portion of FIA includes: (1) auxiliary census data from remote 

sensing and administrative records at Phase 1; (2) measurements of fi eld plots at 

FIA Phase 2; and (3) more expensive measurements of forest health on a relatively small Phase 3 sub-sample of Phase 2 fi eld plots.

RRE may be applied to multilevel sampling in a sequential recursive fashion 

(Bierman 1977) by combining two modules (e.g., H in Eq. 8) at a time until all 

modules are used in the fi nal sample survey estimate. The sequential recursive 

character of RRE is analogous to object-oriented programming. RRE combines 

two, and only two, objects at a time. An object might be a vector constant from 

a population census, a population vector estimate from a single sample phase or 

stage, or the vector output from a previously executed object. RRE receives two 

input objects. In Kalman fi lter terms, one object is the full state-vector, and the 

other is the measurement vector. The vector dimensions of each module can vary, 

but every pair of modules in an object are related through a linear transformation 

with known constants (e.g., H in Eq. 8). In the following, the linear transformation is an indicator matrix composed of zeros and ones that maps the state-vector 

onto the measurement vector. RRE optimally combines these two vectors (Eq. 

10) into a more precise estimate of the full population vector. The output vector is 

then available as one of the two input objects into the next RRE recursion.

The sequential recursive concept has precedence in more traditional approaches to estimation in complex sample surveys. (See Särndal and others (1992:148) 

for an example with three-stage sampling.) Again, using the FIA forest health 

monitoring example, RRE might start by combining the estimates from Phase 1 

and 2 objects. The outcome is more effi cient estimates of the variables measured 

at Phase 2. Then, the composite estimate for Phase 2 variables could be combined 

with the estimates based on the forest health monitoring variables that are measured with the small Phase 3 sample. The result is a second composite estimate 

that sequentially assimilates relevant information from all three phases.

If the correlations and associations are suffi ciently strong among the fi eld measurements from Phase 2 and Phase 3, then the relatively precise Phase 2 estimates 

will improve the precision from the sparse sample at Phase 3. Furthermore, if the 
20 USDA Forest Service RMRS-GTR-239. 2010.

correlations and associations are suffi ciently strong between the remotely sensed 

measurements at Phase 1 and the fi eld measurements at Phase 2, then the remotely 

sensed auxiliary data will improve the estimation precision for variables that use 

the Phase 2 protocols, which, in turn, will improve the estimation precision for 

the forest health monitoring variables collected at Phase 3. Because the RRE is 

a special case of the Kalman fi lter, the fi nal composite estimate is optimal in the 

sense of minimum variance and EBLUE, and it would be the maximum likelihood estimate if all multivariate sampling errors are Gaussian (Jazwinski 1970; 

Maybeck 1977). The fi nal composite estimates of forest health variables incorporate all gains in effi ciency available from the remotely sensed data from Phase 1 

and the fi eld data from Phase 2.

Depending on how the estimation problem is formulated, RRE can constrain 

linear combinations of estimates to equal known constants from one or more 

auxiliary sources (Czaplewski in prep.[b]). Examples of such constants include a 

census of remotely sensed pixel values that completely cover the sampled population, including the prevalence of each category or the total of each remotely 

sensed, continuous variable in the population. Other examples include census 

statistics on population totals from offi cial administrative records such as areal 

extent of geopolitical units (e.g., counties and National Forests) and large water 

bodies (e.g., “Census water”). This feature qualifi es RRE as a calibration estimator 

(Estevao and Särndal 2004; Särndal 2007). Furthermore, RRE can directly apply 

inequality constraints (Doran 1997; Simon and Chia 2002) such as a population 

total that must be greater than or equal to zero, namely, non-negative. In addition, 

depending on how the estimation problem is formulated, RRE can combine auxiliary statistical estimates of unknown population constants from multiple phases 

and/or stages of probability sampling. If estimators for multivariate, multilevel 

sampling can be recast into the structure of a recursive Kalman fi lter, then the 50 

years of theory and practice with the Kalman fi lter can be immediately transferred 

to estimation in complex sample survey designs.

Next, specifi c designs, which are not intended to be exhaustive, are considered 

in more detail. Table 1 is a short guide to the mathematical notation, which is 

somewhat deep in order to consistently cover diverse elements within a variety 

of complex sampling designs. These examples are followed by other applications 

that use the fi nal results from RRE to make other types of estimates, such as smallarea, synthetic estimators and model-based estimators for missing data. Some of 

these examples use non-linear transformations of population vector estimates, 

and multivariate Taylor series are used to estimate their covariance matrices. A 

recursive sequential object-oriented approach simplifi es fi rst- and second-order 

Taylor series approximations for transformations that use addition, subtraction, 

multiplication, and division operators.

Two-Phase Sampling for Continuous Variables with a 

Sample at Phase 1

This section recasts the generalized presentation of RRE (Eqs. 9 through 11) 

into an example that is more explicitly relevant to multilevel sample surveys, 

namely, a special case of two-phase sampling, in which: (1) a simple random 

sample of the Phase 1 object is measured with a multivariate, remotely sensed 

protocol (e.g., photo-interpretation of sample aerial photography, denoted M = 1); 

and (2) a second independent, simple random sub-sample of the Phase 1 object 

(i.e., Phase 2 sample) that is further measured with a multivariate fi eld protocol, denoted M = 2, in addition to the multivariate photo-interpretation protocol 

(M = 1). Also, this section provides more specifi c details from the perspective of 

probability sampling (e.g., Särndal and others 1992; Knottnerus 2003; de Gruijter 

and others 2006).
USDA Forest Service RMRS-GTR-239. 2010. 21

Every i

th and jth sampling unit from the Phase 1 or Phase 2 sample in this example is considered an infi nitesimally small point, and both protocols (M = 1 and 

M = 2) measure each sample point from a small point support region in the immediate vicinity of that sample point. This uses the perspective of de Gruijter and 

others (2006:74-75) for sampling continuous populations in space, such as forests, in which the “target universe ... (is) an infi nitely large population of possible 

sampling locations, and the distinction between sampling with replacement and 

sampling without replacement is immaterial. ... (Any) correlations between observations (caused by non-replacement) are negligible in this context, as are fi nite 

Symbol 

Matrix 

dimensions Description 

Row Column

[ ] M=a j y 1 qM=a

Vector of qM=a variables measured with 

protocol M = a in PSU j

j

»

1⁄4

o «

¬

a

=

=

M b

M a

y

y

1 qM=a+

qM=b

Vector of qM=a variables measured with 

protocol M = a and qM=b variables 

measured with protocol M = b in PSU j

[ ]

[ ]

[ ]

[ ] j i

i

j

n

i

i j

a i

n

j

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

=

= =

= =

=

= =

= =

¦

M c 1

M b 1

1

M bC M b ,

M 1

ˆ

y

y

y y

y

1 

qM=a+

2qM=b+

qM=c

Vector of qM=a variables measured with 

protocol M = a at SSU = 1 in PSU = j, and 

the vector mean of qM=b variables 

measured with protocol M = b at all nj

SSUs in cluster plot PSU = j, and qM=b

variables measured with protocol M=b at 

SSU = 1 in PSU = j, and qM=c variables 

measured with protocol M = c at SSU = 1 

in PSU = j

= s=d

=

»

1⁄4

o «

¬

a

M b

M a

ˆ

ˆ

t

t

1 qM=a+

qM=b

Design-based sample estimate for the 

vector of population totals for the qM=a

variables measured with protocol M = a, 

and the qM=b variables measured with 

protocol M = b from sample s = d

[] [ ]

[ ] [] »

»

1⁄4

o

«

«

¬

a

» =

1⁄4

o «

¬

a

= = = = =

= = = = =

= =

=

s d s d

s d s d

s d

M b M a M b

M a M a M b

M b

M a

ˆ ˆ ˆ , ˆ ˆ

ˆ, ˆ ˆ ˆ ˆ

ˆ

ˆ ˆ

C t t V t

V t C t t

t

t

V

qM=a+

qM=b

qM=a+

qM=b

Design-based sample estimate for the 

covariance matrix for the vector of 

population totals for the qM=a variables 

measured with protocol M = a, and the 

qM=b variables measured with protocol M = 

b from sample s = d, where 

[ ] M=b M=a s=d ˆ, ˆ ˆ C t t is the qM=b-by-qM=a crosscovariance matrix between estimates for 

variables measured with the M = a and M 

= b protocols 

M b s { } d ,e

M a

ˆ

ˆ

= =

=

»

1⁄4

o «

¬

a

t

t

1 qM=a+

qM=b

RRE estimate for the vector of population 

totals for the qM=a variables measured with 

protocol M = a, and the qM=b variables 

measured with protocol M = b from the 

combination of estimates from samples s 

= d and s = e

Table 1. Abbreviated summary of notation.
22 USDA Forest Service RMRS-GTR-239. 2010.

population corrections.” The infi nite-population perspective is simpler (Mandallaz 

2008:61) and arguably more natural for continuous spatial populations. FIA shares 

this same perspective (Scott and others 2005). It is this independent and invariant 

perspective of de Gruijter and others that justifi es treatment of the large Phase 1 

sample as independent of the smaller Phase 2 sample. This differs from the perspective in fi nite population sampling, in which the Phase 2 sample is considered 

a probability sub-sample of the Phase 1 sample, which is not independent and not 

necessarily invariant (e.g., Särndal and others 1992:Chapter 9).

Phase 1

Phase 1 is a simple random point sample (s = 1) from a spatial population 

with an infi nite number of points but of known area A. The sample size (ns=1) is 

large relative to Phase 2 (below). A small support-region (de Gruijter and others 

2006:74-75) for each sample point i is measured with the Phase 1 protocol (e.g., 

photo-interpretation), which is denoted with the M = 1 subscript. The multivariate 

sample survey estimates are denoted by the qM=1-by-1 vectors and qM=1-by-qM=1

covariance matrix:

[ ] [ ]

( )

[ ] [ ]

( ) [ ] ( ) [ ] [ ] ( ) [ ] [ ]

( )

( ) »

»

»

»

1⁄4

o

«

«

«

«

¬

a

−

′ − −

=

=

=

= =

∈ =

= = = = = =

= =

= = = =

∈ =

=

=

= =

¦

¦

1 1

2

M 1 M 1 1 M 1 M 1 1

2

M 1 1

M 1 1 M 1 1

1

M 1

1

M 1 1

1

ˆ ˆ

ˆ ˆ

ˆ ˆ

1 ˆ

s s

j s

j s j s

s

s s

j s

j

s

s

n n

A

A

n

y y y y

V t

t y

y y

 (19)

where the constant A in Eq. 19 is the known size of the sampled population (e.g., 

25 million acres), the qM=1-by-1 vector [y] represents qM=1 photo-interpreted variables, and the qM=1-by-1 vector [t] represents the corresponding qM=1 population 

totals.

For example, photo-interpreters inexpensively apply the M = 1 measurement 

protocol to a 1-acre support-region for each point in the large Phase 1 sample. The 

support-region covers an area approximately equivalent in size to one sub-plot 

in the four-point cluster plot used FIA fi eld measurements (Bechtold and Scott 

2005). The support-region is centered on the sample point. The number of photointerpreted variables may be numerous (e.g., 5<qM=1<1000), and they may be a 

mixture of both binary and continuous measurements.

Different remotely sensed categorical variables need not be cross-classifi ed into 

a single categorical variable as with stratifi cation (see “Categorical Variables”). 

Polychotomous categorical variables with k categories (k>2) are converted into 

(k) binary variables (e.g., de Gruijter and others 2006:69). Furthermore, binary 

variables need not be mutually exclusive or exhaustive. In addition, if the support-region straddles multiple categorical conditions, then the corresponding 0-1 

binary variables easily become continuous variables (bounded by 0 and 1) that 

represent the proportion of the support-region included in each binary category.

Phase 2

Phase 2 is an independent simple random sample (s = 2) that is measured with 

both the inexpensive M = 1 protocol (remote sensing) and the expensive M = 2 

protocol (fi eld measurements). The qM=2-by-1 vector [yM=2]i in Eq. 20 represents 

these qM=2 variables. The Phase 2 sample size (ns=2) is small relative to the Phase 

1 sample (ns=2<<ns=1).
USDA Forest Service RMRS-GTR-239. 2010. 23

Let the Phase 2 estimates be the (qM=1 + qM=2)-by-1 vectors and covariance 

matrix (qM=1 + qM=2)-by-(qM=1 + qM=2):

 

( ) [ ]

( )

( )

M 2 2

M 1

M 2 2

M 1

2 1

2 M 2

M 1

2

M 2 2

M 1

2 M 1,2

ˆ

ˆ

ˆ

ˆ

, where 1

ˆ

ˆ ˆ

= =

=

= =

=

= =

∈ = =

=

=

= =

=

= =

»

1⁄4

o «

¬

a » =

1⁄4

o «

¬

a

<< »

1⁄4

o «

¬

a =

»

1⁄4

o «

¬

a =

¦

s s

s s

j s s j

s

s

A

n n

n

y

y

t

t

y

y

y

y y

( ) [ ] ( ) [ ]

( )

( )

( ) [ ] ( ) [ ][ ]

( ) [ ][ ] ( ) [ ] »

1⁄4

o «

¬

a =

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

¬

a

−

°¿

°

3⁄4

1⁄2

° ̄

°

®

­

»

1⁄4

o «

¬

a ′ − ′ −  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a − »

1⁄4

o «

¬

a

=  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

= = = = =

= = = = =

= =

∈ =

= = = =

= =

=

=

=

= =

=

¦

M 2 M 1 2 M 2 2

M 1 2 M 1 M 2 2

2 2

2

M 1 M 1 M 2 M 2

M 2 2

M 1

M 2

M 1

2

2 M 2

M 1

ˆ ˆ ˆ , ˆ ˆ

ˆ , ˆ ˆ ˆ ˆ

1

ˆ ˆ ˆ

ˆ

ˆ

ˆ ˆ

s s

s s

s s

j s

j j

j s

s n n

A

C t t V t

V t C t t

y y y y

y

y

y

y

t

t V

 (20)

The last term in Eq. 20 defi nes a partition of the sample covariance matrix. 

C in the partitioned covariance matrix denotes the qM=2-by-qM=1 cross - covariance 

sub-matrix between the qM=2 estimated populations totals measured with the 

M = 2 protocol and the qM=1 variables measured with the M = 1 protocol from the 

Phase 2 sample.

The M = 2 protocol represents the standard FIA Phase 2 fi eld protocol, which 

results in standardized measurements of tree and stand characteristics within the 

1/6-acre support-region. The support-region in the Phase 2 sample is centered on 

sub-plot 1 in the standard FIA fi eld plot, and the support-region is composed of 

all four 1/24-acre FIA sub-plots (Bechtold and Scott 2005:29). The M = 1 protocol (i.e., photo-interpretation) used in the Phase 2 sample is exactly as described 

above under Phase 1, including the 1-acre support-region approximately centered 

on the same sample point used in the M = 2 protocol. As will be demonstrated, 

improvements in statistical effi ciency with two-phase sampling depend upon 

the degree of covariance between variables measured with expensive and lessexpensive protocols (e.g., fi eld data and remote sensing). Therefore, it is important that the M = 1 protocol be applied to the same site as the M = 2 protocol, 

which requires minimization of registration errors between remotely sensed measurements and fi eld measurements.

Recursive Restriction Estimator (RRE)

The Recursive Restriction Estimator (RRE) offers one means to combine the 

independent estimators from Phase 1 (Eq. 19) and Phase 2 (Eq. 20) into a more 

effi cient estimator. This is accomplished by structuring the estimation problem 

in the context of the static multivariate Kalman fi lter. This is the key concept 

in simplifying the estimation problem in complex sample designs. Because the 

statistical properties of the Kalman fi lter are well understood, at least in certain 

disciplines, and because the fi lter is fully developed as an optimal linear estimator, 

the sample survey problem is largely solved by recasting it as a discrete multivariate stochastic process (Särndal 1992:21) using the Kalman fi lter as a static 
24 USDA Forest Service RMRS-GTR-239. 2010.

linear model (Maybeck 1979: Section 3.11) and then applying well-developed 

results for the Kalman fi lter to readily solve the problem. Much of that recasting 

has already been performed as an antecedent to Eqs. 9 through 12. The remaining 

steps follow.

First, the Phase 1 and Phase 2 samples are independent by defi nition; hence, 

C(1) = 0, and the special case of the Kalman fi lter in Eq. 12 applies.

Second, consider state-vector in the Kalman fi lter. In the case of two-phase 

sampling, state-vector may be defi ned as the population totals for all variables of 

interest from both the M = 1 protocol (photo-interpretation) and the M = 2 protocol (FIA fi eld data). In this case, n in Maybeck’s defi nition of the state-vector 

equals (qM=1 + qM=2). In the context of the current section, the state-vector tX(1) in 

Eq. 12 is estimated from Phase 2 sample (s = 2) and the estimators in Eq. 20.

Third, defi ne the measurement vector tZ(1) in Eq. 12 as the estimated population 

totals for the remotely sensed variables from the Phase 1 sample (s = 1), with the 

corresponding estimators given in Eq. 19.

Fourth, a linear model must be defi ned that describes how the measurement 

vector t

Z(1) relates to the state-vector tX(1) in Eq. 12. This model is given in Eq. 

8 as tZ(1) = H tX(1), where H is a conformable matrix of known constants. In the 

current context, H is simply a matrix of ones and zeros that extracts the Phase 2 

estimated totals of the photo-interrelated variables (M = 1 protocol) from the full 

state-vector and its corresponding partition from the covariance matrix for the full 

state-vector.

In summary, these steps yield a multivariate estimator for the special case of 

two-phase sampling considered here that is structured as the Kalman fi lter in 

Eq. 12:

( ) ( ) ( )

() () ( ) ( )

[ ] ( ) () ()

M1 M1

1 1

M2 M2 2 2

1 1 M1 M1 1 1

M 1

1 1 M 1 1

M 2 2

M 1

M 1 2

M 2 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆˆ ˆ , , ˆ

ˆ ˆ ˆ

X X

s s

Z Z s s

X Z s

s

s

s

= =

= = = =

= = = =

=

= = = =

=

= = = =

ao ao § ·

= = «» «»  ̈  ̧  ̈  ̧ ¬1⁄4 ¬1⁄4 © 1

= = ao ao ¬1⁄4 ¬1⁄4

§ · a o = ==  ̈  ̧ « » a o ¬ 1⁄4 © 1 « » ¬ 1⁄4

a o a o = « » ¬ 1⁄4 « » ¬ 1⁄4

=

t t

t Vt V

t t

t t Vt V t

t

H I 0 Ct t C t 0

t

t

t H

t

[ ]

( )

[ ]

M 1

M 1 2

M 2 2

M 1 M 1

M 2 M 2 2 2

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

s

s

s s

=

= = = =

= =

= = = =

§ · a o a o =  ̈  ̧ « » ′ ¬ 1⁄4  ̈  ̧ © 1 « » ¬ 1⁄4

ao § · a o a o « » =  ̈  ̧ « » « » « »  ̈  ̧ ¬ 1⁄4 « » ¬ 1⁄4 ¬ 1⁄4 © 1

t

V t HV H

t

t t I

I 0 I 0V t t 0

 (21)
USDA Forest Service RMRS-GTR-239. 2010. 25

Here, the H matrix has dimensions qM=1-by-(qM=1 + qM=2). The cross-covariance 

matrix in Eq. 21 because the Phase 1 sample is, by design in this exam ple, independent of the Phase 2 sample. From Eqs. 12, 19, 20, and 21, RRE (Kalman fi lter) 

that combines estimates from the Phase 1 (s = 1) and Phase 2 (s = 2) samples, 

denoted s = {1,2}, is:

{ }

[ ]

{ } [] [] M 1 1 M 1 2

M 2 2

M 1

M 2 2

M 1

M 1 1

M 2 2

M 1

M 2 1,2

M 1

ˆ ˆ ˆ

ˆ

ˆ

ˆ ˆ ˆ

ˆ

ˆ

ˆ

= = = =

= =

=

= =

=

= =

= =

=

= =

=

» + −

1⁄4

o «

¬

a =

°¿

°

3⁄4

1⁄2

° ̄

°

®

­

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a » + −

1⁄4

o «

¬

a » =

1⁄4

o «

¬

a

s s

s

s

s

s s

K t t

t

t

t

t

K t H

t

t

t

t

 (22)

Equation 22 is a multivariate extension to the familiar univariate composite 

estimator in Eq. 2. In the current context, and using the partitions of t he Phase 2 

covariance matrix in Eq. 20, the (qM=1 + qM=2)-by-qM=1 weighting matrix K (Eqs. 

12 and 22) may be expressed as:

( ) [ ]

( ) [ ] { } ( ) [ ] ( ) [ ]

( ) [ ][ ] { } ( ) [ ] ( ) [ ] »

»

1⁄4

o

«

«

¬

a

+

+ =

°¿

°

3⁄4

1⁄2

° ̄

°

®

­ ′ +  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

°¿

°

3⁄4

1⁄2

° ̄

°

®

­ ′  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a =

−

= = = = = = =

−

= = = = = =

−

= =

= =

=

= =

=

1

M 2 M 1 2 M 1 2 M 1 1

1

M 1 2 M 1 2 M 1 1

1

M 1 1

M 2 2

M 1

M 2 2

M 1

ˆ ˆ ˆ ˆ ˆ , ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆˆ ˆ

s s s

s s s

s

s s

C t t V t V t

V t V t V t

H V t

t

t H H V

t

t K V

 (23)

Equation 23 is the multivariate extension of the scalar weight in the univariate 

composite estimator (Eqs. 3 and 4). Equation 23 might not be numerically stable, 

and it is very important to use more numerically stable methods that are reviewed 

in the section “Robust Numerics.”

The partition of the covariance matrix between the Phase 1 and Phase 2 estimators used in Eq. 23 is defi ned in Eq. 20. Combining Eqs. 22 and 23:

{ }

[] [] ( ) ( ) [ ]

( ) [ ] { } [] []

[ ] [ ][ ] ( ) ( ) [ ]

( ) [ ] { } [] [] »

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

«

¬

a

−

°

¿

°

3⁄4

1⁄2

°

 ̄

°

®

­

+

+

−

°

¿

°

3⁄4

1⁄2

°

 ̄

°

®

­

+

+

» =

1⁄4

o «

¬

a

= = = =

−

= =

= =

= = = = =

= = = =

−

= =

= =

= = = =

= =

=

M 1 1 M 1 2

1

M 1 1

M 1 2

M 2 2 M 2 M 1 2

M 1 1 M 1 2

1

M 1 1

M 1 2

M 1 2 M 1 2

M 2 1,2

M 1

ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ , ˆ ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ

ˆ

ˆ

s s

s

s

s s

s s

s

s

s s

s

t t

V t

V t

t C t t

t t

V t

V t

t V t

t

t

 (24)

and the covariance matrix estimator for the vector estimate in Eq. 24 is:
26 USDA Forest Service RMRS-GTR-239. 2010.

RRE in Eqs. 22 to 25 improves estimates for remotely sensed variables (protocol M = 1) in addition to the variables measured in the fi eld (protocol M = 2). 

However, the typical purpose of the Phase 1 sample is variance reduction for the 

population estimates from the Phase 2 sample and the M = 2 measurement protocol. The estimates of variables measured with the M = 1 protocol might not be 

relevant after their role in variance reduction is met. Only those sub-matrices in 

Eqs. 20 through 25 that apply to the Phase 2 protocol (M = 2) might be needed. 

Those qM=2-dimensional partitions are readily extracted from the vector of estimated population totals and its estimated covariance matrix as follows:

[ ] { } [ ]

{ }

[ ] [ ][ ] ( ) ( ) [ ]

( ) [ ] { } [] []

»

»

»

1⁄4

o

«

«

«

¬

a

−

°

¿

°

3⁄4

1⁄2

°

 ̄

°

®

­

+

= +

»

1⁄4

o «

¬

a =

= = = =

−

= =

= =

= = = = =

= =

=

= =

M 1 1 M 1 2

1

M 1 1

M 1 2

M 2 2 M 2 M 1 2

M 2 1,2

M 1

M 2 1,2

ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ , ˆ ˆ ˆ

ˆ

ˆ ˆ

s s

s

s

s s

s

s

t t

V t

V t

t C t t

t

t

t 0 I

[ ] { } ( ) [ ]

{ }

( ) [ ] ( ) [ ][ ] ( ) [ ]

( ) [ ] ( ) [ ] [ ]

»

»

»

1⁄4

o

«

«

«

¬

a ′

°

¿

°

3⁄4

1⁄2

°

 ̄

°

®

­

+

= −

»

1⁄4

o

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a =

= = =

−

= =

= =

= = = = =

= =

=

= =

M 2 M 1 2

1

M 1 1

M 1 2

M 2 2 M 2 M 1 2

M 2 1,2

M 1

M 2 1,2

ˆ , ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ , ˆ ˆ ˆ ˆ

ˆ

ˆ ˆ ˆ ˆ

s

s

s

s s

s

s

C t t

V t

V t

V t C t t

I

0

t

t V t 0 I V

 (26)

Because the regression estimator and RRE are both minimum variance linear 

predictors, it is no surprise that Eq. 26 has a multivariate structure somewhat 

{ }

( )

( )

M1 M1 M2

M1 M1 M1

M2 M2 M2 1,2 2 2

M 1

M 1 2

M 2 2

ˆˆ ˆ ˆˆ ˆ ˆˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ

ˆ ˆ

q qq

ss s

s

s

= ==

== =

×

== = == =

=

= = = =

=

= =

§ · ao ao ao §· §·  ̈  ̧ «» «» «» = −  ̈ ̧  ̈ ̧ a o ¬ 1⁄4  ̈  ̧  ̈ ̧  ̈ ̧ © 1 ¬1⁄4 ¬1⁄4 ¬1⁄4 ©1 ©1

§ · a o a o = −  ̈  ̧ « » a o « » ¬ 1⁄4  ̈  ̧ ¬ 1⁄4 © 1 « » ¬ 1⁄4

a o ¬ a o − ¬ 1⁄4

=

M 1

M 1 s 2

tt t

V V KI 0 V

tt t

t

V KV t C

t

V t

Vt I

( )

( ) () () ( )( )

( )

( ) ( )

ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ

− −

= = =

= = = =

= = = =

−

= =

= =

= =

a oa o ­1⁄2 ­1⁄2 a o « »« » °° °° 1⁄4 ¬1⁄4

®3⁄4 ®3⁄4 ao ao − ¬1⁄4 ¬1⁄4 °° °° + + ao ao  ̄¿  ̄¿ ¬1⁄4 ¬1⁄4 ¬ 1⁄4¬ 1⁄4

a o ­ 1⁄2 a o « » ° ° ¬ 1⁄4 ′ − a o ® 3⁄4 ¬ 1⁄4 ° ° + a o  ̄ ¿ ¬ 1⁄4 ¬ 1⁄4

1 1

M 1 s 2 s 2

M1 M1 s2 s2

M 1 M 1 s 1 s 1

1

M 1 s 2

M 1 s 2

M 1 s 1

V t

Vt IVt C

Vt Vt

V t

CI V t

V t

( ) ( )

( )

( ) M1 M2 2

ˆ ˆ

ˆ ˆ ˆ ˆ

ˆ ˆ

where , ˆ ˆ ˆ ˆ s

−

= =

= =

= =

= ==

a o « » « » « » « » « » « » a o ­ 1⁄2 a o « » « » ° ° ¬ 1⁄4 « a o − ′ » « » ® 3⁄4 ¬ 1⁄4 « » ° ° + a o « ¬ 1⁄4 » « » ¬ 1⁄4  ̄ ¿ «¬ »1⁄4

= a oa o ¬ 1⁄4¬ 1⁄4

1

M 1 s 2

M2 s2

M 1 s 1

V t

Vt C C

V t

CC t t

(25)
USDA Forest Service RMRS-GTR-239. 2010. 27

analogous to the univariate regression estimator (e.g., Cochran 1977:Section 7.3). 

Knottnerus (2003:356) developed the same estimator from the perspective of the 

conditional restriction estimator.

The effi ciency of two-phase sampling depends on the linear relationship between the variables measured with the M = 1 protocol (e.g., photo-interpretation) 

and the variables measured with the M = 2 protocol (e.g., FIA fi eld data). The 

strength of this relationship is quantifi ed from the Phase 2 sample estimate of the 

cross-covariance matrix C in Eq. 26. If there is little or no correlation between the 

M = 1 and M = 2 variables, then C~0, and the composite estimate of the M = 2 

variables in Eq. 26 equals that of the Phase 2 sample (s = 2) alone (Eq. 20). This 

means that the Phase 1 sample of M = 1 variables would produce little or no gain 

in statistical effi ciency for estimating the M = 2 variables. However, if the crosscovariance matrix C is relatively large, and the covariance matrix for population 

totals from the M = 1 measurements in Phase 1 (s = 1) are relatively small, then 

the composite variance in Eq. 24 can be substantially less than the covariance 

matrix for the Phase 2 sample alone (Eq. 20).

The next example is a minor modifi cation of the above example. Rather than 

a sample of points at Phase 1, the next example uses a full-coverage census of 

remotely sensed pixels at Phase 1.

Two-Phase Sampling for Continuous Variables with a 

Census at Phase 1

Multivariate two-phase sampling with a sample at Phase 1 (previously discussed) is clos ely related to the two-phase sampling that uses a census at Phase 1, 

which is the subject of this section. The following example is an alternative to the 

situation in FIA, where remotely sensed Landsat data are used with a post-stratifi -

cation estimator. Czaplewski (in prep.[b]) considers this example in substantially 

more detail and includes a discussion on the equivalence of RRE to direct designbased calibration estimators in sample surveys (Estevao and Särndal 2004) that 

are nearly design-unbiased (Särndal 2007).

In order to conform to notation in more complex examples that follow, let the 

census of pixels be denoted s = U, indicating that the sample is the complete set 

of pixels that completely cover the sampled population U. The remote sensing 

protocol used to measure each pixel is denoted by M = 0. If Phase 1 in the previous example was a census rather than a sample, then [ ] tM s1 = = 1 t would exactly 

equal tM=1, and the covariance matrix for Phase 1 would be the qM=1-by-qM=1 zero 

matrix (i.e., V([ ] ) t

t

M s = = 1 1 = 0). In the following example, the Phase 1 sample 

s = 1 is replaced by the s = U census, and the M = 1 photo-interpretation protocol 

is replaced by the M = 0 spaceborne sensor protocol for all pixels in the sampled 

population. Therefore, [ ] , ([ ] ) t tt t t M sU M M sU == = == 0 00 = = V 0, and Eqs. 22 

to 25 evolve into:

{ }

[ ]

{ }

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a −

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a = ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

°¿

°

3⁄4

1⁄2

° ̄

°

®

­

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a » + −

1⁄4

o «

¬

a » =

1⁄4

o «

¬

a

= =

=

= =

=

= =

=

= =

=

= =

= =

=

= =

=

M 2 2

M 0

M 2 2

M 0

M 2 ,2

M 0

M 2 2

M 0

M 0

M 2 2

M 0

M 2 ,2

M 0

ˆ

ˆ ˆ ˆˆ ˆ ˆ ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

s U s s

s

s U

s U s

t

t

K H V

t

t

V

t

t

V

t

t

K t H

t

t

t

t

 (27)

where
28 USDA Forest Service RMRS-GTR-239. 2010.

( ) [ ]

( ) [ ][ ] ( ) [ ]

( ) [ ][ ] ( ) [ ] »

»

1⁄4

o

«

«

¬

a

=

»

»

1⁄4

o

«

«

¬

a

=

°¿

°

3⁄4

1⁄2

° ̄

°

®

­ ′  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

°¿

°

3⁄4

1⁄2

° ̄

°

®

­ ′  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a =

=

−

= = = = =

−

= =

= = =

= =

−

= =

=

= =

=

1

M 2 M 0 2 M 0 2

1

M 0 2

M 2 M 0 2

M 0 2

1

M 2 2

M 0

M 2 2

M 0

ˆ ˆ ˆ , ˆ ˆ

ˆ ˆ ˆ , ˆ ˆ

ˆ ˆ

ˆ

ˆ ˆ ˆˆ ˆ

s s

s

s

s

s s

C t t V t

I

V t

C t t

V t

H

t

t H H V

t

t K V

H [I 0]

The Phase 1 census in Eq. 27 is denoted by s = U, and the measurement protocol 

used for the full-coverage satellite data is denoted by M = 0 rather than the Phase 

1 protocol M = 1. This minor shift in nomenclature for the “M” protocol facilitates 

more complex estimators that follow. It is assumed that V t ([ ] ) M s2 = = 0 t is full 

rank, i.e., the matrix inverse is feasible for the partition of the sample covariance 

matrix that corresponds to the census variables. Knottnerus (2003:330, 333) uses 

this same approach, which is called a regression model with linear restrictions on 

the regression coeffi cients, in the Restriction Estimator.

The partition of RE corresponding to the auxiliary census will exactly equal 

the vector census constant [tM=0]s=U:

{ } ( ) [ ][ ] ( ) [ ] [ ]

[ ] [ ] [ ]

[ ] [ ][ ] [ ] ( ) ( ) [ ] ( ) [ ] [ ]

[ ]

[ ] [ ][ ] [ ] ( ) ( ) [ ] ( ) [ ] [ ] »

»

1⁄4

o

«

«

¬

a

+ − =

»

»

1⁄4

o

«

«

¬

a

+ −

+ − =

°¿

°

3⁄4

1⁄2

° ̄

°

®

­

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a − »

»

1⁄4

o

«

«

¬

a

» +

1⁄4

o «

¬

a » =

1⁄4

o «

¬

a

= = = =

−

= = = = = = =

= =

= = = =

−

= = = = = = =

= = = = = =

= =

=

− = =

= = = = = = =

=

= =

=

M 0 M 0 2

1

M 2 2 M 2 M 0 2 M 0 2

M 0

M 0 M 0 2

1

M 2 2 M 2 M 0 2 M 0 2

M 0 2 M 0 M 0 2

M 2 2

M 0

1 M 0

2 M 2 M 0 2 M 0 2 M 2

M 0

M 2 ,2

M 0

ˆ ˆ ˆ ˆ , ˆ ˆ ˆ

ˆ ˆ ˆ ˆ , ˆ ˆ ˆ

ˆ ˆ

ˆ

ˆ

ˆ ˆ ˆ , ˆ ˆ ˆ

ˆ

ˆ

ˆ

s s s s U s

s U

s s s s U s

s s U s

s

s U

s U s s s

t C t t V t t t

t

t C t t V t t t

t t t

t

t

t H

C t t V t

I

t

t

t

t

 (28)

Likewise, the corresponding partition of the RE covariance matrix equals zero, 

with zero cross-covariance for the off-diagonal partition.
USDA Forest Service RMRS-GTR-239. 2010. 29

Because it is the s tudy variables that are of primary interest and not the auxiliary 

variables, the partitions of the RE for the study variables are extracted from Eqs. 

28 and 29 as:

[ ] { } [ ] [ ][ ] [ ( ) ( ( ) [ ] )]( ) [ ] [ ]

[ ] { } ( ) ( ) [ ] ( ) [ ][ ] ( ( ) [ ] ) ( ) [ ][ ] 2 M 2 M 0

1

M 2 ,2 M 2 2 M 2 M 0 2 M 0 2

M 0 M 0 2

1

M 2 ,2 M 2 2 M 2 M 0 2 M 0 2

ˆ , ˆ ˆ ˆ ˆ ˆ , ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ , ˆ ˆ ˆ ˆ

= = =

−

= = = = = = = = =

= = = =

−

= = = = = = = = =

′ = −

= + −

s s U s s s

s U s s s s U s

V t V t C t t V t C t t

t t C t t V t t t

 (30)

Simon and Chia (2002) prove that the RE in Eqs. 27, 28, and 30 satisfi es the vector 

constraint [] [] t tt M sU M M s == = == 0 { , 2} 0 2 = ! 0 t t ; is unbiased (i.e., E[ ] tM = 2 t s 

= {U, 2} = tM = 2); has a smaller error covariance than the unconstrained sample 

survey vector estimator (e.g., trace ([ ] ) V t t t

M s U2 = = 2 {,} # trace ([ ] ) V tM = 2 s 2 = t t ; 

and is the BLUE. Czaplewski (in prep.[b]) further demonstrates that the diagonal 

partition of the covariance matrix for the variables in Eq. 16 is R = 0, and the offdiagonal cross-correlation matrix partition between the study variables and the 

auxiliary census variables is exactly zero, i.e., Ct t 0 t ([ ],[ ]) t t M M sU = == 2 0 ( ,2) = . 

In effect, the auxiliary census vector acts as a constraint on RE. Exact agreement 

with the census vector is imposed by the RE without post-stratifi cation. The estimated covariance matrix in Eq. 30 is unbiased (conditional upon the sample), 

and it is not an approximation, as with post-stratifi cation (e.g., Cochran 1977:135; 

Särndal and others 1992:266; Scott and others 2005).

The RE partition in Eq. 30 contains all the relevant information available in 

the auxiliary census variables, and the information provided by those census variables is unnecessary for any subsequent recursions with the RRE. In other words, 

the RE has fi ltered out all relevant information in these census auxiliary variables. 

As discussed in the previous section, the effi ciency of this sampling design and 

its estimator (Eq. 30) remains dependent upon the correlations and associations 

between the remotely sensed satellite measurements (M = 0) and the FIA fi eld 

measurements (M = 2).

Recall that the covariance matrix V t ([ ] ) M s2 = = 0 t t in Eq. 27 is assumed 

to be full rank. However, because the sum of all elements in vector estimate 

1[ ] t

t

M s = = 0 2 = A, its covariance matrix is positive-semidefi nite and not full 

rank. At least one of the census constraints is redundant.

{ } ( ) [ ][ ] ( ) [ ]

( ) [ ]

( ) [ ][ ]

( ) [ ]

( ) [ ]

( ) [ ][ ]

( ) [ ][ ]

( ) [ ][ ]

( ) [ ][ ]

( ) [ ]

( ) [ ][ ] ( ( ) [ ] ) ( ) [ ][ ]

( ) [ ] ( ) [ ][ ] ( ( ) [ ] ) ( ) [ ][ ] »

»

1⁄4

o

«

«

¬

a

′ −

=

»

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

«

¬

a

′ − −

′ − ′ − =

»

»

1⁄4

o

«

«

¬

a

»

»

1⁄4

o

«

«

¬

a

− ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a = ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

= = =

−

= = = = = = =

= = =

−

= = = = =

= =

= = =

= = =

= = =

= = =

= =

= =

= = =

= =

−

= = = = = = =

=

= =

=

2 M 2 M 0

1

M 2 2 M 2 M 0 2 M 0 2

2 M 2 M 0

1

M 2 M 0 2 M 0 2

M 2 2

M 2 M 0 2

M 2 M 0 2

2 M 2 M 0

M 0 M 2 2

M 0 2

M 0 2

M 2 M 0 2

M 0 2

1

2 M 2 M 0 2 M 0 2 M 2

M 0

M 2 ,2

M 0

ˆ , ˆ ˆ ˆ ˆ ˆ , ˆ ˆ ˆ ˆ

ˆ , ˆ ˆ ˆ ˆ ˆ , ˆ ˆ

ˆ ˆ

ˆ , ˆ ˆ

ˆ , ˆ ˆ

ˆ , ˆ ˆ

ˆ , ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ , ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ , ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

s s s s

s s s

s

s

s

s

s

s

s

s

s

s U s s s

0 V t C t t V t C t t

0 0

C t t V t C t t

V t

C t t

C t t

C t t

C t t

V t

V t

C t t

V t

C t t V t

I

t

t

V

t

t

V

(29)
30 USDA Forest Service RMRS-GTR-239. 2010.

Numerically Stable Algorithm

Equation 27 is infeasible because the inverse of the covariance matrix 

V t ([ ] ) M s2 = = 0 t t is not full rank whenever the auxiliary vector contains proportions or totals for a dichotomous transformation of a polychotomous categorical 

variable. In more general terms, the Kalman fi lter is notoriously vulnerable to 

numerical round-off errors as the measurement error covariance matrix becomes 

small, especially as recursions accumulate. The challenges posed by numerical 

errors to optimal estimation, including RRE, and practical solutions are discussed 

in the section “Robust Numerics”.

In the special case of a measurement vector of known census constants, as in 

this section, a slightly different algorithm applies a vector of equality constraints 

that is numerically robust, even with a positive-semidefi nite covariance matrix 

(Simon and Chia 2002). Because the constraints are a vector of known census 

constants tM=0, i.e., V(tM=0) = 0, they are mutually independent, and each i

th scalar 

census value (t

M=0)i

 of the qM=0-by-1 constraint vector tM=0 may be applied separately in a sequential recursive procedure.

Let hi be the 1-by-(qM=0 + qM=2) vector containing row i of the qM=0-by-(qM=0 +

qM=2) measurement matrix H in Eq. 27. The estimator that sequentially applies all 

qM=0 constraints, i = {1,2,..., qM=0}, is defi ned as:

{ } { }

( )

{ } 1 M 2 ,2

M 0

M 0

1 M 2 ,2

M 0

M 2 ,2

M 0

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

− = =

=

=

− = =

=

= =

=

§ ·

»

»

1⁄4

o

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

+ −  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a =  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

i s U

i i i

i s U i s U

t

t

t

k h

t

t

t

t

 (31)

where

{ } M 2 2

M 0

0 M 2 ,2

M 0

ˆ

ˆ

ˆ

ˆ

= =

=

= =

=

»

1⁄4

o «

¬

a =  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

s U s t

t

t

t

{ }

( )

{ }

( )′ − »

»

1⁄4

o

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a = −  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

− = =

=

= =

=

i i

i s U

i i

i s U

I k h

t

t I k h V

t

t V

1 M 2 ,2

M 0

M 2 ,2

M 0

ˆ

ˆ ˆ ˆˆ ˆ

 (32)

where

{ }

{ }

i

i s U

i

i s U

i

i h

t

t

V

h

t

t

h V

k ′  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

 ̈

 ̈

©

§

′  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

=

− = =

=

− = =

= 1 M 2 ,2

M 0

1 M 2 ,2

M 0

ˆ

ˆ ˆ

ˆ

ˆ ˆ

1

{ }

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a =  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

= =

=

= =

=

M 2 2

M 0

0 M 2 ,2

M 0

ˆ

ˆ ˆ ˆˆ ˆ

s U s t

t V

t

t V

 (33)

ki

 is the (qM=0 + qM=2)-by-1 column vector of optimal (EBLUE) weights, and (t

M=0)i

is the i

th element of the census constraint vector [tM=0]. The fi nal RRE after qM=0

recursions through Eqs. 31, 32, and 33 equals:
USDA Forest Service RMRS-GTR-239. 2010. 31

{ }

( )

{ }

¦ =

=

=

− = =

=

=

= =

=

= =

=

»

»

1⁄4

o

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a » + −

1⁄4

o «

¬

a =  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a M 0

M 0

1

1 M 2 ,2

M 0

M 0

M 2 2

M 0

M 2 ,2

M 0

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ q

i

i s U

i i i

s q s U

t

t

t k h

t

t

t

t

 (34)

{ }

( ) { } ( ) ( ) { }  ̧

1

·  ̈

©

§ ′ −  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a = −  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a ∏ = ∏ =

=

=

= =

=

=

= =

=

M 0 M 0

M 0

1,2, ,

M 2 2

M 0

, ,2,1

M 2 ,2

M 0

ˆ

ˆ ˆ ˆ

ˆ ˆ i q i i

s

i q i i

q s U

  I k h

t

t

I k h V

t

t

V (35)

The fi nal RRE for the qM=2-by-1 vector of population estimates for the qM=2 study 

variables equals the linear transformations of Eqs. 34 and 35 with the qM=2-by-

(qM=0 + qM=2) indicator matrix Hy = [0|I]:

[ ] { }

M 2 { } ,2

M 0

M 2 ,2 ˆ

ˆ ˆ

s U

s U y

= =

=

= = »

1⁄4

o «

¬

a = t

t

t H (36)

[ ] { } ( )

{ }

y

q s U

y U q s H

t

t V t H V ′

»

»

»

1⁄4

o

«

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a =

=

=

= =

=

= =

M 0

M 0

M 2 ,2

M 0

M 2 ,2 ˆ

ˆ ˆ ˆ ˆ (37)

Each i

th recursion (Eqs. 31 to 33) has an associated scalar residual difference 

between the auxiliary census total (tM=0)I This residual may be standardized with 

its estimate variance:

( )

{ }

{ }

i

i s U

i

i s U

i i

i

t

r

h

t

t

h V

t

t

h

′ »

»

1⁄4

o

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a −

=

− = =

=

− = =

=

=

1 M 2 ,2

M 0

1 M 2 ,2

M 0

M 0

ˆ

ˆ ˆ

ˆ

ˆ

 (38)

If all assumptions are correct, then ri in Eq. 38 is expected to have a zero mean 

with unit variance. If its realized value is inconsistent with this expectation (e.g., 

ri is larger than two standard deviation units [ri<-2 or ri>2]), then one or more assumptions are suspect. It is also possible that a rare extreme value for the sample 

estimate of the i

th auxiliary variable was observed by chance. This may be most 

common for categorical auxiliary variables that are rare in the population. One ad 

hoc rule would be to omit the i

th auxiliary variable with a relatively large residual 

(e.g., |ri|>2) from the RRE. This rule would provide some insurance against overzealous application of remotely sensed and other geospatial auxiliary data, which 

often can be cross-classifi ed in numerous ways, especially when intersected with 

geopolitical boundaries (e.g., states, counties, and National Forests).

The fi nal (qM=0)th recursion of the RRE in Eqs. 31 to 33 might be affected by the 

sequence 1 ≤ i ≤ qM=0, in which the scalar census auxiliary variables are applied. 

This might be especially true if the ad hoc rule above is used to censor certain 

scalar auxiliary variables that have unexpectedly large residuals (Eq. 38). The fi rst 

set of auxiliary variables might be geopolitical boundaries so that areal summaries 

of population estimates agree with the known area of each geopolitical entity. The 

next set might be common and important remotely sensed classifi cations, such as 

a binary forest/non-forest classifi cation for each pixel. All pixels under this binary 

auxiliary variable might be cross-classifi ed by geopolitical boundaries. The auxiliary variables with the largest extent might be applied fi rst. Similar to methods 
32 USDA Forest Service RMRS-GTR-239. 2010.

used by Statistics Sweden (Särndal and Lundström 2008), some sort of stepwise 

approach might fi rst apply those auxiliary variables that are most effective in increasing the precision of FIA study variables. Further research is warranted.

When census constants are imposed as constraints, the sequential algorithm 

in Eqs. 31 to 37 are numerically more effi cient than the usual algorithm in Eq. 

30 (Maybeck 1979:375; Grewal and Andrews 2001:226). At least as important, 

the sequential algorithm in Eqs. 31 to 37 is more easily implemented in database 

software because it replaces a single matrix inversion with a sequence of qM=0

scalar inverses. Matrix algebra software is not required, which makes implementation more feasible within database software used by government programs like 

FIA. Regardless, all computations and data storage should use double precision 

numerics to reduce insidious round-off errors. The sequential algorithm does not 

eliminate redundant measurements; therefore, it adds no new information. Rather, 

it simply accommodates redundancies and avoids matrix inversion through a convenient and robust numerical solution.

RRE as the Optimal Linear Calibration Estimator

The seminal paper by Deville and Särndal (1992) signaled the emergence 

of calibration estimators in the sample survey literature as a means to improve 

effi ciency with auxiliary data. The calibration estimator can achieve the same effi -

ciency with a smaller sample size as the Horvitz-Thompson estimator. Therefore, 

the calibration estimator is relevant to small-area estimation where small sample 

sizes limit estimation precision. In their paper, titled Borrowing Strength is Not 

the Best Technique Within a Wide Class of Design-Consistent Domain Estimators, 

Estevao and Särndal (2004) consider direct estimators of small-domain parameters with fi xed auxiliary census information. Other than their fi nite-population 

perspective, this is exactly the subject of this section.

If the number of PSUs in a domain is suffi ciently large, then direct domain (or 

sub-population) estimators are feasible. Direct domain estimators make use of 

the study variable, that is, the variable of interest, and auxiliary variables that are 

solely within the domain (Rao 2005). (Indirect estimators, which borrow strength 

from outside the domain, are considered under “Non-Linear Pseudo-Estimators” 

below.) Estevao and Särndal (2004) conclude “for a fi xed set of auxiliary information, the minimum asymptotic design-based variance is obtained with a direct 

estimator, derived by calibration rather than by regression fi tting ... and any estimator in the class that attempts to borrow strength is less precise.”

This conclusion appears to include a different type of small-domain composite 

estimator, which combines an indirect synthetic model-based estimate with the 

direct design-based estimator for the small-domain. The objective is to reduce the 

inherent risk of bias with the model-based synthetic estimator. Examples of this 

approach include Lui and Cumberland (1991), Longford (1999), Fuller and Rao 

(2001), and Francisco (2003). However, based on the cited fi ndings of Estevao 

and Särndal (2004), this type of composite estimator is not given further attention 

in the remainder of this report.

In the class of direct domain estimators considered by Estevao and Särndal 

(2004:657), the asymptotically optimal estimator of a scalar study variable yd+

from the sample s in domain d is given (with their notation) as:

( )( ) ( )  ̧

1

·  ̈

©

§  ̧ − 1·  ̈

©

§ = + − ¦¦ − ′ ¦¦∈ ∈

−

∈ ∈

+ ⊕ + ⊕

k s s l

k l kl Cl dk

k s s l

d d C d k l kl Cl Ck y y x x a a a x x a a a x y

1

ˆ ˆ ˆ π π (39)
USDA Forest Service RMRS-GTR-239. 2010. 33

where

Ud ⊆ U Domain d of the finite population U = {1,...,k,...,N} 

UC ⊆ U Calibration group C within the finite population U = {1,...,k,...,N} 

y y k s dk = δ dk k : ∈ The observed scalar value of the study variable for PSU k in sample s 

in domain d (įdk= 1 for k, a member of Ud = 0, otherwise) 

¦∈

⊕ =

k s

d k dk y a y π ˆ Horvitz-Thompson (HT) estimator of the domain total for scalar study 

variable in domain d

k k a = 1 π Sampling weight (i.e., expansion factor) for PSU k

Ck x

J-by-1 vector of auxiliary information for PSU k in calibration group C

(i.e., population control group) 

xC + J-by-1 vector of auxiliary information in calibration group C

¦∈

⊕ =

k s

d k dk x a x π ˆ HT estimator of the domain total for the J-by-1 vector of auxiliary 

information in domain d, where [ ] d⊕ = C+ x x π E ˆ

Equation 39 may be rewritten as:

 ( ) ( ) π + ⊕π

−

d+ = d⊕ + ′Cd C C − d y y C V x x x x ˆ ˆ ˆ ˆ ˆ 1

where 

( )  ̧

1

·  ̈

©

§ = ¦¦ − ′

k s ∈ ∈ s l

VCx ak al akl xClxCk ˆ

HT estimator for the J-by-J non-singular covariance 

matrix for the vector of estimated totals for the auxiliary 

variables in calibration group C (Särndal and others 

1992:43, 188) 

( )  ̧

1

·  ̈

©

§ = ¦¦ − k s ∈ ∈ s l

Cd y k l kl Cl dk C a a a x y x ˆ

HT estimator for the J-by-1 cross-covariance vector for 

the estimated totals for the vector of auxiliary variables x

in calibration group C and the scalar study variable y in 

domain d (Särndal and others 1992:43, 188) 

 (40)

Equation 40 can be expressed in a different matrix expression, in which the dimensions are enlarged from J to (J + 1) to form:

( )

( )  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a −

»

»

1⁄4

o

«

«

¬

a

′ +»

1⁄4

o «

¬

a =

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a » −

1⁄4

o «

¬

a

′ +»

1⁄4

o «

¬

a = »

1⁄4

o «

¬

a

⊕

⊕

+

−

⊕

⊕

⊕

⊕

+ −

⊕

⊕

+

+

π

π

π

π

π

π

π

π

d

d

C C

Cd y

C

d

d

d

d

C

d Cd y C

d

d

d

y y

y y y

ˆ

ˆ ˆ ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ ˆ ˆ

ˆ

ˆ

1

1

x

V x H

C

x V

x

x H

C V

x x I

x

x

x

x x

 (41)

where the J-by-(J+1) indicator matrix H = [I|0].With further algebraic embellishment, Eq. 41 equals:
34 USDA Forest Service RMRS-GTR-239. 2010.

 

( )

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

+ − »

1⁄4

o «

¬

a =

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a − »

»

1⁄4

o

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§ ′  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§ ′  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

+»

1⁄4

o «

¬

a =

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a −  ̧

 ̧

1

·

 ̈

 ̈

©

§ ′

»

»

1⁄4

o

«

«

¬

a

′ »

»

1⁄4

o

«

«

¬

a

′ +»

1⁄4

o «

¬

a =»

1⁄4

o «

¬

a

⊕

⊕

+

⊕

⊕

⊕

⊕

+

−

⊕

⊕

⊕

⊕

⊕

⊕

⊕

⊕

+

−

⊕ ⊕

⊕

+

+

π

π

π

π

π

π

π

π

π

π

π

π

π

π

π π

π

d

d

C

d

d

d

d

C

d C

d

d C

d

d

d

d

d

C

Cd y d

C Cd y

Cd y

C

d

d

d

d

y y

y y y y

y y v y y

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ ˆ ˆˆ ˆ ˆˆ

ˆ

ˆ

ˆ ˆ ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

1

1

x

K x H

x

x

H x H

x

H H V

x

V

x

x

H x H

C

V C

H

C

x x V

x

x x

x

x

where 

( )»

»

1⁄4

o

«

«

¬

a

′ = ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

⊕ ⊕

⊕

π π

π

Cd y d

C Cd y

d C

d

y vˆ yˆ ˆ

ˆ ˆ

ˆ

ˆ ˆ

x

x x

C

x V C

V

( ) »

»

1⁄4

o

«

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a

′ =»

1⁄4

o

«

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a

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1

·

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§

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1⁄4

o «

¬

a

⊕ ⊕

⊕

Cd y

C

Cd y d

C Cd y

d C

d

y v y x

x

x

x x

C

V

0

I

C

V C

H

x

V ˆ

ˆ

ˆ ˆ ˆ

ˆ ˆ

ˆ

ˆ ˆ

π π

π

»

»

1⁄4

o

«

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a

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1

·

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§

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o «

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a

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§ ′  ̧

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1

·

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§

»

1⁄4

o «

¬

a =

−

⊕

⊕

⊕

⊕

1

ˆ

ˆ ˆ ˆˆ ˆ H

x

H H V

x

K V

d C

d

d C

d

y y π

π

π

π

[ ] »

1⁄4

o «

¬

a =

+

+

+

d

d

d

y

y ˆ ˆ 0 0 1

x 

 (42)

Equation 42 is algebraically identical to the RE for the state-vector in Eq. 27, except the study variable is a scalar (y) rather than a vector (t

M=2) and:

[ ] M 0 2 ˆ ˆ ⊕ = = = d s x t π

Sample survey estimate of the domain totals for 

the vector of auxiliary variables 

[ ] d + = s=U = M 0 x t Known census statistics for the domain totals for 

the vector of auxiliary variables 

[ ] [ ] M 2 2 M 2 2 ˆ ˆ ˆ ⊕ = = = = = = d s s y t t π

Sample survey estimate of the scalar domain total 

for the study variable 

M 2 2

M 0

ˆ

ˆ

ˆ

ˆ

= =

=

⊕

⊕

»

1⁄4

o «

¬

a =»

1⁄4

o «

¬

a

s d

d

y t

x t

π

π The [(J + 1) = (qM=0 + qM=2)]-by-1 column vector of 

sample survey estimates 

[ ] M 2 { } ,2 ˆ ˆ d s U y + = = = t

The scalar calibration estimate of the study 

variable in domain d, which is exactly equal to the 

RRE estimate 

Therefore, the RRE is algebraically identical to the optimal direct calibration 

estimator for domain d in which the domain total (e.g., a census) is known exactly without error for one or more auxiliary variables (e.g., Ud

 = U) for one or 

more calibration groups (Estevao and Särndal 2004:649), and the domain membership for each sampled unit is known without classifi cation error. The domain 

census constants “must not be out-of-date, erroneous total” (Estevao and Särndal 

2004:650).

With a univariate study variable (y) and a numerically stable inverse covariance matrix, the results from the RE in Eqs. 27 or 39 may be represented as a 
USDA Forest Service RMRS-GTR-239. 2010. 35

system of calibration weights, wk = akgk, for each sampled PSU = k (Estevao and 

Särndal 2004:651). Using the notation of (Estevao and Särndal 2004:651, 657),

( ) ( ) ( )  ̧

1

·  ̈

©

§ − ′ = = + − ¦∈

−

+ ⊕

l s

k k k k C d C k l kl Cl dk w a g a x x V a a a x y x

1 ˆ ˆ π (43)

These will be the optimal calibration weights for a single study variable, 

e.g., (ydk)a, but they are sub-optimal for any other study variable, e.g., ( ydk ) b, except in the rare case in which their cross-covariance matrices are identical, e.g., 

( )( ) CCdxy a = CCdxy b t t . Otherwise, each study variable requires a different system 

of optimal scalar calibration weights from Eq. 43. The section “Optimal PSU 

Expansion Values” develops a vector system of expansion values that is optimal 

for each study variable in the RRE state vector [y].

One simpler ad hoc method for a single scalar calibration weight wk

 with a 

vector of L study variables [y] is a linear transformation with a 1-by-L matrix for 

the fi rst principal component from a Principal Component Analysis on all PSUs 

k ∈ s , yielding a scalar study variable (ydk)

PC=1 unique to each PSU. This scalar 

variable, which would contain much of the information available in a multivariate variable of interest, would be used solely to compute the calibration weight 

in Eq. 43.

There is a fundamental problem with the calibration estimator, regardless of 

the chosen study variable of interest. Estevao and Särndal (2004:657) make the 

following observation regarding the optimal calibration estimator: “In estimation 

for the whole population U, the asymptotically optimal estimator has been carefully examined in, for example, Casady and Valliant (1993), Montanari (1998, 

2000), and Montanari and Ranalli (2002). It is known to be unstable, especially 

for designs more complex than SRS. Here we encounter the asymptotically optimal estimator in the context of domain estimation. The discussion in the cited 

references is relevant here too. ... A prudent approach is to use ... (a sub-optimal 

sampling weight) ... in all cases.”

As frequently recognized throughout this report, the RE, i.e., the static Kalman 

fi lter, is vulnerable to numerical errors, especially if the domain totals for the 

auxiliary variables (e.g., xd+ = [tM = 0]s = U ) are known with perfect or high precision. Similarly recognized are the numerical solutions that are well developed in 

the engineering literature on the Kalman fi lter. When the familiar calibration and 

regression estimators for complex sample surveys are seen from a different perspective, i.e., as special cases of the static Kalman fi lter, then numerical solutions 

instantly become available from 50 years of engineering experience. Equations 

31 to 37 are examples.

In the current context, a census of vector constants for geopolitical domains, 

such as the areas of counties and public lands, are available from administrative 

records. Each sampled PSU is known to be a member of one of these domains. 

Such constraints not only assure summaries of statistical estimates are consistent with these widely known administrative records, but it is possible they can 

improve accuracy of sample survey estimates for other study variables (e.g., 

Van Deusen 2005). In addition, remotely sensed auxiliary census variables 

might further improve effi ciency of the estimators for the study variables within 

a calibration group, which may be identical to a study domain. This might utilize multiple sets of overlapping or non-exhaustive domains, namely, complex 

auxiliary information (Estevao and Särndal 2006). Examples include domains 

based on polychotomous classifi cations of ecoregions, ecofl oristic zones, climatic 

zones, land forms, landscape types, and land ownership maps. They may also 

include classifi cations of land cover and forest conditions with remotely sensed 

data. The recursive algorithm in Eqs. 31 to 37 can be sequentially applied to each 

set of independent census domains. This fortunate attribute of RRE further permits off-line optimization of prediction models for each individual dichotomous 
36 USDA Forest Service RMRS-GTR-239. 2010.

category of any polychotomous variable, which is familiar practice with remotely 

sensed predictor variables and software. Improvements through remote sensing 

techniques should, in principle, improve statistical accuracy of population and 

domain estimates. The various sets of independent calibration groups can be more 

broadly defi ned than domains of study, which can assure suffi cient sample size 

within each group (Lehtonen and others 2003). They need not be cross-classifi ed, 

which reduces the problem of very small-domains that contain too few PSUs for 

direct domain estimation. Additional research is needed to best use the very detailed auxiliary data possible with airborne and spaceborne sensors. For example, 

which auxiliary variables provide the most gain in statistical precision for FIA 

study variables without undo risks to numerical stability?

Three-Phase Sampling for Continuous Variables

Three-phase sampling “can be a very effective design for the estimation of 

regional and national forest cover type frequencies ... with data gathered in each 

phase at a different resolution (scale)” (Magnussen 2003). Extension to three or 

more sampling phases is expedited by the sequential approach inherent to estimation within the RRE. For example, consider a three-phase design that starts with a 

census of satellite pixels (s = U) that is measured with M = 0 remote sensing protocol. The next phase is photo-interpretation of a large simple random sample of 

points that is measured with both the M = 1 photo-interpretation protocol and the 

M = 0 remote sensing protocol. The fi nal phase is a small, simple random sample 

of points measured with the M = 2 FIA fi eld protocol in addition to the M = 0 and 

M = 1 protocols. This example duplicates much of the previous section on “TwoPhase Sampling for Continuous Variables with a Census at Phase 1”. However, 

the purpose of the current section is to demonstrate more fully the sequential application of the RRE to more complex sample survey designs.

The fi rst step is a design-based estimate for population totals with the M = 0 

remote sensing protocol and the M = 1 photo-interpretation protocol from the 

Phase 1 sample (s = 1). The (qM=0 + qM=1)-by-1 vector estimate and its (qM=0 + qM=1)-

by-(qM=0 + qM=1) covariance matrix are:

( )

( ) ( ) [ ]

[ ]

( ) [ ]

[ ]

( )

»

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

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¬

a

°

°

°

¿

°

°

°

3⁄4

1⁄2

°

°

°

 ̄

°

°

°

®

­ ′

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1

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«

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−

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a

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1

·

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§

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−

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− =  ̧

 ̧

1

·

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©

§

»

»

1⁄4

o

«

«

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a

»

1⁄4

o «

¬

a =

»

»

1⁄4

o

«

«

¬

a

¦

¦

∈ =

= =

= =

=

=

= =

= =

=

=

= = = =

=

∈ = =

=

= = =

=

1

M 1 1

M 0

M 1

M 0

M 1 1

M 0

M 1

M 0

1 1

2

1 M 1

M 0

1 M 1

M 0

1 M 1 1

M 0

ˆ ˆ

ˆ 1

ˆ ˆ

ˆ

ˆ

j s

s

s U

j

s

s U

j

s s s

j s j s s

A A n n

A

n

A

y

t

y

y

y

t

y

y

t

t

V

y

y

t

t

 (44)

The sample covariance matrix in Eq. 44 uses the census values for the population 

means from the M = 0 remote sensing protocol (Eq. 48), rather than the corresponding estimates from the Phase 1 sample, to improve the estimated variances 

and covariances.

The second step is a design-based estimate for population totals with the M = 0 

remote sensing protocol, the M = 1 photo-interpretation protocol, and the M = 2 

FIA fi eld data protocol from the small Phase 2 sample (s = 2). The (qM=0 + qM=1 

+ qM=2)-by-1 vector estimate and its (qM=0 + qM=1 + qM=2)-by-(qM=0 + qM=1 + qM=2) 

covariance matrix are:
USDA Forest Service RMRS-GTR-239. 2010. 37

( )

¦∈ =

=

=

=

=

= =

=

=

»

»

»

1⁄4

o

«

«

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¬

a

=

»

»

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1⁄4

o

«

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a

2

M 2

M 1

M 0

2

2 M 2

M 1

M 0

ˆ

ˆ

ˆ

j s

j

s

s

n

A

y

y

y

t

t

t

( )

( )

( ) [ ]

[ ]

[ ]

[ ]

( ) [ ]

[ ]

[ ]

[ ]

[ ]

[ ] ( ) [ ] »

»

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=

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§

+

+

+

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«

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a

− =

 ̧

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1

·

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©

§

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»

»

1⁄4

o

«

«

«

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a

= =

=

=

=

=

=

=

= =

=

= =

= =

= =

=

= =

= =

=

= =

=

=

=

= =

= =

= =

=

= =

= =

=

= =

=

=

=

= =

∈ =

= =

=

= ¦

M 2 2

M 1

M 0

M 2

M 2

M 1

M 0

2 M 1

M 0

M 2 2

M 1 2

1 2

1

M 1 1

1 2

2

M 0

M 2

M 1

M 0

M 2 2

M 1 2

1 2

1

M 1 1

1 2

2

M 0

M 2

M 1

M 0

2 2

2

2

2 M 2

M 1

M 0

ˆ ˆ ˆ

ˆ , ˆ ˆ

ˆ , ˆ

ˆ ˆ ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

1 ˆ ˆ

ˆ

ˆ

ˆ

s

s

s

s

s s

s

s

s s

s

s U

j

s

s

s s

s

s

s s

s

s U

j

s s

j s

s

n n

n

n n

n

A

n n

n

n n

n

A

n n

A

V t

t

t C t

t

t

t C

t

t V

y

y

y

t

y

y

y

y

y

y

t

y

y

y

t

t

t

V

 (45)

The sample covariance matrix in Eq. 45 uses the census values for the population means from the M = 0 remote sensing protocol and a composite estimate of 

the M = 1 means from the Phase 1 and 2 samples, rather than the corresponding 

estimates from the Phase 2 sample alone, to improve the estimated variances and 

covariances. The fi nal term in Eq. 45 defi nes a matrix partition of the sample 

covariance matrix.

The third step combines the (qM=0 + qM=1)-by-1 Phase 1 estimate (Eq. 44, s = 1) 

with the (qM=0 + qM=1 + qM=2)-by-1 Phase 2 estimate (s = 2, Eq. 45) with RRE, similar to Eq. 24. The notation corresponding to the Kalman fi lter in Eq. 12 is:

( ) ( ) ( )

() () ( )

[ ] () () H I 0 C( ) t t 0

t

t

V t V

t

t

t

t

t

t

V t V

t

t

t

t

= =

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a =

»

»

1⁄4

o

«

«

¬

a =

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

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 ̈

 ̈

©

§

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

=

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

=

= =

=

= =

=

= =

=

=

= =

=

=

1 1

M 1 1

M 0

1

M 1 1

M 0

1

2 M 2

M 1

M 0

1

2 M 2

M 1

M 0

1

ˆ, ˆ ˆ

ˆ

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ ˆ ˆ

ˆ

ˆ

ˆ

ˆ

X Z

s

Z

s

Z

s

X

s

X

 (46)
38 USDA Forest Service RMRS-GTR-239. 2010.

The H matrix has dimensions (qM=0 + qM=1)-by-(qM=0 + qM=1 + qM=2). From Eqs. 12 

and 46, the composite estimator is:

 ̧

 ̧

 ̧

 ̧

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1

·

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©

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¬

a ′ +

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1

·

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′

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«

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a

=

−

= =

=

= =

=

=

= =

=

=

1

M 1 1

M 0

2 M 2

M 1

M 0

2 M 2

M 1

M 0

ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

s

s s

t

t H V

t

t

t

H H V

t

t

t

K V

 { }

 ̧

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1

·

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§

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1⁄4

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«

«

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a

» −

1⁄4

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¬

a

+

»

»

»

1⁄4

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«

«

«

¬

a

=

»

»

»

1⁄4

o

«

«

«

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a

= =

=

=

= =

=

= =

=

=

= =

=

=

2 M 2

M 1

M 0

M 1 1

M 0

2 M 2

M 1

M 0

1,2 M 2

M 1

M 0

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

s

s

s s t

t

t

H

t

t K

t

t

t

t

t

t

 

{ } »

»

»

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1⁄4

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«

«

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a

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1

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«

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−

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«

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a

=

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1

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1⁄4

o

«

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a

= =

=

=

= =

=

=

= =

=

=

2 M 2

M 1

M 0

2 M 2

M 1

M 0

1,2 M 2

M 1

M 0

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

s s s t

t

t

K H V

t

t

t

V

t

t

t

V

 (47)

The composite estimator, denoted s = (1,2) in Eq. 47, is the minimum variance 

estimate of population totals for variables measured with all three protocols (remotely sensed satellite data, photo-interpreted data and fi eld data) given the large 

Phase 1 sample and the smaller Phase 2 sample but not the census of remotely 

sensed data. Equation 47 might not be numerically stable, and it is very important 

to use numerically more stable methods that are reviewed in the section “Robust 

Numerics.” The fi rst three steps are analogous to the sequential approach used by 

Magnussen (2003).

The fourth step is to enumerate the qM=0-by-1 vector population totals for the 

variables measured with M = 0 (remote sensing) from the census of remotely 

sensed pixels in the full-coverage satellite imagery:

[ ] [ ]

V( ) [ ] t 0

t y

=

=

= =

∈

= = ¦ =

s U

j U

s U j

M 0

M 0 M 0

ˆ ˆ

ˆ

 (48)

The qM=0-by-qM=0 covariance matrix in Eq. 48 is the zero matrix because census 

totals are known exactly.

The fi fth step is to combine the (qM=0 + qM=1 + qM=2)-by-1 composite estimate 

from the Phase 1 and Phase 2 samples, denoted s = (1,2) in Eq. 47, with the qM=0-

by-1 census vector, denoted s = U in Eq. 48, again, similar to Eq. 24. The notation 

corresponding to the Kalman fi lter in Eq. 12 is:
USDA Forest Service RMRS-GTR-239. 2010. 39

( )

{ }

( ) ( )

{ }

( ) [ ] ( ) ( )

[ ] () () H I 0 C( ) t t 0

t t V t 0

t

t

t

V t V

t

t

t

t

= =

= =

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

1⁄4

o

«

«

«

¬

a

=

»

»

»

1⁄4

o

«

«

«

¬

a

=

= =

= =

=

=

= =

=

=

1 1

1 M 0 1

1,2 M 2

M 1

M 0

1

1,2 M 2

M 1

M 0

1

ˆ, ˆ ˆ

ˆ ˆ ˆ ˆ

ˆ

ˆ

ˆ

ˆ ˆ ˆ

ˆ

ˆ

ˆ

ˆ

X Z

Z s U Z

s

X

s

X

 (49)

The H matrix has dimensions qM=0-by-(qM=0 + qM=1 + qM=2). From Eqs. 12 and 49, 

the composite estimator is:

{ } { }

{ }

[ ] { } [ ] ( )  ̧

1

·  ̈

©

§′

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

1⁄4

o

«

«

«

¬

a

=

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

′ +

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

1⁄4

o

«

«

«

¬

a

′

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

1⁄4

o

«

«

«

¬

a

=

−

= =

= =

=

=

−

= =

=

=

= =

=

=

1

M 0 1,2

1,2 M 2

M 1

M 0

1

1,2 M 2

M 1

M 0

1,2 M 2

M 1

M 0

ˆ ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

s

s

s s

H V t

t

t

t

V

H 0

t

t

t

H H V

t

t

t

K V

{ } { }

[ ]

{ }

{ }

[] [] { } ( ) M 0 M 0 1,2

1,2 M 2

M 1

M 0

1,2 M 2

M 1

M 0

M 0

1,2 M 2

M 1

M 0

,1,2 M 2

M 1

M 0

ˆ ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

= = = =

= =

=

=

= =

=

=

= =

= =

=

=

= =

=

=

+ −

»

»

»

1⁄4

o

«

«

«

¬

a

=

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

1⁄4

o

«

«

«

¬

a

+ −

»

»

»

1⁄4

o

«

«

«

¬

a

=

»

»

»

1⁄4

o

«

«

«

¬

a

s U s

s

s

s U

s U s

K t t

t

t

t

t

t

t

K t H

t

t

t

t

t

t

{ } { } { }  ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

1⁄4

o

«

«

«

¬

a

−

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

1⁄4

o

«

«

«

¬

a

=

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

1⁄4

o

«

«

«

¬

a

= =

=

=

= =

=

=

= =

=

=

1,2 M 2

M 1

M 0

1,2 M 2

M 1

M 0

,1,2 M 2

M 1

M 0

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

s U s s t

t

t

K H V

t

t

t

V

t

t

t

V

 (50)

The composite estimator in Eq. 47, denoted s = (U,1,2), is the minimum variance estimator given the Phase 1 and Phase 2 samples and the census of remotely 

sensed pixel data. Equations 44 to 50 may be expanded, as in Eqs. 20 to 27, to 

reveal analogies between RRE and more traditional univariate regression estimators (e.g., Särndal and others 1992). Again, numerically stable, but algebraically 

equivalent methods must be used to apply Eq. 50. These methods are discussed in 

the section “Robust Numerics.”
40 USDA Forest Service RMRS-GTR-239. 2010.

The fi nal step is to extract the portions of the composite estimator in Eq. 47 

that contain the estimated population totals for variables measured with the FIA 

fi eld protocol (M = 2):

 

[ ] { } [ ]

{ } ,1,2 M 2

M 1

M 0

M 2 ,1,2

ˆ

ˆ

ˆ

ˆ

s U

s U

= =

=

=

= =

»

»

»

1⁄4

o

«

«

«

¬

a

=

t

t

t

t 0 0 I

( ) [ ] { } [ ]

{ }

»

»

»

1⁄4

o

«

«

«

¬

a

°

°

¿

°

°

3⁄4

1⁄2

°

°

 ̄

°

°

®

­

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

1⁄4

o

«

«

«

¬

a

=

= =

=

=

= =

I

0

0

t

t

t

V t 0 0 I V

,1,2 M 2

M 1

M 0

M 2 ,1,2

ˆ

ˆ

ˆ

ˆ ˆ ˆ

s U

s U

 (51)

A similar sequential process (Eqs. 44 to 51) may be used with four or more 

phases. Knottnerus (2003:359) uses the same approach to complex designs. The 

dimensions of the vectors and covariance matrices will increase because the 

estimates for all protocols are required until the very last step in the estimation sequence. The sequential process may also be used with multiple sources of census 

data, such as areas by geopolitical categories (e.g., counties and land ownership 

types) or other full-coverage, remotely sensed data (e.g., classifi cations of pixel 

data from the MODIS satellite and stand management geospatial information system independently produced by a National Forest). Knottnerus (2003:364) and 

Czaplewski (in prep.[b]) give more details.

A Potentially More Robust but Less Effi cient Estimator

A potentially more robust but less effi cient estimator for a multiphase design 

is possible by reducing the matrix dimensions at each of the sequential steps describ e d earlier in this section. Magnussen (2003) found that RRE, when applied to 

three-phase sampling of categorical variables, is not necessarily robust and consistent compared with a sub-optimal sequential estimator. Performance of RRE 

decreases as the multivariate dimensions become large relative to sample sizes. 

Dimensions may be decreased by sequentially considering only those variables 

shared by two levels at one time in the multilevel sampling design. For example, 

consider the three-phase estimator in Eqs. 44 to 51 that requires covariance matrices with dimensions of (qM=0 + qM=1 + qM=2)-by-(qM=0 + qM=1 + qM=2). A description 

of the following recursions is offered in prose form above (“The Basic Idea”).

The RRE that combines the fi rst two sampling phases (Eqs. 44 to 47) can be 

modifi ed to produce a (qM=0 + qM=1)-by-1 vector estimate of population totals for 

the qM=1 photo-interpreted (M = 1) variables.

( ) ( ) ( )

( ) [ ] ( ) ( )

[ ] () () H I 0 C( ) t t 0

t t V t 0

t

t

V t V

t

t

t

= =

= =

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a = »

»

1⁄4

o

«

«

¬

a

=

= =

= =

=

= =

=

1 1

1 M 0 1

M 1 1

M 0

1

M 1 1

M 0

1

ˆ, ˆ ˆ

ˆ ˆ ˆ ˆ

ˆ

ˆ ˆ ˆ ˆ

ˆ

ˆ ˆ

X Z

Z s U Z

s

X

s

X

 (52)
USDA Forest Service RMRS-GTR-239. 2010. 41

Incorporating the structure of Eq. 52 into RRE (Eq. 12) yields the fi rst step in the 

alternative sequential approach:

 

[ ] ( ) [ ]  ̧

1

·  ̈

©

§′  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a = −

= =

= =

= 1

M 0 1

M 1 1

M 0 ˆ ˆ ˆ

ˆ ˆ s

s

H V t

t

t K V

 { }

( ) [] [] M 0 M 0 1

M 1 1

M 0

M 1 ,1

M 0 ˆ ˆ ˆ

ˆ

ˆ

ˆ

= = = =

= =

=

= =

=

» + −

1⁄4

o «

¬

a » =

1⁄4

o «

¬

a

s U s

s U s

K t t

t

t

t

t

{ }

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a −

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a = ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

= =

=

= =

=

= =

=

M 1 1

M 0

M 1 1

M 0

M 1 ,1

M 0

ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ

s U s s t

t K H V

t

t V

t

t V

 (53)

Numerically reliable algorithms must be used to apply Eq. 53, and these methods 

are briefl y covered in the section “Robust Numerics”.

Since the population totals for the qM=0 variables from the M = 0 census of satellite pixels are constants that are exactly known a priori by defi nition, they may 

be deleted from Eq. 53 in subsequent steps:

 

[ ] { } [ ]

M 1 { } ,1

M 0

M 1 ,1 ˆ

ˆ ˆ

s U

s U

= =

=

= = »

1⁄4

o «

¬

a = t

t

t 0 I

[ ] { } ( ) [ ]

{ }

»

1⁄4

o «

¬

a ′

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a =

= =

=

= = I

0

t

t V t 0 I V

M 1 ,1

M 0

M 1 ,1 ˆ

ˆ ˆ ˆ ˆ

s U

s U

 (54)

Equation 54 is an estimate of population totals for variables measured with the 

M = 1 photo-interpretation protocol that is made more precise than the M = 0 census of classifi ed satellite pixels. This fi rst sequence invo lves covariance matrices 

with dimensions (qM=0 + qM=1)-by-(qM=0 + qM=1) in Eq. 53 rather than (qM=0 + qM=1 + 

qM=2)-by-(qM=0 + qM=1 + qM=2) in Eqs. 44 to 51.

The Phase 2 sample vector estimate, with dimension (qM=1 + qM=2), and the 

suffi cient qM=1-dimensional statistics from Eq. 54 are inputs to the next sequential 

step:

( ) ( ) ( )

( ) [ ] { } ( ) ( ) [ ] { } ( )

[ ] () () H I 0 C( ) t t 0

t t V t V t

t

t

V t V

t

t

t

= =

= =

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a = »

»

1⁄4

o

«

«

¬

a

=

= = = =

= =

=

= =

=

1 1

1 M 1 ,1 1 M 1 ,1

M 2 2

M 1

1

M 2 2

M 1

1

ˆ, ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ

ˆ ˆ ˆ ˆ

ˆ

ˆ ˆ

X Z

Z s U Z s U

s

X

s

X

 (55)

Incorporating the structure of Eq. 55 into RRE (Eq. 12) yields the fi nal step in the 

alternative sequential approach:
42 USDA Forest Service RMRS-GTR-239. 2010.

[ ] { } ( )

1

M 1 ,1

M 2 2

M 1

M 2 2

M 1 ˆ ˆ ˆ

ˆ ˆ ˆˆ ˆ

−

= =

= =

=

= =

=

°¿

°

3⁄4

1⁄2

° ̄

°

®

­ ′ +  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

°¿

°

3⁄4

1⁄2

° ̄

°

®

­ ′  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a = s U

s s

H V t

t

t H H V

t

t K V

 

{ }

[ ] { } { } [ ]

[ ] { } [ ]

M 2 { } ,1,2

M 1

M 2 ,1,2

M 1 ,1 M 1 2

M 2 2

M 1

M 2 ,1,2

M 1

ˆ

ˆ ˆ

ˆ ˆ ˆ

ˆ

ˆ

ˆ

s U

s U

s U s

s U s

= =

=

= =

= = = =

= =

=

= =

=

»

1⁄4

o «

¬

a =

» + −

1⁄4

o «

¬

a » =

1⁄4

o «

¬

a

t

t

t 0 I

K t t

t

t

t

t

{ }

[ ] { } [ ]

{ }

»

1⁄4

o «

¬

a ′

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a =

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a −

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a = ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

= =

=

= =

= =

=

= =

=

= =

=

I

0

t

t V t 0 I V

t

t K H V

t

t V

t

t V

M 2 ,1,2

M 1

M 2 ,1,2

M 2 2

M 1

M 2 2

M 1

M 2 ,1,2

M 1

ˆ

ˆ ˆ ˆ ˆ

ˆ

ˆ ˆ ˆˆ ˆ ˆ

ˆ ˆ

s U

s U

s U s s

 (56)

Numerically stable methods, which are covered in the section “Robust Numerics,” 

must be used with Eq. 56.

Equations 52 to 56 start with a combination of the Phase 1 census (s = U) 

and the Phase 2 photo-interpreted sample (s = 1) and then a combination of this 

multivariate result (denoted s = U,1) with the Phase 3 sample (s = 2) to produce 

the fi nal estimate (denoted s = U,1,2). The sequence could be reversed, starting 

with a combination of Phase 2 (s = 1) and Phase 3 (s = 2) and then combining the 

composite vector estimate (s = 1,2) with the Phase 1 census of classifi ed pixels 

(s = U). In the absence of numerical errors, the results should be identical, regardless of the particular sequence.

The three-phase estimator in Eq. 51 differs from the alternative estimator in Eq. 

56, the latter of which uses matrices with smaller dimensions. Magnussen (2003) 

posited that smaller dimensions improve consistency and robustness of the threephase estimator, although the estimator might be less effi cient. Improvements to 

robustness require sacrifi cing sample data on variables measured with the satellite protocol (M = 0) with the third-phase sample (s = 2). The sample size at the 

third phase is typically small, and the sacrifi ce might have little practical signifi cance, especially if it proves more robust. However, the concerns reported by 

Magnussen might be partially caused by numerical errors that are inherent to any 

digital computer with fi nite word length. This is a well-known hazard with the 

Kalman fi lter, especially when known constants (e.g., census of classifi ed satellite 

pixels or administrative records that completely cover the sampled population) 

are incorporated. There is abundant engineering literature on more numerically 

robust solutions to the Kalman fi lter estimator and, hence, RRE. Solutions most 

often involve the matrix square roots of the covariance matrices and orthogonal 

transformations that substitute simultaneous multivariate solutions with sequential univariate solutions. Maybeck (1979:Chapter 7) provides a concise overview 

of these methods, and Bierman (1977) devotes an entire book to the subject. The 

hazard exists for any RRE applied to large, multilevel, multivariate sampling designs. This subject is briefl y discussed in the section “Robust Numerics.”
USDA Forest Service RMRS-GTR-239. 2010. 43

Single-Phase-Sampling for Continuous Variables with 

Cluster Plots

The multiphase designs considered until this point presuppose that the separate phases are composed of independent samples. However, this supposition is 

invalid for other feasible designs. One such case is the single-stage cluster design (Särndal 1992:125, 304), which is also termed trakt sampling (Mandallaz 

2008:53). This special case of cluster sampling is considered from the perspectives 

of an infi nite population, point sampling, and multiple collocated support-regions 

for each sample point. An example is modeled after the Nevada Photo-based 

Inventory Pilot (NPIP), which is an FIA initiative under its 2007 strategic plan 

(Frescino and others 2009a; Frescino and others 2009b) and is briefl y described 

in the “Introduction”.

Start with a simple random point-sample s of a continuous population in space. 

Each sample point j is centered on an FIA fi eld plot. Each sample point is measured with a relatively inexpensive protocol (M = 1), namely photo-interpretation 

of high-resolution aerial photography, and the relatively expensive FIA fi eld protocol (M = 2). The M = 1 protocol, and the support-region to which it is applied, 

are described as an example for Eq. 19. The qM=1-by-1 vector [y M=1]

1,j represents 

qM=1 photo-interpreted variables at the sample point j. The M = 2 protocol and 

support-region are described as an example for Eq. 20. The qM=2-by-1 vector 

[y M=2]j represents qM=2 continuous variables measured by an FIA fi eld crew at 

sample point j.

So far, this exactly describes the Phase 2 sample in the previous section on 

two-phase sampling. However, two-phase sampling assumes a large independent 

Phase 1 sample that is measured solely with the M = 1 protocol (e.g., photointerpretation). In this section, the Phase 1 portion is omitted and is replaced with 

a related design based on a cluster plot. The primary motivation behind cluster 

plots is to increase the number of photo-interpreted measurements with little extra 

incremental cost.

For example, let each sample point j in sample s be circumscribed by a relatively large but proximate support-region. This large support-region supplements 

the much smaller support-region described in the previous paragraph. In the NPIP 

example, the large support-region is 50 acres, and it is measured with photointerpretation (M = 1) of nj = 49 points that are systematically distributed over the 

50-acre support-region. The protocol used for each point (i, j), i = {1,2,...,49} 

in cluster plot j exactly matches the M = 1 protocol described in the previous 

paragraph. In fact, one of the 49 points, defi ned as i = (1, j), is at the center of the 

50-acre support-region, and it is identical to the j

th sample point in the previous 

paragraph.

The qM=2-by-1 vector mean of measurements from the nj = 49 points in the 

large support-region is another quantitative descriptor of the sample point j. It is 

denoted as protocol M = 1C, which refers to application of the M = 1 protocol to 

a large support-region, which may be considered a cluster plot:

[ ] [] j

n

i

j i j n

j

y M=1C = ¦ y M=1 , (57)

Had the problem been modeled as a fi nite population with two-stage sampling, 

then the points (i, j) would be considered “second-stage sampling units” (Särndal 

and others 1992:125).

The objective is to improve the estimated population totals for the M = 2 

variables with the estimated population totals for the M = 1 variables. In fi negrained, heterogeneous landscapes at the scale of the large support-region, the 

estimated population totals for photo-interpreted variables measured with the 
44 USDA Forest Service RMRS-GTR-239. 2010.

M = 1 protocol should be more precise with the large support-region (Eq. 57) 

compared to the single M = 1 measurement at the center of the support-region, 

namely, sample point (1, j). The primary value of the M = 1 and M = 2 variables 

measured at sample point (1, j) is estimation of qM=2-by-qM=1 sample crosscovariance matrix between the fi eld (M = 2) and photo-interpreted (M = 1) 

variables. If the cross-covariances are relatively large and the qM=1-by-1 vector

estimate of the population totals for the photo-interpreted variables is relatively 

precise, then the qM=2-by-1 vector estimate of the population totals for the FIA 

fi eld variables will be more precise.

Start with the (2×qM=1 + qM=2)-by-1 design-based vector estimator for all variables measured at the point sample s and its (2×qM=1 + qM=2)-by-(2×qM=1 + qM=2) 

sample covariance matrix:

 

[ ]

[ ]

[ ]

¦

¦

∈

= =

= =

=

= =

=

=

=

=

=

=

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a =

 ̧

 ̧

1

·

 ̈

 ̈

©

§ =

»

»

»

1⁄4

o

«

«

«

¬

a

=

»

»

»

1⁄4

o

«

«

«

¬

a

j s

j

i

i

j

n

i

i j

s

s s

n

n

A A

j

M 2 1

M 1 1

1

M 1C M 1 ,

M 2

M 1

M 1C

M 2

M 1

M 1C

1

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

y

y

y y

y

y

y

t

t

t

( )

( ) [ ] ( ) [ ][ ]

[ ]

( ) [ ] ( ) [ ][ ] ( ) [ ][ ]

( ) [ ][ ] ( ) [ ] ( ) [ ][ ]

( ) [ ][ ] ( ) [ ][ ] ( ) [ ] »

»

»

1⁄4

o

«

«

«

¬

a

′ ′

′ ′

′ ′

=

»

»

»

1⁄4

o

«

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§ ′ »

1⁄4

o «

¬

a

′ ′

=

′

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

1⁄4

o

«

«

«

¬

a

−

»

»

»

1⁄4

o

«

«

«

¬

a

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

1⁄4

o

«

«

«

¬

a

−

»

»

»

1⁄4

o

«

«

«

¬

a

− =

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

1⁄4

o

«

«

«

¬

a

= = = = =

= = = = =

= = = = =

=

=

=

=

=

= = = =

∈

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

¦

M 2 M 1C M 2 M 1 M 2

M 1 M 1C M 1 M 1 M 2

M 1C M 1C M 1 M 1C M 2

M 2

M 1

M 1C

M 2

M 1

M 1C M 1C M 1 M 2

M 2

M 1

M 1C

M 2

M 1

M 1C

M 2

M 1

M 1C

M 2

M 1

2 M 1C

M 2

M 1

M 1C

ˆ ˆ ˆ , ˆ ˆ ˆ , ˆ ˆ

ˆ , ˆ ˆ ˆ ˆ ˆ , ˆ ˆ

ˆ , ˆ ˆ ˆ , ˆ ˆ ˆ ˆ

ˆ

ˆ ˆ ˆ , ˆ

ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

1 ˆ

ˆ

ˆ

ˆ

C t t C t t V t

C t t V t C t t

V t C t t C t t

t

t

t V

t

t C

V t C t t t

y

y

y

y

y

y

y

y

y

y

y

y

t

t

t

V

j s

s j s j

s s

s

n n

A

 (58)

The last two terms in Eq. 58 defi ne partitions of the sample covariance matrix. 

This estimator for the sample covariance matrix ignores heteroscedasticity among 

cluster plots, which simplifi es variance estimates (e.g., Särndal 1992:153). This 

simplifi cation is the norm with FIA estimators (Scott and others 2005).

Next, use the more complete version of the Kalman fi lter in Eqs. 9 through 11 

that accommodates non-zero covariances between the state- and measurementvectors. The following maps the correspondence among notation in Eq. 58 and 

Eqs. 9 through 11, and the remainder of this section.
USDA Forest Service RMRS-GTR-239. 2010. 45

( ) ( ) ( )

( ) [ ] ( ) ( ) ( ) [ ]

[ ] () () ( ) [ ] s

X Z

Z s Z s

s

X

s

X

 ̧

 ̧

1

·

 ̈

 ̈

©

§ ′

»

»

1⁄4

o

«

«

¬

a = =

= =

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a =

»

»

1⁄4

o

«

«

¬

a =

=

=

=

= =

=

=

=

=

M 1C

M 2

M 1

1 1

1 M 1C 1 M 1C

M 2

M 1

1

M 2

M 1

1

ˆ , ˆ

ˆ ˆ ˆ , ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ

ˆ ˆ ˆ ˆ ˆˆ ˆ

t

t

t

H I 0 C t t C

t t V t V t

t

t

V t V

t

t

t

 (59)

The H matrix in Eq. 59 has dimensions qM=1-by-(qM=1 + qM=2). By substitution of 

the equalities in Eq. 59 into Eq. 9, the (qM=1 + qM=2)-by-qM=1 Kalman gain matrix 

K equals:

[ ]

[ ]

( ) [ ]

[ ] [ ]

( ) [ ] [ ]

( ) [ ] ( ) [ ][ ]

( ) [ ][ ] ( ) [ ]

( ) ( ) [ ]

( ) [ ]

( ) [ ][ ]

( ) [ ][ ]

1

M 1C M 1

M 1 M 1C

M 1C

M 1

M 2 M 1 M 2 M 1C

M 1 M 1 M 1C

1

M 1C M 1 M 2

M 1C

M 2

M 1

M 1C

M 2

M 1

M 1C

M 2

M 1

M 2

M 1

ˆ , ˆ ˆ

ˆ , ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ , ˆ ˆ ˆ , ˆ ˆ

ˆ , ˆ ˆ ˆ ˆ

ˆ ˆ , ˆ ˆ

ˆ , ˆ

ˆ ˆ

ˆ ˆ

ˆ

ˆ ˆ

ˆ , ˆ

ˆ ˆ

ˆ

ˆ ˆ

−

= =

= =

=

=

= = = =

= = =

−

= = =

=

=

=

=

=

=

=

=

=

=

=

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

 ̈

 ̈

 ̈

©

§

+ ′

+ ′

+

»

»

1⁄4

o

«

«

¬

a

′ + ′

+ ′ =

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

 ̈

 ̈

 ̈

 ̈

 ̈

 ̈

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

»

»

»

»

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

«

«

«

«

«

«

«

¬

a

»

1⁄4

o

«

¬

a

+ ′ ′

 ̧

 ̧

1

·

 ̈

 ̈

©

§ ′

»

»

1⁄4

o

«

«

¬

a

+

+

»

1⁄4

o

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§ ′

»

»

1⁄4

o

«

«

¬

a

+

»

1⁄4

o

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a

=

s

s

s

s

s s

s s

s

s

s

s

s

s

C t t

C t t

V t

V t

C t t C t t

V t C t t

0

I

C t t t

t

t

t

I 0 C

V t

0

I

t

t

I 0 V

t

t

t

C

0

I

t

t

V

K

 (60)

Numerically reliable algorithms must be used to apply Eq. 60, and these methods 

are briefl y covered in the section “Robust Numerics”.

From Eqs. 10, 59, and 60, the (qM=1 + qM=2)-by-1 vector estimate of population 

totals and its covariance matrix for all photo-interpreted (M = 1) and FIA fi eld 

variables (M = 2), denoted M = (1,1C), is:

( )

( ) [ ][ ]

( )

[ ] [ ]  ̧

 ̧

1

·

 ̈

 ̈

©

§

 ̧

 ̧

1

·

 ̈

 ̈

©

§ ′

»

»

1⁄4

o

«

«

¬

a

+  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a

+  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a =  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a

+ −

»

»

1⁄4

o

«

«

¬

a =

»

»

1⁄4

o

«

«

¬

a

=

=

=

=

=

=

=

= =

=

= =

=

=

= =

=

s s s s

C s s

s

M 1C

M 2

M 1

M 2

M 1

M 2

M 1

M 2 M 1,1C

M 1

M 1 M 1

M 2

M 1

M 2 M 1,1C

M 1

ˆ , ˆ

ˆ ˆ ˆˆ ˆ ˆˆ ˆ ˆˆ ˆ

ˆ ˆ ˆ

ˆ

ˆ

ˆ

t

t

t

C

t

t

K I 0 V

t

t

V

t

t

V

K t t

t

t

t

t

 (61)
46 USDA Forest Service RMRS-GTR-239. 2010.

Equations 58 to 61 may be expanded, as in Eqs. 20 to 27, to reveal analogies 

between RRE and more traditional univariate regression estimators (e.g., Särndal 

and others 1992).

The example in Eq. 61 uses the M = 1C photo-interpretation protocol, which 

is identical to the M = 1 protocol applied to sample point (1, j), except that the 

M = 1C protocol is the mean of the nj

 point samples within the large supportregion (i.e., cluster plot). In addition, other remote sensing protocols for measuring the large support-region are possible. Examples include MODIS satellite data 

(Frescino and others 2009b), with 15- to 60-acre pixels; earth-orbiting satellite 

Light Detection and Ranging (LiDAR) laser data; and airborne LiDAR data. If 

these supplemental measurements from other sensors are correlated with photointerpretations, then they will improve the precision of estimated population totals 

for the photo-interpreted variables, which, in turn, will increase precision of the 

estimated population totals for the M = 2 FIA fi eld variables.

Two-Phase Sampling for Continuous Variables with 

Cluster Plots

The fi nal example considered here is relevant to the vision expressed in the FIA 

Strategic Plan (USDA Forest Service 2007). The following example uses fullcoverage pixel data from a spaceborne sensor (protocol M = 0); a large sample 

of relatively inexpensive, 50-acre cluster plots measured with the M = 1 photointerpretation protocol (in addition to the M = 0 protocol); and a small sample of 

50-acre cluster plots that are fully measured with the M = 1 photo-interpretation 

protocol with a single 1/6-acre FIA plot within the 50-acre cluster plot measured 

with the M = 2 FIA fi eld protocol. Spaceborne sensor data (M = 0 protocol) exists 

for the small sample of cluster plots that contain an FIA fi eld plot; however, those 

M = 0 data are not used so as to reduce state-vector dimensions, avoid numerical 

problems, and avoid sampling zeros. This is similar to the non-optimal, but more 

robust, approach used in the example of “Three-Phase Sampling for Continuous 

Variables”.

The following example is set in the Interior West and interior portions of Alaska 

in the United States where, in the absence of major disturbance events, ambient 

stand dynamics (tree growth, regeneration, and mortality) are relatively slow and 

predictable. These changes can only be reliably measured with fi eld observations, 

such as the FIA fi eld protocol, although photo-interpretation of time-series of 

high-resolution aerial photography can provide valuable auxiliary data. However, 

rapid and substantial changes in forest conditions over large areas of forest land 

are caused by land management treatments, changes in land use, and episodic tree 

mortality from insects, diseases, wildfi res and other agents. These disturbancerelated changes can be reliably, although not perfectly, measured with interpretation of multitemporal, high-resolution aerial photography that is accurately 

registered to a large permanent photo-plot (e.g., a 50-acre cluster plot). This situation is the motivation for NPIP (Frescino and others 2009a; Frescino and others 

2009b), which is led by the Interior West Forest Inventory and Analysis program. 

The FIA study by Lister and others (2009) has some similarities. The following 

example uses inexpensive large photo-plots to improve statistical effi ciency with 

Landsat auxiliary data and increase the sample size of 50-acre photo-plots in order 

to improve estimates of change. Accurate and effi cient monitoring of changes in 

land cover and land use is a short-term priority for FIA research and development.
USDA Forest Service RMRS-GTR-239. 2010. 47

Assume each permanent, 1-acre FIA fi eld plot, with sampling intensity of 

1 plot per 6000 acres, is circumscribed by 50-acre cluster plot. Each cluster 

plot is measured through photo-interpretation (M = 1) of a systematic sample of 

49 points. Every year, an interpenetrating, 15-percent sub-sample of these cluster 

plots is imaged with high-resolution aerial photography and measured with the 

M = 1 protocol. Each year, an independent, 5-percent sub-sample is measured 

with both the FIA fi eld protocol (M = 2) and the M = 1 protocol with new, highresolution aerial photography. This sums to 20 percent of all FIA plots measured 

each year, which fully complies with the authorities legislated in 1998 Farm Bill 

(USDA Forest Service 1998). In this hypothetical example, every FIA fi eld plot, 

forested and otherwise, is measured once every fi ve years with high-resolution 

aerial photography (M = 1 protocol), although every forested plot is measured 

only once every 20 years by FIA fi eld crews (M = 2 protocol). For purposes of 

estimation with new data from a single year, the Phase 1 sample includes that 

15 percent of all sample points that are imaged during the year with aerial photography alone and measured with the M = 1 protocol, and the Phase 2 sample 

includes that 5 percent of all sample points measured during that same year with 

both interpretation of aerial photography (M = 1 protocol) and FIA fi eld crews 

(M = 2 protocol).

Each sample point j in the Phase 1 and Phase 2 samples is associated with a 

unique county and ownership category, and the resulting vector of binary indicator variables is denoted [yM=0]i=1,, where the i = 1 subscript signifi es point 1 of the 

49 points in the 50-acre cluster plot, which is located at the center-point of the 

1/6-acre FIA fi eld plot and the center of the 50-acre cluster plot. Finally, multispectral measurements from spaceborne sensors and other full-coverage geospatial 

census data are used to predict vegetation characteristics for each pixel that covers 

the sampled population (M = 0). These pixels may be registered to each of the 49 

individual points in the 50-acre cluster plot, or summaries of the pixel values for 

that cluster plot.

The sample of large photo-plots in this example offers additional opportunities 

to improve empirical models that predict forest characteristics using full-coverage, remotely sensed data from earth-observing satellites. Compared to 1/6-acre 

FIA fi eld plots, large photo-plots are more accurately registered to spaceborne 

sensors, such as those aboard the Landsat and Aqua earth-orbiting satellites. The 

spatial scales of 30-m (0.2-acre) or 250-m (16-acre) satellite pixels are more compatible with 50-acre photo-plots than the cluster of four 1/24-acre FIA subplots in 

the FIA fi eld plot. Compatibility of scale affects success of prediction models that 

use remotely sensed data (Xu and others 2009). Accurate registration between satellite imagery and training data is important for fi tting accurate prediction models 

(Pontius 2000; Verbyla and Boles 2000; Carmel and others 2001; Halme and 

Tomppo 2001; Czaplewski and Patterson 2003; Czaplewski 2005; Ashok and others 2007; McRoberts 2010). For example, Frescino and others (2009b) improved 

predictions of forest characteristics, at least to a modest degree, with 15-acre 

MODIS pixels by fi tting models with photo-interpretation of 50-acre photo-plots 

in Nevada rather than measurements of FIA fi eld plots. Registration accuracy also 

affects the strength of the cross-covariances among variables measured with different fi eld and remote sensing protocols, hence, the gains in statistical effi ciency 

that are possible with complex sampling designs. Therefore, a modifi ed M = 1C 

protocol is defi ned as the aggregate sum of M = 1 photo-interpreted measurements 

for each of the 49 points in the 50-acre photo-plot.
48 USDA Forest Service RMRS-GTR-239. 2010.

Sequential application of RRE provides a suitable estimator for this complex 

example. First, compute the Phase 1 sample estimator as:

[ ]

[ ] ( )

( ) [ ] ( ) ( )

¦

¦ ¦

∈ = = =

=

=

=

= =

=

=

=

= =

= =

∈ =

=

= =

= =

= = =

=

= =

=

′

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a

» −

1⁄4

o «

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a

» −

1⁄4

o «

¬

a

− =

»

»

»

1⁄4

o

«

«

«

¬

a

=  ̧

 ̧

1

·

 ̈

 ̈

©

§ =

»

»

1⁄4

o

«

«

¬

a =

»

»

1⁄4

o

«

«

¬

a

1 M 1C 1

M 0

M 1C

M 0

M 1C 1

M 0

M 1C

M 0

1 1

2

M 1C 1

1

1

M 1C M 1 ,

M 0 1

1 M 1C 1

M 0

M 1C 1

M 0

ˆ

ˆ

ˆ

ˆ

1

ˆ ˆ

1

ˆ

ˆ

ˆ

ˆ

j s j s j s s s

s

j s

j

j

n

i

i j

i

s s s

n n

A

n n

A A j

y

y

y

y

y

y

y

y V t

y y

y

y

y

t

t

 (62)

This requires registration of 30-m Landsat pixels to the entire 50-acre photo-plot 

but not any of the 49 individual, photo-interpreted points. Measurements of the 

250 Landsat pixels that cover the 50-acre photo-plot are summed into a single 

M = 0 measurement for each PSU. Likewise, photo-interpreted measurements for 

each of the 49 points in a 50-acre photo-plot are summed to a single measurement 

(M = 1C) of the entire photo-plot. Recall that this Phase 1 estimate uses the annual 

15 percent interpenetrating sub-sample of all FIA plots.

Second, compute the Phase 2 sample estimator from the 5 percent annual 

sub-sample:

 

[ ]

[ ]

[ ]

[ ]

( )

¦ ¦ ∈ =

= =

= =

=

= =

= =

=

= =

=

=

=

= =

=

=

=

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

=

 ̧

 ̧

1

·

 ̈

 ̈

©

§ =

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

=

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

2

M 2 1

M 1 1

1

M 1C M 1 ,

M 0 1

2

M 2 2

M 1

M 1C

M 0

M 2 2

M 1

M 1C

M 0

ˆ 1

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

j s

j i

i

j

n

i

i j

i

s

s s

n

n

A A

j

y

y

y y

y

y

y

y

y

t

t

t

t

( )

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

( )

¦∈ =

= =

=

=

=

= =

= =

=

= =

= =

=

=

=

= =

= =

=

= =

= =

= =

=

=

=

′

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

−

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

−

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

− =

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

2

M 2 2

M 1

M 1C

M 0

M 2 1

M 1 1

M 1C

M 0 1

M 2 2

M 1

M 1C

M 0

M 2 1

M 1 1

M 1C

M 0 1

2 2

2

2 M 2

M 1

M 1C

M 0

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

1

ˆ

ˆ

ˆ

ˆ

ˆ

j s

j s i

i

i

j s i

i

i

s s

s

n n

A

y

y

y

y

y

y

y

y

y

y

y

y

y

y

y

y

t

t

t

t

V

 (63)

This estimate includes the M = 0 and M = 1C measurements for the entire 50-

acre photo-plot, the M = 1 photo-interpretations for each of the 49 points in each 

50-acre photo-plot, and the M = 2 measurement of the 1/6-acre FIA fi eld plot at 

the center of the 50-acre photo-plot.
USDA Forest Service RMRS-GTR-239. 2010. 49

Third, apply the composite estimator that combines the vector estimates from 

the Phase 1 sample (Eq. 62) and the Phase 2 sample (Eq. 63). The primary purpose 

is to improve the estimated population totals for the photo-interpreted (M = 1) 

variables and the geopolitical areal variables (M = 0) by combining the Phase 1 

and Phase 2 sample estimators, although this will also improve estimates of those 

FIA fi eld variables (M = 2) that are correlated with remotely sensed and geopolitical variables. The notation corresponding to the Kalman fi lter in Eq. 12 is:

( ) ( ) ( )

( ) ( ) ( )

[ ] () () H I 0 C( ) t t 0

t

t

V t V

t

t

t

t

t

t

t

V t V

t

t

t

t

t

= =

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a =

»

»

1⁄4

o

«

«

¬

a =

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

=

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

=

= =

=

= =

=

= =

=

=

=

= =

=

=

=

1 1

1 M 1C

M 0

1

M 1C 1

M 0

1

2 M 2

M 1

M 1C

M 0

1

M 2 2

M 1

M 1C

M 0

1

ˆ, ˆ ˆ

ˆ

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ

ˆ ˆ ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

X Z

s

Z

s

Z

s

X

s

X

 (64)

The H matrix has dimensions (qM=0 + qM=1)-by-(qM=0 + 2×qM=1 + qM=2). From 

Eqs. 12 and 64, the composite estimator is:

 ( )

[ ]

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

»

»

»

»

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1⁄4

o

«

«

«

«

«

¬

a

» −

1⁄4

o «

¬

a

+

»

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o

«

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«

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«

¬

a

=

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»

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1⁄4

o

«

«

«

«

«

¬

a

= =

=

=

=

= =

=

= =

=

=

=

= =

=

=

=

M 2 2

M 1

M 1C

M 0

M 1C 1

M 0

M 2 2

M 1

M 1C

M 0

M 2 1,2

M 1

M 1C

M 0

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

s

s

s s t

t

t

t

I 0

t

t K

t

t

t

t

t

t

t

t

 ( )

[ ]

2 M 2

M 1

M 1C

M 0

2 M 2

M 1

M 1C

M 0

1,2 M 2

M 1

M 1C

M 0

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

= =

=

=

=

= =

=

=

=

= =

=

=

=

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

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−

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a

=

 ̧

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1

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©

§

»

»

»

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1⁄4

o

«

«

«

«

«

¬

a

s s s t

t

t

t

K I 0 V

t

t

t

t

V

t

t

t

t

V

[ ]

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

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§

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«

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a

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1

·

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¬

a

+»

1⁄4

o

«

¬

a

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1

·

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©

§

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«

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a

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o

«

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a

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1

·

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©

§

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

=

−

= =

=

= =

=

=

=

= =

=

=

=

1

1 M 1C

M 0

2 M 2

M 1

M 1C

M 0

2 M 2

M 1

M 1C

M 0

ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

s

s s

t

t

V

0

I

t

t

t

t

I 0 V

0

I

t

t

t

t

K V

 (65)
50 USDA Forest Service RMRS-GTR-239. 2010.

Fourth, apply the estimator for cluster plots, similar to Eq. 61, except use 

the composite estimate from equation Eq. 65 rather than the sample estimate 

of the same vector of population totals from Eq. 58. The purpose is to improve 

estimat ed population totals for photo-interpreted variables by combining the photointerpreted measurements made at the sample point (M = 1, i = 1) and the mean 

of the photo-interpreted measurements made for the surrounding cluster plot 

(M = 1C, i = 1,2,...,nj

), where nj = 49 in this example. The notation corresponding 

to the Kalman fi lter in Eq. 12 is:

 

 

 

   

 

M 0 M 0

1 M 1 1 M 1

M 2 M 2

1,2 1,2

M 0 M 0

M 1C M 1C

M 1 M 1

M 2 M 2 1,2

ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ

 

 

 

 

 

 

 

a o § · a o

« »  ̈  ̧ « » « »  ̈  ̧ « »

« »  ̈  ̧ « » ¬1⁄4 © 1 ¬ 1⁄4

a o a a o « » a o « « » « » « » « « » « » « » « « » « » « » « ¬ 1⁄4 ¬ 1⁄4 « » « ¬ 1⁄4 ¬

X X

s s

s

t t

t t Vt V t

t t

t t

I000 I000 t t

00I0 0 0 I 0V t t

000I 000I t t

 

   

> @

 

       

> @

1,2

1 M 1C 1 M 1C 1,2 1,2

M 0 M 0

M 1C M 1C

M 1 M 1

M 2 M 2 1,2 1

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ

 

 

 

 

 

 

§ ·o a o  ̈  ̧» « »  ̈  ̧» « »  ̈  ̧» « »  ̈  ̧» « »  ̈  ̧ © 1»1⁄4 « » ¬ 1⁄4

 ao a o ¬ 1⁄4 ¬ 1⁄4

a o § · a o « »  ̈  ̧ « » « »  ̈  ̧ « » « »  ̈  ̧ « » « »  ̈  ̧ « »  ̈  ̧ « » ¬ 1⁄4 « » © 1 ¬ 1⁄4

s

Z Z s s

s s

I00

000

0I0

00I

t t Vt V t

t t

t t 0I00 0 I 0 0V t t

t t

 

> @

   

 

 

,2

M 0

1 1 M 1 M 1C

M 2

1,2

M 0

M 1C

M 1

M 2

1,2

ˆ

ˆ ˆ ˆˆ ˆ ˆ , ,

ˆ

ˆ

ˆ ˆ ˆ

ˆ

 

 

 

 

 

 

 

a o

« »

« »

« »

« » « » ¬ 1⁄4

§ · a o  ̈  ̧ « » a o c  ̈  ̧ « » ¬ 1⁄4  ̈  ̧ « » © 1 ¬ 1⁄4

 § · a o a o a o  ̈  ̧ « » « » « »  ̈  ̧ « » « » « »  ̈  ̧ « » « » « »  ̈  ̧ « » « » ¬ 1⁄4  ̈  ̧ © 1 « » ¬ 1⁄4 « » ¬ 1⁄4

X Z

s

s

0

I

0

0

t

Ct t C t t

t

H 0I0 t 0

I000

t I

0 0 I 0V

t 0

000I

t 0

 (66)
USDA Forest Service RMRS-GTR-239. 2010. 51

The H matrix in Eq. 66 has dimensions qM=1-by-( qM=0 + qM=1 + qM=2). From Eqs. 9 

through 11 and Eq. 66, the composite estimator is:

 ( )

( ) ( )

[ ] ( )

( )»

»

»

»

1⁄4

o

«

«

«

«

¬

a

»

»

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1⁄4

o

«

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a

+ −

»

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o

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a

=

»

»

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1⁄4

o

«

«

«

¬

a

= =

=

=

= =

= =

=

=

= = =

=

=

1,2 M 2

M 1

M 0

M 1 1,2

1,2 M 2

M 1

M 0

M 1,1C

M 2 1,2

M 1

M 0

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

s

C s

s s t

t

t

K t H

t

t

t

t

t

t

( )

( ) ( ) ( )

[ ]

( )»

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=

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=

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= = =

=

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1,2

M 1C

M 2

M 1

M 0

1,2 M 2

M 1

M 0

1,2 M 2

M 1

M 0

M 1,1C

M 2 1,2

M 1

M 0

ˆ ,

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

s s s s

t

t

t

t

C

t

t

t

K H V

t

t

t

V

t

t

t

V

 ( )

[ ]

( )

( )

( ) [ ] ( )

[ ]

( )

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( )  ̧

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1

·

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«

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a

=

−

=

=

=

=

=

=

=

=

=

=

= =

= =

=

=

=

=

=

=

=

= =

=

=

1

1,2

M 2

M 1

M 0

M 1C

1,2

M 1C

M 2

M 1

M 0

M 1C 1,2

1,2 M 2

M 1

M 0

1,2

M 1C

M 2

M 1

M 0

1,2 M 2

M 1

M 0

ˆ

ˆ

ˆ

, ˆ ˆ

ˆ ,

ˆ

ˆ

ˆ

ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ

ˆ ,

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

H

t

t

t

C t

t

t

t

t

H C

V t

H

t

t

t

H V

t

t

t

t

C

H

t

t

t

V

K

s

s

s

s

s

s

 

 (67)

Numerically reliable algorithms must be used to apply Eq. 67. These methods are 

briefl y covered in the section “Robust Numerics”.
52 USDA Forest Service RMRS-GTR-239. 2010.

Fifth, use RRE to constrain the vector estimate of population totals for areal 

variables to exactly match the census totals for geopolitical variables (e.g., area 

by county and ownership type), which are exactly known from administrative 

records (Särndal and others 1992:230). The notation corresponding to the Kalman 

fi lter in Eq. 12 is:

( )

( )

[ ]

( ) ( )

( )

[ ]

( ) [ ] ( ) ( )

[ ] () () H I 0 C( ) t t 0

t t V t 0

t

t

t

V t V

t

t

t

t

= =

= =

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

=

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

=

= =

= = =

=

=

= = =

=

=

1 1

1 M 0 1

M 1,1C

M 2 1,2

M 1

M 0

1

M 1,1C

M 2 1,2

M 1

M 0

1

ˆ, ˆ ˆ

ˆ ˆ ˆ

ˆ

ˆ

ˆ

ˆ ˆ ˆ

ˆ

ˆ

ˆ

ˆ

X Z

Z s U Z

s

X

s

X

 (68)

The H matrix in Eq. 68 has dimensions qM=0-by-(qM=0 + qM=1 + qM=2). From Eqs. 12 

and 68, the composite estimator is:

 

( )

[ ] ( )

[ ]

[ ]

( )

[ ]»

»

»

»

1⁄4

o

«

«

«

«

¬

a

»

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»

1⁄4

o

«

«

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¬

a

+ −

»

»

»

1⁄4

o

«

«

«

¬

a

=

»

»

»

1⁄4

o

«

«

«

¬

a

= = =

=

=

= =

= = =

=

=

=

= =

=

=

M 1,1C

M 2 1,2

M 1

M 0

M 0

M 1,1C

M 2 1,2

M 1

M 0

M 1,1C

,1,2 M 2

M 1

M 0

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

s

s U

U s s t

t

t

K t H

t

t

t

t

t

t

( )

[ ] ( )[ ] ( )M [ ] 1,1C

M 2 1,2

M 1

M 0

M 1,1C

M 2 1,2

M 1

M 0

M 1,1C

M 2 ,1,2

M 1

M 0

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

=

= =

=

=

=

= =

=

=

=

= =

=

=

 ̧

 ̧

 ̧

 ̧

1

·

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§

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o

«

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a

−

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1

·

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o

«

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a

=

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1

·

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1⁄4

o

«

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«

¬

a

s U s s t

t

t

K H V

t

t

t

V

t

t

t

V

 

( )

[ ] ( )

[ ]  ̧

 ̧

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1

·

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§

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«

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′

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′

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§

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o

«

«

«

¬

a

=

−

= = =

=

=

= = =

=

=

1

M 1,1C

M 2 1,2

M 1

M 0

M 1,1C

1,2 M 2

M 1

M 0

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ ˆ H

t

t

t

H H V

t

t

t

K V

s s

 (69)

Equations 62 to 69 may be expanded, as in Eqs. 20 to 27, to reveal analogies 

between RRE and more traditional univariateregression estimators (e.g., Särndal 

and others 1992).

Numerically reliable algorithms must be used in these various steps. These 

methods are briefl y covered in Eqs. 31 to 37 for a census of auxiliary data and, 

more generally, in the section “Robust Numerics”.

Finally, extract the matrix partitions in Eq. 69 that represent the estimated population totals for the variables measured with the FIA fi eld protocol M = 2.
USDA Forest Service RMRS-GTR-239. 2010. 53

 

[ ] ( )

[ ] [ ]

( )

M [ ] 1,1C

M 2 ,1,2

M 1

M 0

M 1,1C

,1,2 M 2

ˆ

ˆ

ˆ

ˆ

= = =

=

=

=

= =

»

»

»

1⁄4

o

«

«

«

¬

a

=

s U

s U

t

t

t

t 0 0 I 

( ) ( )

[ ] [ ]

( )

[ ]

»

»

»

1⁄4

o

«

«

«

¬

a

′

′

 ̧

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1

·

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©

§

»

»

»

1⁄4

o

«

«

«

¬

a

=

=

= =

=

=

=

= =

I

0

0

t

t

t

V t 0 0 I V

M 1,1C

M 2 ,1,2

M 1

M 0

M 1,1C

,1,2 M 2

ˆ

ˆ

ˆ

ˆ ˆ ˆ

s U

s U

 (70)

This example actually incorporates dual, two-phase sampling designs. One 

is the Phase 1 sample of cluster plots that are measured solely with photointerpretation (M = 1C) and classifi ed by county and ownership type (M = 0), 

which is given in Eq. 62. Phase 1 in the second two-phase design uses the administrative records for area by county and ownership type (M = 0), which is 

considered an exact census. Because of Eq. 69, the sum of all areal variables (e.g., 

area by land cover and forest type) in Eq. 70 within an ownership-type and within 

a county will exactly equal the administrative census statistics for that land area. 

Sequential application of RRE permits designs that use a variety of overlapping 

sampling phases and stages and different types of collocated support-regions that 

are treated as cluster plots.

The section titled “Multivariate Vector of Auxiliary Variables” describes the 

numerous types of remotely sensed auxiliary data that may be accommodated 

in this type of complex design with cluster plots, sub-sampling, and censuses of 

the sampled population. The next section covers some mathematical features of 

those auxiliary data that are part of a classifi cation system of categorical discrete 

variables.

Categorical Variables

All estimators considered above assume the predictor variables are continuous, such as remotely sensed estimates of standing wood volume. However, 

remote sensing variables may be discrete, such as predictions of forest type and 

la nd cover type categories. In addition, many FIA variables are discrete, such as 

categories of various forest conditions. The units of measure on these discrete 

variables are generally “proportion of the population,” which ranges between 0 

and 1. Equivalently, the units might be hectares or acres, which range between 0 

and the total area of the population.

The estimators above can directly accommodate any categorical variable as an 

auxiliary predictor variable, a population variable, or both. Any polychotomous 

categorical variable with c categories may be transformed into c binary dichotomous variables (e.g., de Gruijter and others 2006:69; Knottnerus 2003:130; Zhang 

2000). Within the estimators already presented, these dichotomous variables may 

be included as sub-vectors, along with continuous variables in the yM=0’ vector. 

Likewise, CM=2’ categories in the FIA fi eld protocol M = 2’ may be converted 

into CM=2’ dichotomous variables, and these may be inserted as sub-vectors in the 

yM=2’ measurement vector. The resulting RRE will retain their minimum variance 

property. However, the statistical effi ciency of estimates might be improved by 

capturing the CM=0’×CM=2’ cross-classifi cation of categorical protocols M = 0’ and 

M = 2’. Otherwise, the estimators above only include the CM=0’ + CM=2’ margins of 

this cross-classifi cation in yM=0’ and yM=2’. This enhancement is considered here.
54 USDA Forest Service RMRS-GTR-239. 2010.

For an FIA variable and a remotely sensed variable that are both polychotomous categorical variables, a contingency table can be built that includes statistics 

for the cross-classifi cation by these two protocols. If both categorical variables 

have equivalent classifi cation systems, then CM=0’ = CM=2’, and the resulting square 

C M=0’

×CM=2’ contingency is directly related to an error matrix or confusion matrix 

in the remote sensing literature (Story and Congalton 1986).

In contrast, the classifi cation systems for the M = 0’ (e.g., spaceborne remote 

sensing) and M = 2’ (e.g., FIA) protocols can differ in the number of categories and their defi nitions. The cross-classifi cation with both protocols yields a CM=0’×CM=2’ contingency table. Gallego and Bamps (2008) use the term “fi ne scale 

profi les” for the resulting rectangular contingency table to avoid misinterpretation 

as an error or confusion matrix in the remote sensing lexicon. Card (1982) and 

Czaplewski and Catts (1992) demonstrate that such a contingency table is the basis for multivariate inverse calibration (Tenenbein 1972; Brown 1982; Czaplewski 

and Catts 1992) or direct calibration (Dymond 1992; Gallego and Bamps 2008).

It is possible to imbed this contingency table within any of the above estimators 

for complex sampling designs. Joint classifi cation probabilities are estimated by 

augmenting the M = 2’ measurement vector with the cross-classifi cation with both 

the remotely sensed (M = 0’) and fi eld (M = 2’) protocols. This simply requires restructuring the CM=0’×CM=2’ elements in the cross-classifi cation contingency table 

as the (CM=0’×CM=2’)-by-1 sub-vector of [yM=2’]

i

 for sample unit i in Eqs. 20, 45, 56, 

58, or 63.

Czaplewski (in prep.[b]) provides a detailed example of this application, in 

which census statistics from full-coverage, remotely sensed data serve as constraints (Simon and Chia 2002). He offers solutions to numerical round-off 

errors that can cause unreliable results. There are also well-developed alternatives to numerical hazards, such as U-D factorization (Bierman 1977; Maybeck 

1979:Chapter 7). Numerical problems are not always apparent from the fi nal 

results, and numerical errors can exceed random estimation errors (Bierman 

1977), which obviates the gains in statistical effi ciency with the Kalman fi lter. 

It is strongly recommended that the numerically robust methods presented by 

Maybeck and Bierman be used as standard procedure.

The inclusion of the contingency table for all cross-classifi cations can create 

a large number of state variables. For example, if there are 12 categories in the fi eld protocol and 17 categories in the remote sensing protocol, 204 variables 

must be estimated. However, only one margin of the contingency table might be 

required for FIA statistical reports, such as the estimated areas for each of the 12 

categories classifi ed with the fi eld protocol. The remaining 192 variables are a 

nuisance in some sense. The Schmidt-Kalman fi lter is an alternative (Jazwinski 

1970; Grewal and Andrews 2001). The Schmidt-Kalman fi lter does not specifi -

cally estimate the nuisance variables, thus reducing the dimensions of the vector 

estimates. However, this advantage is gained at a cost. The solution is not necessarily optimal, and estimates of the desired state variables might not be as effi cient 

as a fi lter that directly estimates all nuisance variables. In the special case of multiphase sampling with a homogeneous PSU where each PSU can be classifi ed into 

one and only one category in each classifi cation system, the margins of the contingency table are suffi cient statistics for constrained RRE (Czaplewski in prep.[b]).

There are well-developed alternatives available that apply constraints upon the 

Kalman fi lter solution (Doran 1997; Simon and Chia 2002). See the thorough 

and modern reviews by Gupta and Hauser (2007) and Gupta (2008) for linear 

equality constraints that might be imposed so that the estimated total for the remotely sensed categorical variables exactly equals the census totals. Also, it is 

possible with numerous rare categories that some estimated population totals will 

be negative. The likely cause is application of classifi cation systems that are too 

detailed relative to the sample size. One pragmatic solution is to collapse the 

detail of a hierarchical classifi cation system to be more realistic given limited 

sample sizes. Another alternative is application of inequality constraints that force 
USDA Forest Service RMRS-GTR-239. 2010. 55

relevant estimated state variables to be non-negative. Doran (1997), Simon and 

Chia (2002), Knottnerus (2003:352), Gupta and Hauser (2007) and Gupta (2008) 

provide a useful introduction to these estimators.

Robust Numerics

As repeatedly discussed above, numerical precision of the multivariate estimator merits special attention. Numerical precision is a function of the word length 

for a digital computer, and it is often referred to as round-off error. The subtraction operator, when applied to two large numbers or matrices, is especially prone 

to signifi cant round-off error. The vector residual in Eq. 10 is an example. Also, 

matrix inversion is problematic for an ill-conditioned covariance matrix, as in Eq. 

9 for the gain matrix K. The covariance matrix in Eq. 9 will be ill-conditioned 

whenever sub-sets of auxiliary variables are highly codependent. The dichotomous transformation of a polychotomous, remotely sensed categorical variable is 

a common example.

For example, Knottnerus (2003:236) found that small errors in decomposition 

of the Horvitz-Thompson estimator can invoke large errors in variance estimates. 

Estevao and Särndal (2004:657) warn that the calibration estimator is numerically 

unstable, especially for designs more complex than Simple Random Sampling. 

Magnussen (2003) found that RRE, when applied to three-phase sampling of 

categorical variables, is not necessarily robust and consistent compared with a 

sub-optimal sequential estimator. Magnussen posited that numerical stability can 

be improved if multivariate dimensions are decreased by sequentially considering 

only those variables shared by two levels at one time in the multilevel sampling 

design. For example, the three-phase estimator in Eqs. 44 to 51 requires covariance matrices with dimensions of (qM=0 + qM=1 + qM=2)-by-(qM=0 + qM=1 + qM=2) at 

all sequential steps, where qM=i denotes the number of variables estimated with 

protocol M = i. The alternative three-phase estimator in Eqs. 52 to 56 involves 

dimensions of (qM=0 + qM=1)-by-(qM=0 + qM=1) at one step and (qM=1 + qM=2)-by-(qM=1 

+ qM=2) at another step. This requires modest sacrifi ce of statistical effi ciency 

because it ignores the M = 0 measurements available for the small Phase 3 subsample, but the sacrifi ce is rewarded with improved robustness and consistency. 

This same strategy applies to any sampling design that employs three or more 

phases, stages, or measurement protocols within a cluster plot.

However, an additional solution is possible. Numerical issues with the multivariate Kalman fi lter are well known and extensively studied as components of 

the divergence problem. Bierman (1977) devotes an entire book to the subject. 

Maybeck’s (1979) Chapter 7 is a succinct and thorough coverage of the subject. 

Comprehensive references on the applied Kalman fi lter all deal with this issue 

(e.g., Grewal and Andrews 2001; Bar-Shalom and others 2001; Chui and Chen 

2009). The reliability of the Kalman fi lter is also vulnerable to spurious observations, outliers, or inaccurate models (Guttman and Lin 1995).

For example, inversion of a qM=1-by-qM=1 covariance matrix is required in Eq. 

26. The dimensions of this covariance matrix may become large with three or 

more sampling phases or stages; detailed, remotely sensed census data; and areal 

constraints based on geopolitical domains. (Dimensions required in this matrix inversion are unaffected by the dimensions of the variables measured with the M = 2 

protocol, i.e., FIA fi eld data.) More seriously, the covariance matrix for auxiliary 

variables can be singular, making the matrix inversion infeasible (Särndal and 

others 1992:241). In especially pathological cases, the magnitude of numerical 

round-off errors can become large relative to the magnitude of random estimation 

errors (Bierman 1977:1).

Bierman (1977:22) cites one ad hoc method to detect signifi cant numerical errors: solve the system of matrix equations with single-precision numerics 

(e.g., 32-bit), and then solve it again with double-precision (e.g., 64-bit). If the 
56 USDA Forest Service RMRS-GTR-239. 2010.

differences in the vector results are of pragmatic signifi cance, then numerical errors are a problem that must be addressed.

In the early days of Kalman fi ltering, when 8-bit computers were state-of-theart, numerous solutions to these numerical challenges were devised. The square 

root transformation of a covariance matrix has about twice the numerical precision compared to matrix operations with untransformed covariance matrices 

(Bierman 1977:91), which may be further improved with more numerically 

precise digital computers (e.g., double-precision with 64-bit processors versus 

8-bit) and orthogonal transformations such as the Cholesky decomposition, the 

Householder orthogonal transformation, the Givens orthogonal transformation, 

and the Gram-Schmidt orthogonalization. One especially effective solution is the 

U-D factorization method, which is a modifi ed Gram-Schmidt orthogonalization (Bierman 1977:148). Stengel (1986:357), Grewal and Andrews (2001) and 

Bar-Shalom and others (2001) also discuss U-D factorization and its derivation. 

The Square Root Information Filter (SRIF) is another alternative that surpasses the U-D factorization in some applications (Bierman 1977:69). Knottnerus 

(2003:281) shows the relationship between the numerically stable Gram-Schmidt 

orthogonalization and augmented regression (Blundell and Robin 1999) in recursive applications.

Most Kalman fi ltering algorithms based on these numerically robust methods 

orthogonalize the vector of auxiliary population totals with a linear transformation, such as the inverse square root of the corresponding covariance matrix. The 

resulting vector of transformed auxiliary statistics for the population is expected 

to have a unit covariance matrix, meaning each element of the transformed vector 

has zero covariance with all other elements of the transformed vector. Therefore, 

each element of the transformed auxiliary vector may be separately applied to 

improve precision of estimated population totals for the FIA study variables. The fi nal vector estimate incorporates all elements of the transformed auxiliary vector 

after all elements are processed in this sequence of recursions.

Boggs and others (1995) note that these types of algorithms are 

not very widely used, because of the erroneous perception that 

these factorization techniques are too complicated compared to the 

conventional (Kalman fi lter), use too much computer storage, and 

involve too much computation. This (incorrect) perception is due 

in large part: (1) to an incomplete understanding ... of algorithms 

that are heavily dependent on advanced numerical analysis (Golub 

and Van Loan 1996, Lawson and Hanson 1974), and (2) on the use 

of ineffi cient computer ... implementations (to process state-vectors 

with) ... up to a few thousand variables (Jiang and Ghil 1993).

Boggs and others (1995) expect that Bierman’s SRIF offers even further improvements. Furthermore, these methods are amenable to implementation in database 

software without the need for matrix algebra software. Implementation in corporate database software is convenient for a large production statistical program 

such as FIA (McRoberts and others 2004).

In unpublished Monte Carlo simulations by the author, the U-D factorization 

has been fast and very successful in the numerical sense, with covariance matrices 

having qM=1 = 200 remotely sensed and geopolitical census variables, even without specialized matrix algebra software. This confi rms that even larger problems 

might be numerically feasible and robust. Czaplewski (in prep.[b]) presents other 

numerically robust yet simple methods with auxiliary census variables.

Czaplewski (in prep.[a]) applies the Kalman fi lter as a time-series estimator 

for FIA panel data (Patterson and Reams 2005). The multivariate innovation sequence is associated with a time-series of covariance matrices. These must be 

inverted to compute the multivariate weights (i.e., Kalman gain matrices) that 

combine the vector of expected observations with the realized observations. The 
USDA Forest Service RMRS-GTR-239. 2010. 57

covariance matrices are severely rank-defi cient, and many full-rank covariance 

matrices remain ill conditioned.

Simon and Chia (2002) provide stable methods suitable for auxiliary constants 

such as a census of pixels from full-coverage “wall-to-wall” spaceborne sensors 

(e.g., Landsat, SPOT, IRS, and MODIS); these methods apply to both continuous 

and categorical predictions from remotely sensed data. Some variables are nonnegative by defi nition (e.g., biomass must have a value greater than or equal to 

zero), but it is possible that the static Kalman fi lter can produce infeasible negative estimates, even with stable numerics. This problem is more common with 

rare attributes (e.g., forest clearings caused by small fi res or clearcuts). Doran 

(1997), Simon and Chia (2002), and Knottnerus (2003:379) offer relevant inequality constaints that can eliminate infeasible results.

Czaplewski (in prep.[a]) successfully implements the dynamic Kalman fi lter under these adverse conditions. He uses the LDL’ square root fi lter given by 

Bar-Shalom and others (2001:308-317). The LDL’ factorization uses the Linear 

Algebra PACKage (LAPACK) routine for the LU decomposition (Anderson 

and others 1999), which is a freely available FORTRAN 90 routine applicable 

to singular covariance matrices. Results from the LU decomposition are further 

factored into the LDL’ structure with methods given by Golub and Van Loan 

(1996: algorithm 4.1.2) for a symmetric nonsingular matrix. Any singular covariance matrix is pivoted so that the fi rst diagonal partition is nonsingular and well 

conditioned. The remaining partition of the covariance matrix represents nearly 

redundant information that is simply omitted from the Kalman update estimator. 

The pivoting algorithm retains any portion of the innovation covariance matrix 

directly corresponding to state-variables that are associated with auxiliary observations. The remainder of the pivot priorites use LAPACK output. The LDL’ 

square root fi lter requires orthogonalization of the observation matrix. This is 

done with the inverse square root of the covariance matrix for the auxiliary variables using Method 2 in Higham (2002:263).

The estimators given above, and many more related estimators that are envisioned for future research, are merely abstract matrix equations at this point. 

Important details such as numerical stability require further research and development before these methods can be confi dentially incorporated into a government 

statistics program.

Optimal PSU Expansion Values for Each FIA Plot

Scott and others (2005:65) review the historical use of expansion factors in 

FIA.

When periodic inventories and fl at fi les were the FIA standards, it 

was convenient to calculate a small set of expansion factors by which 

individual plot-level or tree-level observations could be converted 

to their population level equivalents. This allowed population 

totals to be obtained via summation, which greatly simplifi ed the 

estimation process. Expansion factors were popular with external 

FIA clients, many of whom used this concept to build their own 

processing systems. The tradeoff for such simplicity is that the use 

of expansion factors precludes the ability to calculate variances. At 

best, the variances of estimators derived from expanded values can 

only be approximated, and these approximations are known to be 

poor (Alegria and Scott 1991). Expansion factors are less practical 

with panelized inventory systems, which are designed to increase 

analytical fl exibility by allowing panels to be combined in a variety 

of ways. Each different panel combination produces a unique set 

of expansion factors, rendering expansion factors associated with 
58 USDA Forest Service RMRS-GTR-239. 2010.

panel systems less stable than those produced by periodic systems. 

The use of expansion factors is discouraged because they prohibit 

accurate variance estimation and they no longer have the advantage of 

simplicity. However, there is still a demand for them, and it may take 

a while to convert processing systems to the estimation procedures 

specifi ed in this chapter. Therefore, FIA will continue to offer 

expansion factors until a demand is no longer apparent.

Calibration weights (Deville and Särndal 1992) are similar to FIA plot expansion factors. Calibration weights are a modifi cation of the design weights 

(i.e., inverse of inclusion probability for each sampled PSU). The modifi cation 

incorporates external information available from the auxiliary data, such as the 

population census of predictions for remotely sensed pixels. Estevao and Särndal 

(2006) discuss calibration weights as a convenient means to improve statistical 

effi ciency with auxiliary information. FIA expansion factors serve the same role 

with post-stratifi cation. FIA uses the temporally indifferent estimator (Patterson 

and Reams 2005), which facilitates the use of expansion factors in FIA databases.

The following demonstrates how the results from RRE for complex designs 

may be expressed as expansion values, much like the expansion factors traditionally used by FIA and calibration weights used in other surveys. Future research 

will investigate dependable methods that use dynamic expansion values with the 

Kalman fi lter.

Let the equal-probability, infi nite-population, design-based, multivariate, 

single-phase estimator for the vector of auxiliary variables (x) and study variables 

(y) be the design-weighted sum of observations for each sampled unit k (i.e., FIA 

plot) in the equal-probability sample s(XY) (i.e., the set of sampled FIA plots):

( )

¦ ( )

¦

»

1⁄4

o «

¬

a » =

1⁄4

o «

¬

a » =

1⁄4

o «

¬

a

»

1⁄4

o «

¬

a » =

1⁄4

o «

¬

a

s XY

XY k

T

XY

T

Y XY

X

s XY

XY XY k

n

A A

n

y

x

y

x

t

t

y

x

y

x

ˆ

ˆ

ˆ

ˆ

1

ˆ

ˆ

 (71)

where n(XY) is the sample size of FIA plots in sample s(XY), A is the total acres in 

the sampled population, and both xk and yk have units of measure such that the 

values per unit area are consistent with the units of A (e.g., 1 acre for a singlecondition category in FIA plot k and volume per acre f or wood volume measurements in plot k). The estimated covariance matrix for the population vector 

estimate in Eq. 71 is defi ned as:

( ) ¦

»

»

»

1⁄4

o

«

«

«

¬

a ′

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a − »

1⁄4

o

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a − »

1⁄4

o

«

¬

a

− » =

1⁄4

o «

¬

a

s

k XY k XY XY XY

T

Y XY

X

n n

A

y

x

y

x

y

x

y

x

t

t V ˆ

ˆ

ˆ

ˆ

ˆ 1

ˆ ˆ 2

 (72)

Further, assume that comparable remotely sensed auxiliary data are available 

from a large independent probability sample s(Z):

( )

¦ ( ) = s Z i

Z

T

Z

n

A

t

ˆ x (73)

( ) ( )( )( ) ¦ ( ) »

1⁄4

o «

¬

a ′ − − − = s Z i Z i Z

Z Z

T

Z

n n

A V t x xˆ x xˆ

1

ˆ ˆ 2

 (74)
USDA Forest Service RMRS-GTR-239. 2010. 59

where X and Z share exactly the same measurement protocol, i.e., xi = zi. If the 

auxiliary data are available for a census of remotely sensed pixels for the entire 

 sampled population, then the population totals for the vector of auxiliary data are 

known exactly as without sampling or enumeration error, i.e., 

V[] 0 t t t

z = .

Expansion Values for FIA Population Estimates

Based on Eq. 16, the optimal RRE given this vector of auxiliary variables has 

the general structure:

RRE ( ) ˆ ˆ ˆˆ Y ZX Y XY XY

a o ao a o = +− ¬ 1⁄4 ¬1⁄4 ¬ 1⁄4 t t Wt t (75)

where the matrix of RRE optimal weights in Eq. 75 is defi ned as the qY-by-qX

matrix W:

[ ]

( ) () () [ ] »

»

1⁄4

o

«

«

¬

a ′

 ̧

1

·  ̈

©

§ = +

=

−1 ˆ ˆ ˆ ˆ ˆ , ˆ ˆ

Y X XY X XY Z C t t V t V t

W 0 I K

Substituting Eq. 71 into Eq. 75:

( ) ( )

( )

RRE

ˆ ˆ T T

kZ k Y s XY s XY

XY XY

k s XY

A A

n n

§ · a o = +−  ̈  ̧ ¬ 1⁄4 © 1

=

¦ ¦

¦

t y Wt x

f

 (76)

where the qY-by-1 vector “expansion value” fk for the kth PSU is:

»

»

1⁄4

o

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§ = + − k

T

Z

k

XY

T

k

n A

A

x

t f y W ˆ

By defi nition, the qY-by-1 vector sum of expansion values in Eq. 76 for all nXY

PSUs will exactly agree with t he optimal RE vector estimate for the sampled 

population, thus allowing generic inference (Opsomer and others 2007). The vector of expansion values for the kth PSU is the sum of the design-weighted qY

-by-1 

vector measurement of the kth PSU (AT

/nXY)yk plus the optimally weighted qX-by-1 

residual vector difference between the auxiliary data for the kth PSU and its corresponding population mean, namely .This is similar in 

structure to Knottnerus’ (2003:395) minimum variance weighting procedure for 

univariate study variables.

The vector of optimal expansion values fk for each sampling unit k in Eq. 76 

has different numerical properties than the scalar expansion factor (AT /n)yk in 

Eq. 71. Unlike the traditional FIA expansion factor, optimal expansion values 

for a sampling unit can have negative elements, although this can be rare or nonexistent. Furthermore, expansion values for forest characteristics (e.g., wood 

volume and forest type) may be non-zero for an FIA plot that is classifi ed as 

non-forest in the fi eld. FIA uses a single multiplicative volume expansion factor 

for all measurements associated with wood volume, such as volume for different 

tree species, tree sizes, and tree conditions. This includes current volume and 

components of volume change (e.g., growth, regeneration, mortality, removals). 

However, the second term in Eq. 76 is an additive adjustment, not a multiplicative 

adjustment. To emulate fully the traditional use of expansion factors by FIA, ad 
60 USDA Forest Service RMRS-GTR-239. 2010.

hoc procedures might be necessary to apply these optimal expansion values from 

RRE. For example, merchantable wood volume measurements across all tally 

trees within PSU k might be summed into a single element in the yk measurement 

vector for plot k. If so, then a single element of f

k

 is the expansion value that applies to volume measurements for all tally trees in sampling unit k, regardless of 

species, size, or condition. That value would need to be prorated to each category 

of wood volume measurement.

Expansion Values for FIA Variance Estimates

The previous section conveniently expressed RRE results as expansion vectors for each PSU. Likewise, the RRE covariance matrix for the population totals 

for both the auxiliary and study variables may be expressed as the sum of values 

for each FIA PSU (e.g., Knottnerus 2003:220, 236). This provides further conveniences in database procedures.

First, combine Eqs. 12 and 72. The resulting covariance matrix equals:

( ) ( ) ( ) ( )

( ) ( ) ( )

RRE

2

ˆ ˆ ˆ

ˆ ˆ

1 ˆ ˆ

XY

T

s

XY XY k k XY XY

A

n n

′ =− − a o ¬ 1⁄4

a o ′ § ·§ · ao ao ao ao « »′ = − − −−  ̈  ̧ ̈  ̧ «» «» «» «» − © 1© 1 ¬1⁄4 ¬1⁄4 «» «» ¬1⁄4 ¬1⁄4 « » ¬ 1⁄4

¦

V t I KH V t I KH

xx xx

I KH I KH

y y y y

 (77)

Extracting the partitions in Eq. 77 specifi c to the study variables Y and defi ning a 

new vector gk for the kth FIA fi eld plot:

( ) RRE RRE [ ] ( ) ˆ ˆ ˆ ˆ Y

k k s

a o = « » ¬ 1⁄4

≈ ¦ ′

0

Vt 0 IVt

I

g g

 (78)

where

[ ]( )  ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

− »

1⁄4

o

«

¬

a = −

k XY XY

T

k

n

A

y

x

y

x

g 0 I I KH ˆ

ˆ

and the sample size of FIA fi eld plots nXY is suffi ciently large such that (nXY-1)≈nXY. 

The vector gk for the kth FIA fi eld plot in Eq. 78 may be rearranged such that:

[ ] [ ][ ]

[ ] ( ) [ ] ( )

[ ] [ ] ( ) k XY k XY

XY

T

k XY k XY

XY

T

k XY k XY XY

T

k

n

A

n

A

n

A

y 0 I K x x y

y y 0 I K x x

y

x

y

x

0 I K 0 I

y

x

y

x

g 0 I

ˆ ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ

= + − −

= − + − +

»

»

1⁄4

o

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

− »

1⁄4

o «

¬

a −

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

− »

1⁄4

o «

¬

a =

 (79)
USDA Forest Service RMRS-GTR-239. 2010. 61

Substituting Eqs. 14 and 75 into Eq. 79, using Z AT t

ˆ as a more precise estimate of the population mean vector for the auxiliary variables than , and then 

combing with Eq. 76:

[ ] ( )

XY

XY

T

k

k XY k XY

XY

T

k

n

A

n

A

f y

g y W x x y

ˆ

ˆ ˆ

= −

= + − −

 (80)

Replacing Eq. 80 into Eq. 78:

( ) RRE ˆ ˆ ˆ ˆ

ˆ ˆ

T T

s k XY k XY Y

XY XY

Y Y XY XY

k k s

XY XY

A A

n n

n n

′ § ·§ · ≈− −  ̈  ̧ ̈  ̧ © 1© 1

′ § ·§ · ao ao ¬1⁄4 ¬1⁄4 =− −  ̈  ̧ ̈  ̧

© 1© 1

¦

¦

Vt f y f y

t t

f f

 (81)

Thus, Eq. 81 expresses the RE covariance matrix for the vector of estimated 

population totals for the FIA study variables as the sum over all FIA PSUs of 

the cross-product of a vector specifi c to each PSU (gk). That vector equals the 

residual difference between the vector of expansion values for the RE for the kth

PSU (fk) (Eq. 76) and the mean design-based vector of expansion values over all 

nXY sampled PSUs.

Further Research

Equations 76 and 81 demonstrate that the optimal RRE population estimates 

and variances for FIA variables may be estimated with a vector of plot-level 

expansion values. Those expansion values use the design-based weights, the designconsistent approximately optimal RRE weights, the observed study variables (yk

), 

and the auxiliary variables (xk) for each sampled PSU k. Unlike traditional FIA 

expansion factors, which cannot be used to estimate variances with post-stratifi -

cation (Scott and others 2005:65), the RRE expansion values may be conveniently 

used in a database environment for FIA core tables and related variance estimation. Furthermore, the memoryless quality of the RRE produces expansion values 

that fully capture all information available in a prior set of ancillary remotely 

sensed data. Thereafter, those remotely sensed data may be (fi guratively) discarded after they are processed by the RRE because they are no longer needed 

for subsequent estimates. This quality is expected to greatly simplify processing, application, and documentation of dynamic expansion values by FIA as new 

time-series of remotely sensed auxiliary data and FIA fi eld data become available. 

However, these weights are based on heuristics, not formal theory. Knottnerus 

(2003:395) develops similar weights from the perspective of Pythagorean regression, which might be further simplifi ed with the matrix inversion lemma. Further 

investigation is needed to gain confi dence in the reliability, numerical stability, 

and processing simplicity of this approach to variance estimation.

Similarly, results from the numerically stable algorithm for use with auxiliary 

census data (Eqs. 34 and 35) may be captured as expansion values at the PSU 

level. However, these expansion values would need to be revised at each individual i

th recursion. Until the end of the recursive sequence, PSU-level expansion 

values would be necessary for the auxiliary variables in addition to the FIA study 

variables. The auxiliary variables may be removed from the state vector after 
62 USDA Forest Service RMRS-GTR-239. 2010.

all relevant information in those auxiliary variables is fi ltered out with the RRE. 

Future research will extend PSU-level expansion values for RE (Eqs. 76 and 81) 

to the more general case of RRE. The section “Robust Numerics” describes more 

general methods to assure numerical stability. It is not yet known how these results can be represented as expansion values at the PSU level.

These results are obtained by treating the sub-optimal matrix of weights K as 

fi xed, which leads to a vector of expansion values for each PSU k that are additive shifts relative to the original direct estimators. This ad hoc approach is fully 

design-consistent when applied to the entire population, but its statistical properties are unknown when applied to small areas. Furthermore, it deviates from more 

traditional approaches (e.g., Särndal and others 1992) in which a single scalar 

regression or calibration weight is applied to all variables measured for a sampled 

PSU. An alternative might explicitly recognize that the optimal K is estimated. 

The estimator in Eq. 76 may be recast as a linear combination of the vector measurements for each sampled PSU k (yk

) with the yk appearing in the estimated 

covariance term in Eq. 77. If that is feasible, then the vector of expansion values 

for each PSU may be replaced by a scalar expansion weight. This would conform 

to traditional regression estimation. However, the resulting scalar weight is not 

necessarily optimal for all study variables, and, in extreme cases, it might be less 

effi cient than the original direct estimator that does not use any auxiliary information (Lundström and Särndal 1999).

Multivariate Vector of FIA Study Variables

The M = 2 protocol produces accurate FIA fi eld measurements (e.g., tree diameter and tree height) that are further transformed into other relevant FIA study 

variables (e.g., tree volume and tree biomass). These variables characterize forest 

structure and composition in terms of tree species, size, wood volume, and biomass; rates of tree volume growth, mortality, and removals; ownership of private 

lands and stewardship responsibilities for public lands; and other indicators of 

forest conditions, health, risks, and hazards (Gillespie 1999). Table 2 summarizes 

these variables in more detail. These variables are cross-classifi ed in multiple 

ways to create detailed FIA core statistical tables (e.g., Reams and McCollum 

2000; Smith 2002).

In all of the examples above, the y vector contains the FIA core variables that 

describe the current population totals tY, or state, of the population. The y vector 

also includes variables that describe the change over time in those same variables, 

i.e., the change in the state of the population. Thus, a dynamic Kalman fi lter can 

include effi cient estimates of changes over time in the sampled population, while 

retaining its memoryless qualities to simplify data management.

Population sub-groups are defi ned by FIA as forest condition classes (e.g., aspen sawtimber stands) with detailed estimates of tree characteristics within each 

condition class (e.g., tree species by various categories of tree size). As a consequence, the number of parameters can increase 500-fold or more for populations 

at the scale of a state (e.g., Pennsylvania, which is approximately 28 million acres 

in size).

There may be literally thousands of such variables, depending on the degree 

of compartmentalization of continuous variables by discrete condition categories, the degree of cross-classifi cation of nominal categorical variables, and the 

number of population groups. For numerical reasons, it is assumed that some 

simplifi cation of the estimation problem reduces the dimension qM=2 of the study 

variables in vector y such that 5<qM=2<5000. At the other extreme, the RRE may 

be structured as a univariate estimator, where each scalar study variable is separately estimated, one at a time, i.e., qM=2 = 1. This latter univariate approach is used 

in the vast majority of sample survey literature. It is also used by FIA (Scottand 
USDA Forest Service RMRS-GTR-239. 2010. 63

others 2005). Future research will investigate the advantages and limitations of 

the multivariate approach, including accurate estimation of large covariance matrices with methods suitable to a production survey system.

Multivariate Vector of Auxiliary Variables

Särndal and others (1992:220, 304, 397) defi ne auxiliary variables to “covary 

with the study variable and thus carry information about the study variable. Such 

covariation is used advantageously in the regression estimator ... (to explain the 

Stand- or site-level conditions 

Number of 

categories Tree-level conditionsa Number of categories 

Land useb,c 5 Tree speciesd

331 

Broad forest typesb,d

29 

Tree size (DBH) 2-inch 

classes 

Detailed forest typesd

136 Tree damage 10 

Stage of stand developmentb

4 Tree quality, value 5 

Stand densityb,e

5 

Wood volume continuous 

Stand originb

2 Growth in wood volume continuous 

Land ownership 10 

Stand age 9 

Stand productivity 7 

Number of trees per acre b,f

continuous 

Wood volume per acre b,f

continuous 

Volume growth per acref

continuous 

Volume mortality per acre b,f

 continuous 

Volume removals per acre b,f

continuous 

a

 FIA measures many other indicators that describe landscapes, habitats, non-tree vegetation, etc. 

b

 Photo-interpretations and photogrammetric measurements with high-resolution imagery are well correlated with 

these field measurements (Aldrich 1979). The correlation is much lower with satellite data and high-altitude (lowresolution) aerial photography. c

 Includes timberland, other forest land, protected forest, non-forest land, and water. 

d

 Any single geographic region of the United States has only 20 to 40 percent of these national categories. 

e

 Includes overstocked, fully stocked, understocked, and non-stocked. 

f

 Totals are produced for thousands of permutations of different tree and forest categories. 

Table 2. Summary of variables measured directly or indirectly with FIA fi eld protocol (Czaplewski 1999).
64 USDA Forest Service RMRS-GTR-239. 2010.

variable of interest, especially) ... as a means of offsetting the scarcity of the 

sample data in small domains.” The associations among auxiliary variables (e.g., 

remotely sensed data) and the study variables (e.g., FIA fi eld measurements and 

transformations) need to be suffi ciently strong. Särndal and others (1992:250) 

recommend correlations among continuous predictor and response variables be at 

least 0.5 or greater. For nominal categorical variables, Czaplewski and Patterson 

(2003) recommend agreements of at least 70 percent or greater.

The principle objective of this paper is to offer highly fl exible statistical approaches capable of using multiple types of remotely sensed data (Czaplewski 

in prep.[b]) to improve statistical estimates and cost-effectiveness of FIA. These 

remote sensing technologies include:

• digital processing of multispectral optical Earth-observing satellite sensors, 

such as Landsat, with 30-m pixels;

• hyper-spectral optical Earth-observing satellite sensors, such as MODIS, with 

250-m to 1000-m pixel resolution (e.g., Huete and others 2002; Tian and others 

2002; Zhan and others 2002; Brown and others 2006; Achard and others 2007);

• LiDAR laser sensors, both airborne and spaceborne; and

• manual interpretation of aerial photography of various scales and resolutions.

All these technologies are briefl y discussed here.

Digital Processing of Multispectral Landsat Data

Digital processing of multispectral Landsat data with 30-m pixel resolution 

has broad-scale applications in extensive monitoring of forest cover. The following is a partial list of the literature that assesses these applications: Horler and 

Ahern 1986; Bauer and others 1994; Cohen and others 1995, 2001; Trotter and 

others 1997; Wynne and Carter 1997; Holmgren and Thuresson 1998; Katila and 

Tomppo 2001; and Lefsky and others 2001. The relatively high spatial resolution 

of these data often permits registration to a support-region at the scale of a 1-acre 

FIA plot, although larger support-regions can also be useful.

The Landsat (M = 0) protocol may include continuous variables, such as remotely sensed estimates of biomass; discrete categorical variables, such as forest 

conditions classes; or model predictions of any measurement made by FIA fi eld 

crews with the M = 2 protocol. The model might be parametric, such as with logistic regressions that predict probability of forest cover with optical satellite data 

as the predictors (e.g., McRoberts and Liknes 2005); might be non-parametric, 

such as with the classifi cation and regression tree method (e.g., Breiman and others 1984; Block and others 2008) with a univariate response variable; or may have 

multivariate response variables, including k-nearest neighbor regression (e.g., 

Katila and Tomppo 2001).

Multispectral data for a Landsat scene, with an effective area of approximately 

150 by 150 km, contains about 25 million Landsat pixels. They can be grouped 

into 4- by 4-pixel Landsat support-regions (3.6 acres or about 1.6 million pixel 

groups) that can be reasonably registered to the 1-acre support-region defi ned 

by the FIA fi eld plot (Czaplewski 2005). The average state in the United States 

requires seven Landsat scenes to cover the state completely. Other geospatial data 

such as descriptors of biophysicial and landscape features are merged with the 

multispectral Landsat data as potential predictor variables (Blackard and others 

2008).

The average state has about 7000 FIA plots, of which about 2000 have forest 

cover. In the annual FIA panel system (Patterson and Reams 2005), 200 to 400 

of these forested plots are grouped into a single panel that is measured during 

a single year. Data from these registered elements are used to train models that 

predict the FIA fi eld measurements for each Landsat pixel. Remotely sensed and 

geospatial data are the predictor variables. The resulting predictions are designated as the M = 0 protocol.
USDA Forest Service RMRS-GTR-239. 2010. 65

Because RRE requires that polychotomous classifi cation systems be transformed into multiple dichotomous variables, the classifi cation categories do not 

necessarily need to be mutually exclusive. For example, classifi cation of pixels 

into an aspen type can be independent from classifi cation of pixels into a cottonwood type. The classifi ers can be independently optimized for each category 

(Czaplewski and Patterson 2001, 2003). A single pixel might be classifi ed into 

more than one category (both aspen and cottonwood) if spectral signatures for 

land cover types overlap to some degree. Also, the classifi cation categories do not 

need to be complete. For example, some spectrally perplexing pixels might not 

be classifi ed into any dichotomous category. One example is pixels obscured by 

clouds or cloud shadows. One limitation is that each category must cover a suffi cient number of more accurately measured FIA fi eld plots.

Landsat data currently support production of annual thematic maps that detect major changes in land cover. Such changes tend to be relatively rare at the 

landscape-scale, and the relatively sparse sample of FIA plots, by themselves, 

might not produce suffi ciently accurate estimates of the attributes of the changed 

sub-population (Lister and others 2009). The wall-to-wall Landsat coverage can 

improve areal estimates for rare features. These changes in land cover are usually important in inventory and monitoring analyses. Therefore, incorporation of 

annual remotely sensed change detection information can be vital to the relevance of any monitoring system (Czaplewski 1999; Czaplewski and others 2005). 

However, the estimators might have to be applied to very large geographic areas, 

perhaps even to multiple states, in order to capture a suffi cient sample size of FIA 

fi eld plots.

Both RRE and post-stratifi cation estimators assume the auxiliary variables 

are independent of the random errors from the sample of PSUs. However, those 

same PSUs are often used as training data to fi t supervised classifi ers or regression models that predict land characteristics (i.e., a thematic map) with remotely 

sensed and other geospatial data as predictor variables. This connection breaks 

the assumed independence. A different sample would produce a slightly different 

thematic map. It is commonly assumed that this dependence is small, and it can 

be ignored. Breidt and Opsomer (2008) provide weak support for this assumption. 

However, it is not necessary if these geospatial variables are directly used without 

fi tting a prediction model. For example, unsupervised classifi cation of multispectral data from Landsat does not require training data, and the results provide truly 

independent nominal auxiliary data. Other relevant geospatial variables may be 

combined with the Landsat data, such as terrain and climate maps. Independent 

continuous auxiliary variables are also available. The Normalized Difference 

Vegetation Index (NDVI) is an example (Tucker 1979). NDVI is defi ned as the 

difference of the red and infrared radiances divided by their sum. It partially normalizes for differences in solar zenith angle and atmospheric conditions. NDVI is 

correlated with green biomass density, which is further correlated with tree stocking and wood volume. No training data are necessary to defi ne NDVI; therefore, 

NDVI is independent of FIA fi eld plots. However, the census of pixel values for 

NDVI could directly serve as auxiliary data to improve population estimates for 

forest biomass and wood volume.

Geopolitical Domains of Study

FIA uses geopolitical domains of study (Särndal and others 1992:5) such 

as counties separated by major types of land ownership or stewardship. 

Administrative records are used to defi ne the total area of each domain, and each 

area is defi ned as an exact constant that is known without error, i.e., a true census. 

FIA currently uses post-stratifi cation to constrain its areal statistical estimates to 

agree exactly with certain administrative records. This often leads to numerous 

strata, many of which have small sample sizes, especially under the annual panel 

system employed by FIA. As an alternative, RRE may be confi gured to impose 

the same statistics as fi xed constraints.
66 USDA Forest Service RMRS-GTR-239. 2010.

Domains of study may be represented in a linear model by “dummy variables” 

that have the value zero if the element is not a member of the intended population group (Särndal and others 1992:260), which, in the context of multivariate 

estimation, are represented in vector form (Särndal and others 1992:171). This is 

exactly analogous to a census of remotely sensed pixels in the previous section.

Photo-Interpretation and Photogrammetry

Photo-interpretation or photogrammetry is used to estimate continuous variables such as total wood volume in the point support-region by major groups of tree 

species and tree size classes, separated by living versus dead trees (e.g., Aldrich 

1979; Lund and others 1997; Czaplewski in prep.[b]). Photo-interpretation has 

an important role in FIA (Catts and others 1987; Oswald 1988; Czaplewski and 

Catts 1992; Frayer and Furnival 1999; Coulston 2008; Frescino and others 2009a; 

Lister and others 2009). Photo-interpretation of high-resolution aerial photography can be accurate for broad characteristics of forest stands and individual trees. 

However, photo-interpretation and photogrammetry are too expensive for wallto-wall full-coverage of the large populations. Likewise, full-coverage acquisition 

of high-resolution and current aerial photography is prohibitively expensive. 

Therefore, a sample of photo-plots is usually acquired and measured, typically 

with cluster plots in a multistage sampling design (e.g., Nusser and others 1998; 

Czaplewski 1999; Magnussen and others 2000; Gillis and others 2005; Lister and 

others 2009). Because the support-region (de Gruijter and others 2006:74-75) for 

each sample point is a well-defi ned area, measurements of all continuous variables can be expressed on a per unit area basis.

Other photo-interpreted variables may be nominal categories such as classifi cation of each sampled support-region into one of 5 to 10 different types of 

land cover (e.g., forest, woodland, shrubland, grassland, cropland, barren lands, 

and water). Another type of photo-interpreted categorization might be predominant land use (e.g., timberland, grazing lands, reserves, parks and recreational 

areas, wildland-urban interface, urban lands, and transportation and utility corridors). Forest cover may be further classifi ed through photo-interpretation by 

broad stand type (e.g., ponderosa pine, lodgepole pine, Douglas fi r, spruce-fi r, and 

cottonwood) and stage of stand development (e.g., seedling/sapling, poletimber, 

sawtimber, and over-mature sawtimber). The support-region for photointerpretation of a nominal category might be the point itself (although a small 

surrounding area is necessary for context) where the measurement is binary; that 

is, point i is interpreted as belonging to category a, i.e., yi,a = 1, or not, i.e., yi,a = 0.

Photo-interpreted measurements need not necessarily have the same units as 

fi eld measurements. For example, photo-interpretations of average tree height 

and crown density and their product in a 1-acre support-region can have a strong 

linear correlation with fi eld measurements of wood volume from a collocated 

1/6-acre FIA fi eld plot. Such a correlation would produce gains in statistical effi ciency, even though wood volume is not being directly photo-interpreted (see 

sub-matrix C in Eq. 26).

Photo-interpretation of high-resolution aerial photography can be used as preliminary reconnaissance to determine if a plot meets the defi nition of stocked 

forest (Frescino and others 2009b; Goeking and Liknes 2009). Field costs can be 

substantially reduced if fi eld crews do not have to visit non-forest plots. However, 

errors in photo-interpretation can bias population estimates (Martinussen and others 2008). Methods similar to those in Eqs. 91 to 99 might be suitable to overcome 

this bias with a design-consistent approach; therefore, any photo-interpreted variables used as pre-fi eld reconnaissance could be a component in the appropriate 

measurement vectors (Eq. 8).

A photo-interpreter might be very confi dent of an interpretation in some situations and less confi dent in other situations. Photo-interpreted variables that 

characterize the support-region might be separated into several variables based on 
USDA Forest Service RMRS-GTR-239. 2010. 67

the degree of certainty of the interpreter. For example, a 0-1 binary variable for 

a particular forest type can be cross-classifi ed by a 0-1 binary “dummy” variable 

for confi dence in an interpretation, which would produce four remotely sensed binary variables: Forest Type A with high confi dence; Forest Type A with moderate 

confi dence; not Forest Type A with high confi dence; and not Forest Type A with 

moderate confi dence. Likewise, accuracy can vary among different photo-interpreters, and dummy variables could separate remotely sensed data into categories 

of higher and lower quality.

The most severe effects of misregistration between support-regions for remotely sensed and fi eld protocols can be at the boundaries between different forest 

conditions, or within fi ne-grained, spatially heterogeneous stands. Misregistration 

effects are fewer within interior of homogeneous stands. These different situations in measurement errors can be identifi ed during the measurement processes 

(Zhu and others 2000) and made available in the estimation step. Even better, 

fi eld crews can accurately locate the sample point on aerial photographs while 

in the fi eld. If photo-interpreters subsequently apply the remote sensing protocol 

centered on the same point on the aerial photograph, then registration accuracy 

can be greatly improved, and the extra effort would be rewarded with increased 

statistical effi ciency. The key principle is to avoid corrupting remotely sensed data 

that are accurate with a relatively small amount of remotely sensed data that are 

not accurate. The effi ciency offered by remotely sensed data decreases rapidly as 

accuracy declines (Czaplewski and Patterson 2001, 2003).

Differences can be considerable in classifi cation and measurement accuracy 

among photo-interpreters. If multiple interpreters with widely different skill 

levels produce the remotely sensed measurements, then there might be some advantages in treating the measurements by each interpreter as a separate variable. 

A randomization process of assigning photo-plots to individual interpreters might 

have some advantages worth consideration.

LiDAR Laser Technology

LiDAR laser technology provides measurements that can be well correlated 

with tree biomass and wood volume (e.g., Means and others 1999; Brandtberg 

and others 2003; Popescu and others 2003; Zimble and others 2003; Sun and 

others 2008; Nelson and others 2009b). A dimensionless index based on characteristics of LiDAR laser pulses and returns, which might not have the units of 

wood volume, might still be well correlated with fi eld measurements of wood volume and biomass density. Gains in effi ciency depend on strong linear correlations 

and associations between remote sensing and fi eld data. These linear relationships 

are quantifi ed by the sample cross-covariance matrix between the remotely sensed 

and fi eld measurements (e.g., sub-matrix C in Eq. 26).

Synthetic Aperture Radar Technology (SAR)

Synthetic aperture radar technology (SAR) also offers promise in providing 

measurements that are reasonably correlated with forest inventory and monitoring 

fi eld data, especially in geographic areas prone to chronic cloud cover that obscures remotely sensed data from optical sensors (Pope and others 1994), which 

includes many tropical and boreal forest ecosystems. Ranson and Sun (1994) 

found correlations of approximately 0.8 between biomass and airborne SAR sensor data for a boreal study area. Success often requires extensive ground-truth 

measurements to build empirical models between forest biomass and SAR backscatter, although there are alternatives (Ranson and Guoqing 1997). Ranson and 

Sun (1997) report similar results for spaceborne SAR data. However, RRE can 

directly use SAR metrics without an external biomass model. Accuracy will improve if the SAR metrics are suffi ciently well correlated with fi eld measurements 

of biomass and other similar variables.
68 USDA Forest Service RMRS-GTR-239. 2010.

Linear, Small-Area, Synthetic Estimator with Census 

Predictors

Matrix functions of multivariate estimates provide opportunities to construct 

model-based estimators for other applications. Estimation for small areas with 

known precision is an excellent example (e.g., Schreuder and others 1993:Chapter 

8.1; Rao 2003:Chapter 1). Many analyses focus on small portions of a sampled 

population, also known as small domains (e.g., Särndal and others 1992:408) or 

small areas (e.g., Rao 2003). As with other large government statistical programs, 

FIA is constantly investigating methods to provide more reliable estimates for 

small geographic areas (e.g., USDA Forest Service 2007; Morin and others 2009; 

Magnussen and others 2009; Roesch 2009).

If the domain is a suffi ciently sampled sub-population, then design-consistent 

methods, such as those presented above, are applicable. Inevitably, many areas 

are so small that they do not contain a suffi cient number of samples to make reliable, design-consistent estimates. One solution available in forest inventory and 

monitoring is relying on full-coverage, remotely sensed pixel data from spaceborne sensors and other geospatial databases (Haslett and Jones 2005). Parametric 

and non-parametric regression models use such data to predict forest characteristics for each and every pixel in the satellite imagery (see “Digital Processing 

of Multispectral Landsat Data”). Statistical enumeration of all multivariate pixel 

predictions within the small area can provide useful predictor data. However, prediction models and the sum of all predictions within the small area can be biased, 

the degree and direction of the bias can be unknown, and the magnitude of the 

bias can be large (Bauer and others 1978; Card 1982; Chrisman 1983; Houston 

and Hall 1984; Hay 1988; Czaplewski 1992; Gallego 2004; Gallego and Bamps 

2008; Iles 2010; Magnusen and others 2010; McRoberts 2010). Also, the statistical reliability of the predictions, which may be quantifi ed with a covariance 

matrix, may be unknown.

If the small area includes a suffi cient number of sample plots that are measured 

with suffi ciently accurate protocols, then an unbiased difference or ratio estimator 

(e.g., Särndal and others 1992:Chapter 6) provides a direct estimator of the covariance matrix, which gauges statistical reliability. Baffetta and others (2009) apply 

this approach in the context of a National Forest inventory. Calibration estimates 

are also relevant (see p. 32).

If the sample size is insuffi cient within a small area, then indirect synthetic estimation, which is “simply the sum of predicted values,” is an alternative (Särndal 

and others 1992:399). McRoberts and others (2007) and McRoberts (2008) 

provide examples relevant to forest inventories. Synthetic estimators are modelbased by defi nition. Model-based estimators can be effi cient, but they are biased 

if the model assumptions are inaccurate (Schreuder and others 1993:Chapter 6.3; 

Lehtonen and others 2003). The risk can be substantial because validation of 

model assumptions is rarely feasible (Estevao and Särndal 2004). For example, 

Blackard and others (2008) found that their non-parametric regression-tree model 

tended to underestimate biomass for test plots that had unusually high amounts of 

biomass and overestimate biomass for test plots that had unusually low amounts 

of biomass. Similar trends might be expected with k-nearest neighbor (kNN) nonparametric regression. This is evidence for conditional bias. If the distribution of 

pixel-level biomass within a small area differs from the population-level distribution, then the synthetic estimator will likely be biased for that small area.

The following example is a model-based, multivariate, synthetic estimator. It accomplishes the same objectives as those considered by McRoberts and 

others (2007), but it uses fewer assumptions and a much simpler model. The 

example uses a complex design of two-phase sampling with cluster plots: a census of full-coverage pixel data from spaceborne sensors and other geospatial 

datasets; a sample of high-resolution, remotely sensed data from airborne sensors for 
USDA Forest Service RMRS-GTR-239. 2010. 69

50-acre cluster plots; and a sub-sample of 1-acre fi eld plots that are collocated 

within the cluster plots (see Eqs. 62 to 70). This design is similar to that used in 

the FIA NPIP study (Frescino and others 2009a). However, the synthetic estimator 

in this example may be applied with any of the sampling designs and population 

estimators of the types covered on 21 through 53 pp.

Visualize an area that is very small with too few samples for a reliable, designconsistent estimate. One relevant case study is the 6000-acre hexagon used by 

FIA for its systematic sampling frame (Reams and others 2005). Each of the approximately 360,000 hexagons in this national sampling frame has a single FIA 

fi eld plot, and only about 30 percent of those plots have forest cover. The same 

hexagons are also used for geospatial presentations and analyses (e.g., Wilson and 

Ibes 2005; Lowe and Cieszewski 2005).

In this example, multispectral measurements from the MODIS sensor (e.g., 

Cohen 2003) are predictors in geospatial models that estimate forest area and 

biomass for each 15-acre MODIS pixel in a full-coverage pixel database (e.g., 

Blackard and others 2008). The remotely sensed predictions for approximately 40 

MODIS pixels are summarized for each collocated 6000-acre hexagon, and the 

resulting summary statistics are input into the hexagon database. Further, assume 

that the hexagons are grouped into ecoregions (Bailey 1995). Summary results for 

a 10,000,000-acre hypothetical ecoregion are given in Table 3.

Population estimates with 

FIA protocol (M = 2) 

Spaceborne protocol (M = 0) Forest cover Non-forest 

cover 

Population 

total 

Forest cover (acres) 2,850,000 150,000 3,000,000

Non-forest cover (acres) 600,000 5,400,000 6,000,000

Boundary (edge) area (acres) 450,000 550,000 1,000,000

Total area 3,900,000 6,100,000 10,000,000

 

Area (acres) by joint classification with spaceborne (M = 0) and 

FIA (M = 2) protocols 

Estimated 

calibration 

coefficient

Forest cover (M = 0) 

Forest cover (M = 2) 2,850,000 3,000,000 0.95

Non-forest cover (M = 2) 150,000 0.05

Non-forest cover (M = 0) 

Forest cover (M = 2) 600,000 6,000,000 0.10

Non-forest cover (M = 2) 5,400,000 0.90

Boundary (edge) area (M = 0) 

Forest cover (M = 2) 450,000 1,000,000 0.45

Non-forest cover (M = 2) 550,000 0.55

 10,000,000 

 

FIA protocol 

(M = 2) 

Spaceborne 

protocol (M 

= 0) 

Forest biomass (dry tons) 220,000,000 200,000,000 1.10

Table 3. Population-level estimates of forest area and biomass from full-coverage, remotely 

sensed data from spaceborne sensors (protocol M = 0) and sample of fi eld data (protocol 

M = 2).
70 USDA Forest Service RMRS-GTR-239. 2010.

Because of misclassifi cation bias and other systematic prediction errors, 

broad-scale predictions of forest area and biomass (e.g., ecoregion estimates) 

with the remotely sensed data will disagree with those from the unbiased sample 

survey estimator. Table 3 provides an example with a 10,000,000-acre hypothetical ecoregion. The remotely sensed data predicts 3,000,000 acres of forest cover 

plus an additional 1,000,000 acres of heterogeneous edge conditions. These are 

partially forested and non-forested, whereas the unbiased sample survey estimator 

predicts 3,900,000 acres of total forest cover.

The remotely sensed area estimates may be calibrated (e.g., Czaplewski and 

Catts 1992) with data from an unbiased estimator. Table 3 provides a simple 

example of the inverse multivariate calibration model (Brown 1982), which is 

also labeled direct calibration (Dymond 1992; Gallego and Bamps 2008). Of the 

3,000,000 acres predicted with remotely sensed data to be forest cover, an estimated 95 percent is truly forest, and the remaining 5 percent is non-forest. Table 

3 includes estimates of calibration coeffi cients for other cover conditions, in addition to biomass estimates. This may also be considered a form of multivariate 

synthetic estimation; see Särndal and others (1992:399) for a brief introduction to 

univariate synthetic estimation.

In matrix notation, the calibration coeffi cients (b) and their estimated covariance matrix are computed as:

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0 0 0 0 0 0

3,000,000

1

200,000,000

550,000

450,000

5,400,000

600,000

150,000

2,850,000

ˆ

1.10

0.55

0.45

0.90

0.10

0.05

0.95

ˆ

ˆ ˆ ˆ ˆ

ˆ ˆ

M 1,1C

,1,2 M 2

M 1,1C

,1,2 M 2

M 1,1C

,1,2 M 2

A

b t

V b A V t A

b A t

s U

s U

s U

 (82)

The non-zero elements in matrix A in Eq. 82 are known constants because they 

are summary statistics from the full census of remotely sensed pixels within the 

ecoregion. Therefore, they are a straightforward linear function of the vector estimate t and its covariance matrix.
USDA Forest Service RMRS-GTR-239. 2010. 71

The following small-area estimator relies on the model that assumes the association between remotely sensed estimates and design-consistent sample survey 

estimates is linear and identical for all 1667 of the 6000-acre hexagons within the 

10,000,000-acre ecoregion. Under this model, the calibration coeffi cients from 

Table 2 and Eq. 82 may be applied to the remotely sensed census statistics for any 

small area within the 10,000,000-acre ecoregion. Examples for two 6000-acre 

hexagons are given in Table 4. The model predictions and their joint covariance 

matrix for all 1667 hexagons within the ecoregion may be simultaneously expressed in matrix notation as:

[ ] { } [ ] { }

[ ] { }

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0 0 0 0 0 0 20,000

0 800 0 4,200 0 1000 0

800 0 4,200 0 1000 0 0

0 0 0 0 0 0 400,000

0 4,000 0 1,500 0 500 0

4,000 0 1,500 0 500 0 0

1.10

0.55

0.45

0.90

0.10

0.05

0.95

ˆ

22,000

4,370

1,630

440,000

1,825

4,175

ˆ

ˆ

ˆ ˆ

ˆ

ˆ

ˆ ˆ

ˆ

ˆ

1667

1

,1,2

1667

1

1667

1

,1,2

1667

1

1667

1

,1,2

1667

1

1667

1

       

 

    

i

i

s U

i

i

i

i

s U

i

i

i

i

s U

i

i

i

i

T

T

b

t

t

T

T

V b

T

T

t

t

b V

T

T

t

t

 (83)

The structure of Eq. 83 produces a covariance matrix for the propagated prediction errors for estimated parameters among all 1667 hexagons. In principle, this 

could directly produce covariance matrices for any linear function of statistics 

from multiple hexagons, such as the sum of estimates for all hexagons that meet 

certain selection c riteria. This capability would further enhance the value of a 

hexagon database for rapid assessments with diverse, small-domain estimates.

Finally, this example may be expanded to be more relevant to FIA. Further, 

assume that a full-coverage geospatial database is composed of pixel data. It contains predictions of the forest composition of each pixel. The kNN model, which 
72 USDA Forest Service RMRS-GTR-239. 2010.

is a form of multivariate non-parametric regression (Härdle 1990), is used to predict all FIA plot-level variables for each and every pixel (Katila and Tomppo 

2001; McRoberts and Tomppo 2007). When used as a model-based estimator for 

population totals, it is biased or, at least, the kNN estimator cannot be shown to be 

design-unbiased. However, there are design-based and model-assisted estimators 

of population totals that can be shown to be design-consistent. These estimators 

can include the same pixel census data within the multilevel sampling design.

Unlike categorical variables, the univariate, small-area estimator for a continuous variable can be based upon the simple ratio between the population estimate 

made with data from the fi eld measurements (protocol M = 2) and population 

estimate made with the kNN model that uses spaceborne, remotely sensed data 

(protocol M = 0). The model in this estimator is simply this ratio multiplied by 

the total of pixel-level, remotely sensed predictions for a small area (e.g., Särndal 

and others 1992:399). The result is a synthetic estimate of the small-area total 

made with the M = 2 fi eld protocol. This is a model-based estimator, in which it 

is assumed that the true ratio for every small area is identical to the ratio for the 

entire population. In the special case of the kNN non-parametric regression, the 

prediction with remotely sensed data protocol (M = 0) and the variable measured 

with the fi eld protocol (M = 2) have identical defi nitions and units of measure. 

The ratio may be considered a calibration coeffi cient that proportionally scales the 

remotely sensed estimates to agree exactly with the unbiased design-consistent 

estimates at the population level. Univariate estimators for different continuous 

variables are readily assembled into a multivariate estimator, an example of which 

is illustrated with Eq. 83 and Table 3.

The multivariate synthetic estimator for a categorical variable is more complex, 

although it is related to the simpler case of continuous variables. The categorical 

model assumes that the misclassifi cation error process is identical for all pixels or, 

at least, over all possible small areas. Rather than a single coeffi cient for calibrating a continuous variable, the sample survey estimator produces a contingency 

table of joint classifi cation probabilities for each category, with one classifi er 

based on remotely sensed data (M = 0) and the other based on fi eld protocol 

(M = 2). The model transforms this contingency table into probabilities conditional on the remotely sensed classifi er (e.g., Eq. 82), much like a probability 

transition matrix. Population estimates used to parameterize this model require 

observations with both the remotely sensed protocol (M = 0) and the fi eld protocol 

(M = 2).

Small-area census with spaceborne protocol (M 

= 0) 

Small-area estimates with FIA protocol (M = 2)

Forest cover Non-forest cover 

Hexagon = 1 

Forest cover (acres) 4,000 0.95 3,800 0.05 200 

Non-forest cover (acres) 1,500 0.10 150 0.90 1,350 

Boundary (edge) area (acres) 500 0.45 225 0.55 275 

6,000 4,175 1,825 

 M = 0 M = 2 

Forest biomass (dry tons) 400,000 1.10 440,000 

Hexagon = q 

Forest cover (acres) 800 0.95 760 0.05 40 

Non-forest cover (acres) 4,200 0.10 420 0.90 3,780 

Boundary (edge) area (acres) 1,000 0.45 450 0.55 550 

6,000 1,630 4,370 

 M = 0 M = 2 

Forest biomass (dry tons) 20,000 1.10 22,000 

Table 4. Example of inverse calibration model applied to two small-areas.
USDA Forest Service RMRS-GTR-239. 2010. 73

Joint classifi cation probabilities are estimated by augmenting the M = 2 

measurement vector containing the cross-classifi cation with both the remotely 

sensed (M = 0) and fi eld (M = 2) protocols. For example, assume there are 26 

categories of land use and land cover types include: lodgepole pine, Engelmann 

spruce, Douglas-fi r, ponderosa pine, western hardwoods, woodland, sagebrush, 

grassland, barren rock, alpine tundra, riparian areas, wetlands, standing water, 

urban zones, and transportation corridors. Also assume each type of forest cover 

is separated into large, medium, small, and non-stocked categories that describe 

stand size. In the special case of the current example, the kNN remote sensing 

protocol (M = 0) and the fi eld protocol (M = 2) can classify every support-region 

based on this 26-category classifi cation system. For each and every pixel, the M = 

0 measurement vector (Eq. 8) contains a segment with 26 elements that represents 

the remotely sensed classifi cation:

[ ]

°

°

°

 ̄

°

°

°

®

­

° ̄

°

®

­

=

= = =

° ̄

°

®

­

=

= = =

»

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=

=

=

0 otherwise

1if pixel classified with M 0 as category Z

0 otherwise

1if pixel classified with M 0 as category A

 where

A

M 0

A

M 0

Z

j

Z

B

j

y

y

y

y

y









y (84)

In the current example, the M = 0 remotely sensed measurement vector (Eq. 8) 

segment in Eq. 84 is used in the sample survey steps (Eqs. 62 and 63) and in all 

sequential steps in the estimator for population totals (Eqs. 64 to 69).

The corresponding measurement vector for the M = 2 fi eld protocol must include the joint classifi cations with both the remote sensing (M = 0) and fi eld 

(M = 2) protocols.

[ ]

°

°

°

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=

∩

∩

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∩ =

∩

∩

∩

∩

=

0 otherwise

and with M 0 as category Z

1if point classified with M 2 as category Z

0 otherwise

and with M 0 as category B

1if point classified with M 2 as category A

0 otherwise

and with M 0 as category A

1if point classified with M 2 as category A

 where 

Z Z

A B

A A

Z Z M 2

Z A

A Z

A B

A A

M 2

y

y

y

y

y

y

y

y

j

j











y (85)

Because there are 26 categories of forest and land cover in this example, there 

will be 26×26 = 676 elements in the M = 2 measurement vector [yM=2]j (see Eq. 8) 

that estimate the 26-by-26 contingency table of joint classifi cation probabilities. 

Equation 85 is used in the Phase 2 estimator (Eq. 63). The sequential estimators in Eqs. 64 to 6 9 are intended to improve the accuracy of the estimated joint 
74 USDA Forest Service RMRS-GTR-239. 2010.

classifi cation with photo-interpretation of high-resolution aerial photography and 

cluster plots (M = 1 and M = 1C).

The fi nal estimates of the joint classifi cation probabilities are transformed into 

conditional transition probabilities, as in Eq. 82, for use in multivariate synthetic 

estimators for small areas (Eq. 83). However, the estimated population totals do 

not require small-area estimation or joint classifi cation probabilities. In order to 

estimate the areas for each type of forest and land cover in the full population, the 

statistics used for joint probabilities are collapsed into the fi eld protocol (M = 2) 

by summing over the joint probabilities cross-classifi ed with the remotely sensed 

protocol (M = 0). Let 1 be a 1-by-26 row vector with all elements equal to 1. The 

estimated population totals for each of the 26 forest and land cover types with the 

FIA protocol (M = 2) is:

[ ] [ ]

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= = ∩ =

= = ∩ =

0 0 I

0

0 0 1

0 1 0

1 0 0

0

I 0 0

V y

0 0 I

0

0 0 1

0 1 0

1 0 0

0

I 0 0

V y

y

0 0 I

0

0 0 1

0 1 0

1 0 0

0

I 0 0

y



   





  

 

  



   











   









M 2 M 2 M 0

M 2 M 2 M 0

ˆ ˆ ˆ ˆ

ˆ ˆ

 (86)

In the current example, the inner partition of the indicator matrix in Eq. 86 would 

be a 26-by-676 matrix that collapses the M = 0 and M = 2 joint classifi cation totals 

into the population totals for the 26 categories of land and forest cover types based 

on the fi eld protocol (M = 2) alone.

To accommodate small-area estimation, the structure of the multivariate estimator remains unchanged for continuous variables (e.g., biomass variable in Eq. 

82). However, the dimensions increase multiplicatively for categorical variables 

as the number of categories increases (e.g., Eq. 85). This increase can be large 

with detailed classifi cation systems. In the current example, 26 categories of land 

use and land cover require 26×26 = 676 variables to estimate the conditional 

probabilities of a point being classifi ed with the fi eld protocol (M = 2) as one of 

the 26 categories, given that the remote sensing protocol (M = 0) classifi ed it into 

one of the 26 categories. Even though most numerical problems with the Kalman 

fi lter are associated with the auxiliary measurement vector (Eq. 8), a large statevector (Eq. 8) can introduce additional numerical problems. However, cells in an 

estimated contingency table whose estimated values equal zero, some of which 

are sampling zeros (e.g., Agresti 2007:54), can be removed from the multivariate 

estimators. The number of zero cells that need not be estimated depends on the 

detail classifi cation system relative to the sample size of points classifi ed with the 

FIA fi eld protocol (M = 2). The number also depends on the accuracy of the M = 0 

classifi cations relative to the M = 2 classifi cations. As classifi cation accuracy increases, more observations are concentrated on the diagonal of the contingency 

table, which represent probabilities of correct classifi cation. Figure 1 provides an 
USDA Forest Service RMRS-GTR-239. 2010. 75

example. It uses the current example and realistic assumptions for the accuracies 

and misclassifi cation probabilities for the 26 categories of land cover and land 

use. If the average classifi cation accuracy is about 70 percent and there are 8000 

FIA plots in the population, then about 70 percent (676×0.70~500) of the cells 

are expected to have non-zero values that must be estimated from the sample. If 

cells that occur rarely are grouped together in a logical fashion, then the number 

of cells that must be estimated decreases to 30 to 50 percent of the total cells (i.e., 

estimates are required for about 200 to 350 of the 676 total cells). Using Figure 

1 with 90 percent accuracy and 1000 FIA plots in the population, only about 15 

percent of the cells (676×0.15~100) would require sample estimates. Figure 1 is 

merely a realistic example that is intended to illustrate a specifi c case, and it does 

not necessarily apply to other cases. Any of these simplifi cations of the multivariate estimator would require restructuring the indicator matrix in Eq. 86.

Because of different management regimes, assume the M = 0 versus M = 2 

classifi cation probabilities are different within two different land ownerships, e.g., 

private versus public lands. The nominal number of categories would increase to 

(26×26)×2 = 1352. However, many of these additional cells will have values of 

zero because the sample size of FIA plots would not change, and the sample size 

within each land ownership category would decrease. Therefore, the number of 

cells requiring sample estimates might not increase dramatically because the frequency of sampling zeros would also increase. The detail of the calibration model 

for categorical variables will be limited by the sample size of FIA fi eld plots, even 

for geographically large populations with thousands of FIA plots. Furthermore, 

auxiliary categorical data might include certain cross-classifi cations that do not 

accurately distinguish among cells in a cross-classifi cation contingency table. 

Those cells may be merged to reduce dimensions further. Research is needed to 

automate such a process.

RRE as a Synthetic Small-Area Estimator

Synthetic estimation in this section is primarily used as an example of a linear 

function of a vector estimate. For example, the RRE could be used to constrain 

a population-level vector estimate to agree with the auxiliary information for a 

small area. Again, using the complex RRE for the NPIP example, extract the partitions that include the full-coverage, spaceborne, remotely sensed variables (M = 0 

protocol) and the FIA fi eld measurements (M = 2 protocol) from Eq. 69.

Figure 1. Example of reduction 

in dimensions of 26-by26 contingency table after 

removing cells with values 

equal to zero and after 

grouping rare cells.
76 USDA Forest Service RMRS-GTR-239. 2010.

( )

[ ] ( )

[ ]

( )

[ ] ( )

[ ]

»

»

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1⁄4

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«

«

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¬

a

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1

·

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§

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a =  ̧

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1

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=

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=

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=

0 I

0 0

I 0

t

t

t

V

0 0 I

I 0 0

t

t V

t

t

t

0 0 I

I 0 0

t

t

M 1,1C

M 2 ,1,2

M 1

M 0

M 1,1C

,1,2 M 2

M 0

M 1,1C

M 2 ,1,2

M 1

M 0

M 1,1C

,1,2 M 2

M 0

ˆ

ˆ

ˆ

ˆ ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

s U

s U

s U

s U

 (87)

Let [tM=0]k be the vector sum of all pixel values in small area k.

( )

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¬

a

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«

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a

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1⁄4

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¬

a

−

=

= =

=

=

= =

=

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= =

=

=

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=

=

=

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=

=

=

0

I

t

t

I 0 V

0

I

t

t

K V

t

t

K I 0 V

t

t

V

t

t

V

t

t K t I 0

t

t

t

t

1

M 1,1C

1,2 M 2

M 0

M 1,1C

1,2 M 2

M 0

M 1,1C

1,2 M 2

M 0

M 1,1C

1,2 M 2

M 0

M 2

M 0

M 1,1C

1,2 M 2

M 0

M 0

M 1,1C

1,2 M 2

M 0

M 2

M 0

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

s s

s s k

s k s k

 (88)

Equation 88 is merely an algebraic operator that constrains a vector estimate. The vector difference between the known totals of the auxiliary data for 

the small area and the estimated population totals for the same auxiliary data 

[ ][ ]

( )

[ ]

»

»

1⁄4

o

«

«

¬

a

»

1⁄4

o «

¬

a −

=

= =

=

=

M 1,1C

1,2 M 2

M 0

M 0 ˆ

ˆ

s k t

t

t I 0

 will be large, which is uncommon in the 

intended application of RE. No claims are made regarding its statistical properties 

or numerical robustness. Alternative methods with the Generalized Restriction 

Estimator for consistent small-domain estimation exist (Sõstra and Traat 2009). 

The performance of the RE in this atypical context requires further investigation.

Non-Linear Pseudo-Estimators

Like the linear transformation covered in the previous section, non-linear transformations are often required. Such transformations are generally straightforward 

for estimation of population parameters or small-area statistics. However, they are 

not straightforward in variance estimation. An example is estimated wood volume 

per unit area of forest, where both total wood volume (t

1

) and total forest area 

(t

2

) are estimates, and their ratio is a non-linear transformation. The population 

estimate of a non-linear transformation is computed in the obvious, straightforward way, but the variance of the transformation typically requires a Taylorseries approximation with complex sampling designs (Tepping 1968; Woodruff 

1971; Krewski and Rao 1981; Wolter 2007).

Särndal and others (1992:173-174, 205-207) term these sorts of non-linear 

transformations pseudo-estimators. Using their notation, defi ne the non-linear 
USDA Forest Service RMRS-GTR-239. 2010. 77

function as θ = f(t

1

,..., tq). Defi ne the pseudo-estimator and its Taylor-series linear 

approximation as:

( )

( ) ( )( )

¦ ( )

¦ ¦¦

=

=

≈ + −

= + − + − − +

=

q

j

j j j

q

j

q

i

q

j

j j j ij j j j j

q

a t t

a t t a t t t t R

t t

1

1

1

ˆ

ˆ ˆ 2

1 ˆ

ˆ f ˆ , , ˆ

π

π π π

π π

θ

θ

θ 

 (89)

where:

( )

( )( )

( )

( )( )

remainder termin the expansion

f , ,

f , ,

, , ˆ , , ˆ

1

2

, , ˆ , , ˆ

1

1 1

1 1

=

∂ ∂

∂ =

∂

∂ =

=

=

R

t t

t t

a

t

t t

a

q q

q q

t t t t i j

q

ij

t t t t j

q

j

π π

π π

 

 





Cochran (1977:319) expounds on the value of this Taylor-series method for approximate variance estimators that use the vector estimate of population totals, 

which is denoted as the 1-by-q vector t in the following.

Using matrix calculus (Deutsch 1965:Chapter 6; Stengel 1986:33; Schott 

2005:Chapter 9; Wolter 2007:Chapter 6), the fi rst-order lin ear Taylor-series approximation of a pseudo-estimator has the following general form:

( )

[ ]

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

−

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

≈ +

≈ +¦ − =

q q

q

q

j

j j j

t

t

t

t

a a

a t t

  

1 1

1

1

ˆ

ˆ

,

ˆ ˆ

π

π

π

θ

θ θ

 (90)

The values of the aj

 terms depend on the application. Examples follow.

Product Estimator: Missing Data Imputation

The product estimator is a useful and simple non-linear transformation. A 

model-based pseudo-estimator that accommodates missing data is used here as 

an example of the product function of multivariate estimates for two random vectors. Th is example presupposes that a small portion of FIA fi eld plots cannot be 

mea sured because they are on private lands for which the landowner denies access 

to FIA fi eld crews. The study question in this example is “What is the estimated 

total biomass accretion rate on all private lands for which access was, or would 

be, denied by the landowner?” Research related to this application is a high priority in FIA short-term research and development.

Biomass accretion at a sample point is determined by re-measuring permanent plots 

in the fi eld. These re-measurements are missing for “denied access” lands because the 

fi eld crews are not permitted to measure the sample point. However, a model-based 

estimate of total biomass accretion can be made for all inaccessible lands.
78 USDA Forest Service RMRS-GTR-239. 2010.

The statistical model in this example assumes biomass accretion per unit area 

per year is the same for both accessible and inaccessible lands that are classifi ed 

as “forest” with remote sensing. In the form of prose, the model estimates biomass 

accretion on inaccessible lands as the product of [total area of such lands] times 

the [accretion per unit area of similar accessible lands].

This example utilizes a complex design that combines spaceborne remote 

sensing technologies (protocol M = 0) and a census of pixels at Phase 1 (s = U) 

and the same remote sensing plus FIA fi eld data (protocols M = 1 and M = 2, 

respectively) at Phase 2 (s = 2). The outcome is the two-phase vector estimate, 

similar to Eqs. 24 to 27, where the covariance matrix for Phase 1 exactly equals a qM=1-by-qM=1 zero matrix.

This example assumes all remotely sensed pixels are classifi ed with the M = 1 

protocol into likely “forest” and “non-forest” categories. The total areas of the 

population within each category (denoted A

f

 and Anf, respectively) are constants 

that are known without error through census of all pixels that cover the population. The remotely sensed “forest” and “non-forest” classifi cations (M = 1) 

include some portion of pixel-level classifi cation errors relative to fi eld measurements with the M = 2 protocol, but the census of pixel classifi cations contain no 

sampling error.

It is further assumed that the Phase 2 measurements (M = 2) include crossclassifi cation of landowner accessibility and remotely sensed classifi cation of 

“forest” cover for each Phase 2 sampling unit. Therefore, a sample estimate is 

available for total area of inaccessible lands that are classifi ed with remotely sensing into the “forest” the category.

The exact area within the population that is classifi ed as “forest” with the 

M = 1 remote sensing protocol from the Phase 1 census of pixels (s = U) is:

[ ]

{ }

¦ [ ] ==

=

=

=

»»1⁄4o

««¬a

=

A

U

i

sU

A

fA

M1

M2

,2

M1

ˆˆ

hytt

h0

 (91)

The estimated area of “denied access” in the entire population (t

1

) is assumed to be 

a single element of the vector estimate. It is extracted by the linear transformation:

( ) [ ] ( )

M2 { } ,2

M1

11

ˆˆ

ˆ

sU

ff

t

=

=

=

»»1⁄4o««¬a

=

tt

0h (92)

where the 1-byqM=2 indicator matrix (h

1

)f is composed of zero elements, except 

the one element (equal to 1) for denied access conditions (M = 2) that are categorized as forest cover with the re mote sensing protocol (M = 1).

The total biomass accretion for the “access granted” segment of the population, which is estimated from the “access granted” fi eld plots, is assumed to be a 

single element of the vector estimate, and it is extracted from the vector estimate 

with the linear transformation:

( ) [ ] ( )

M2 { } ,2

M1

22

ˆˆ

ˆ

sU

ff

t

=

=

=

»»1⁄4o««¬a

=

tt

0h (93)

where the 1-byqM=2 indicator matrix (

h

2

)f in Eq. 93 is composed of zero elements, 

except the one element (equal to 1) for “granted access” conditions (M = 2) that 

are categorized as forest cover with the remote sensing protocol (M = 1) for each 

sample plot in the Phase 2 sample (s = 2).
USDA Forest Service RMRS-GTR-239. 2010. 79

The desired population parameter is the total biomass accretion from lands that 

are expec ted to have “denied access” and that are classifi ed as “forest” cover with 

the spaceborne sensor (M = 1). The presumed model is the product of the [area 

of “denied assess” in the population that is imperfectly classifi ed as forest with 

remote sensing] times [the biomass accretion per unit area of “granted access” in 

“forest” cover as classifi ed with the remote sensing]:

( ) ( ) »

1⁄4

o «

¬

a = f

f

f f A

t

t 2 θ 1 (94)

The pseudo-estimator for the population parameter in Eq. 94 is a constant (Eq. 91) 

times the product of two population estimates:

()( ) f f

f

f t t

A 1 2 ˆ ˆ 1 ˆ  ̧

 ̧

1

·

 ̈

 ̈

©

§

θ = (95)

This model is similar to that of Martinussen and others (2008) for forest area estimates in presence of non-random missing observations.

The product of two estimates in Eq. 95 is a non-linear function for which the 

pseudo-estimator uses the fi rst-order T aylor-series approximation for the variance 

estimator. The partial fi rst derivatives of the non-linear function in Eq. 94 are:

() ()

( )

( )

() ()

( )

( )f

f t t t t f

f f

f

f t t t t f

f f

t

A A

t t

t

a

t

A A

t t

t

a

f

f

1

ˆ ˆ

1 2

2

2

2

ˆ ˆ

1 2

1

1

ˆ 1

ˆ 1

1 1, 2 2

1 1, 2 2

 ̧

 ̧

1

·

 ̈

 ̈

©

§ =  ̧

 ̧

1

·

 ̈

 ̈

©

§

∂

∂ ≈

 ̧

 ̧

1

·

 ̈

 ̈

©

§ =  ̧

 ̧

1

·

 ̈

 ̈

©

§

∂

∂ ≈

= =

= =

 (96)

From Eqs. 89, 90, and 96, the fi rst-order Taylor-series approximation of the nonlinear pseudo-estimator is:

[ ] ( ) ( ) [ ] ( ) ( )

( ) [ ] ( ) ( ) ( ) [ ] ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

[ ] () () ( )

( ) [ ] M 2 { } ,2

2

1

2 1

2 2

2 1 1 1

2 2

1

1 1

2

1 1 1 2 2 2

ˆ ˆ ˆ 1

ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ

s U

f

f

f f

f

f

f f

f f

f

f

f

f

f

f f

f

f

f f

f

f

f

f f f f f f

t t

A

t t

t t

A

t

A

t

t t

A

t

t t

A

t

a t t a t t

» = =

1⁄4

o «

¬

a = +

»

»

1⁄4

o

«

«

¬

a

−

−

»

»

1⁄4

o

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

 ̧

 ̧

1

·

 ̈

 ̈

©

§ = +

−  ̧

 ̧

1

·

 ̈

 ̈

©

§ − +  ̧

 ̧

1

·

 ̈

 ̈

©

§ = +

= + − + −

t

h

h

θ

θ

θ

θ θ

 

 (97)
80 USDA Forest Service RMRS-GTR-239. 2010.

As in Eq. 102, the approximation in Eq. 97 yields an approximate variance estimator for the scalar pseudo-estimator:

( ) ( ) ( ) () ( )

( ) ()

( )

( )

( ) ( ) ( )

{ }

( )

( )

( )

( )

( ) ( ) ( ) ( ) [ ] { } ( ) [ ] ( ) ( ) ( ) ( )

′

»

»

1⁄4

o

«

«

¬

a

+

»

»

1⁄4

o

«

«

¬

a = +

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

 ̧

 ̧

1

·

 ̈

 ̈

©

§

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

©

§

»

1⁄4

o «

¬

a

′

′

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a

»

1⁄4

o «

¬

a

»

1⁄4

o «

¬

a

»

»

1⁄4

o

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

 ̧

 ̧

1

·

 ̈

 ̈

©

§ =

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

 ̧

 ̧

1

·

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a

»

»

1⁄4

o

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

 ̧

 ̧

1

·

 ̈

 ̈

©

§ =

= =

= =

=

f

f

f

f

f

f

f s U

f

f

f

f

f

f

f

f

f

f

f

s U

f

f

f

f

f

f

f

f

f

f f

f f

f

f

f

f

f

A

t

A

t

A

t

A

t

A

t

A

t

A

t

A

t

A

t

A

t

C t t V t

V t C t t

A

t

A

t

V

2

1

1

2

2 M 2 ,2

1

1

2

1

2

2

1

M 2 ,2

M 1

2

2 1 1

1

2

1 2 2

2 1 1 2 1

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ

ˆ

ˆ

ˆ ˆ ˆ ˆ

ˆ

ˆ

ˆ ˆ ,ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ,ˆ ˆ ˆ

h h V t h h

0 h

0 h

t

t V

0 h

0 h

θ

 (98)

Equation 98 is an unbiased variance estimator for the linear transformation in 

Eq. 97, which is accepted as reasonable for the desired non-linear transformation 

in Eq. 95. The second- and higher-order derivatives in Eq. 96 equal zero, i.e., 

R = 0 in Eq. 91. This is the case for all product functions. Therefore, the fi rstorder Taylor-series for the product pseudo-estimator is exact in Eq. 98. It is not 

an approximation. However, in order to calculate the derivatives a1 and a2 in 

Eq. 96, the imperfect estimates in Eqs. 94 and 93 are required. Regardless, the 

pseudo-estimator for biomass accretion domain should not use the approximation 

in Eq. 97 but rather the direct product in Eq. 95.

The variance equation for the pseudo-estimator (Eq. 98) may be combined 

with linear transformations of the vector estimator. For example, “What is the 

estimated biomass accretion for the entire population, regardless of its accessibility condition and its remotely sensed land cover classifi cation?” The answer (θ) 

is the sum of the total estimated accretion from the following four elements in the 

vector estimate: (1) two-phase, design-consistent estimate of accretion based on 

direct fi eld re-measurements of “granted access” plots that have remotely sensed 

classifi cation of “forest” (Eq. 93); (2) two-phase, design-consistent estimate of 

accretion based on direct fi eld re-measurement of “granted access” plots that have 

remotely sensed classifi cation of “non-forest” (similar to Eq. 93); (3) two-phase, 

model-based estimate of accretion from “denied access” plots that have remotely 

sensed classifi cation of “forest” (Eqs. 95 and 98); and (4) two-phase, modelbased estimate of accretion from “denied access” plots that have remotely sensed 

classifi cation of “non-forest” (similar to Eqs. 95 and 98). The estimated scalar 

parameter and its variance are:
USDA Forest Service RMRS-GTR-239. 2010. 81

() () ( ) ( )

[ ]

( )

( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

[ ] { }

[ ] { }

( ) [ ] { } ( ) θ θ

θ

θ

θ θ θ

h V t h

h t

t

h h

h h

h

h

= ′

=

»

»

»

»

»

»

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

«

«

«

«

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

+

 ̧

 ̧

1

·

 ̈

 ̈

©

§

+

=

= + + +

= =

= =

= =

M 2 ,2

M 2 ,2

M 2 ,2

2

1

1

2

2

1

1

2

2

2

2 2

ˆ ˆ ˆ ˆ

ˆ

ˆ

ˆ ˆ

ˆ ˆ 1 1 1 1

ˆ ˆ ˆ ˆ ˆ

s U

s U

s U

nf

nf

nf

nf

nf

nf

f

f

f

f

f

f

nf

f

f nf f nf

V

A

t

A

t

A

t

A

t

t t

 (99)

where

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) »

»

1⁄4

o

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

+ +  ̧

 ̧

1

·

 ̈

 ̈

©

§ = + + + nf

nf

nf

f

f

f

nf

nf

nf

f

f

f

A

t

A

t

A

t

A

t

2

1

2

1

1

2

1

2 ˆ 1 ˆ 1 ˆ ˆ hθ h h h h

and the nf subscript in Eq. 99 denotes the sub-population identifi ed as “non-forest” 

with the remotely sensed data, analogous to the f subscript in Eqs. 91 through 98.

Although not developed here, this approach to missing data may be enhanced 

with high-resolution aerial photography, which was previously identifi ed as the 

M = 1 protocol. This could combine the model-based estimate (Eqs. 91 to 99) 

with a design-based estimate of photo-interpreted variables for the missing-data 

domain. Multivariate estimation facilitates full consideration of the dependencies 

between predictor and response variables. In addition, it would be convenient to 

express this estimator as expansion values (e.g., Lundström and Särndal 1999). 

These enhancements require future research and development.
82 USDA Forest Service RMRS-GTR-239. 2010.

Ratio Pseudo-Estimator: Volume per Estimated Unit Area of 

Forest Cover

The ratio is a common, non-linear transformation. For example, consider the 

ratio of two estimated population totals, such as the ratio of total wood volume (t

1

) 

to total forest area (t

2

), namely, the estimated wood volume per unit area of forest. 

From Eqs. 89 and 90:

( )

( ) ( ) ˆ

ˆ

ˆ ˆ

1

ˆ

where , ˆ 0, ˆ 0 and

ˆ ˆ

ˆ ˆ

2

2

1

2 ˆ ˆ

1 2

2

1 ˆ ˆ 2

1 2

1

2 2

2

2 1

1

2

1

1 1, 2 2 1 1, 2 2 π

π

π π π

π

π

π

π π π π

θ θ

θ

t

t

t

t t

a

t t

t t

a

t t

a t t

t

t

t

t

t t t t t t t t

j

j j j

= − ∂

∂ = ≈ ∂∂ ≈

≠ ≠

= ≈ + −

=

= = = =

=

¦

 (100)

The multivariate Taylor series linear approximation is simply:

( )( )

( ) ( )

( ) ( )

( )

( )»

1⁄4

o «

¬

a

−

−

»

»

1⁄4

o

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§ = + −

−  ̧

 ̧

1

·

 ̈

 ̈

©

§ = + − + −

−  ̧

 ̧

1

·

 ̈

 ̈

©

§ = + − + −

= + − + −

2 2

1 1

2 2

2 2

2

1 1

2

2 2 2

2

1

1 1

2

1 1 1 2 2 2

ˆ

ˆ

ˆ

ˆ

ˆ

1

ˆ ˆ

ˆ ˆ ˆ

1

ˆ ˆ

ˆ ˆ ˆ

1

ˆ ˆ ˆ

t t

t t

t t

t t

t

t t

t

t t

t

t

t t

t

a t t a t t

π

π

π π

π

π

π

π

π

π

π

π

π

π π

θ θ

θ θ

θ

θ θ

 (101)

This simple calculus is easily applied to the variance estimator for thepseudoestimator in Eq. 101:

( ) () ( )

( ) () »

»

»

»

1⁄4

o

«

«

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§

− »

»

1⁄4

o

«

«

¬

a

»

»

1⁄4

o

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§ ≈ −

π

π

π π π

π π π

π π θ

θ θ

2

2

1 2 2

1 1 2

2 2

ˆ

ˆ

ˆ

1

ˆ ˆ ,ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ,ˆ

ˆ

ˆ

ˆ

1 ˆ ˆ

t

t

C t t V t

V t C t t

t t

V (102)
USDA Forest Service RMRS-GTR-239. 2010. 83

As a verifi cation exercise, the Taylor-series approximation agrees with the 

results offered by Särndal and others (1992) in their Eq. 5.6.10. Algebraically 

manipulating Eq. 102:

( ) [ ] () ( )

[ ] ( ) ()

[ ] () ( ) [ ] ( ) ()

[ ] () ( ) ( ) ()

[ ] () ( ) () π π π π

π

π π π π π π

π

π π π

π

π π π

π

π π π

π

π π π

π

π π

θ θ

θ θ θ

θ θ θ

θ

θ

θ θ

2

2

2 1 1 2

2

2

2

2 1 1 2 1 2

2

2 1 2 2

2

2 1 1 2

2

1 2 2

2

1 1 2

2

2 2

ˆ ˆ 2 ˆ ˆ ˆ ,ˆ ˆ ˆ ˆ ˆ

1

ˆ ˆ ˆ ˆ ˆ ,ˆ ˆ ˆ ˆ ,ˆ ˆ ˆ ˆ ˆ

1

ˆ ˆ ,ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ,ˆ ˆ

1

ˆ ˆ ,ˆ ˆ ˆ ˆ ˆ

1

ˆ ˆ ˆ ˆ ˆ ,ˆ ˆ

1

ˆ

ˆ

ˆ

1 ˆ ˆ

V t C t t V t

t

V t C t t C t t V t

t

C t t V t

t

V t C t t

t

C t t V t

t

V t C t t

t

t t

V

= − +

= − − +

−  ̧

 ̧

1

·

 ̈

 ̈

©

§ = − + −

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

−

−

»

»

1⁄4

o

«

«

¬

a

 ̧

 ̧

1

·

 ̈

 ̈

©

§ = −

In practice, the pseudo-estimator is computed directly from the non-linear 

transformation. For example, the pseudo-estimator for the ratio of total wood volume (t

1

) to total forest area (t

2

) from Eq. 100 should be:

, ˆ 0 ˆ

ˆ ˆ 2

2

1 = π ≠

π

π θ t

t

t

 

 (103)

The variance estimator in Eq. 102 is actually a conditionally unbiased estimator of the linear transformation in Eq. 101. It is only an approximation when used 

as a variance estimator for the non-linear transformation in Eq. 100. Regardless, 

the pseudo-estimator for the desired population parameter should use the exact 

non-linear transformation in Eq. 100, with the Taylor-series approximation in 

Eqs. 101 and 102 used only for the associated variance and covariance estimates.

Recursive Object-Oriented Simplifi cation of Taylor-Series 

Approximations

The methods given above for pseudo-estimators greatly facilitate constructi on 

of complex, multivariate, linear Taylor approximations. These can be daunting 

when dealing directly with the variance estimator, but are relatively easy when 

applied directly to the pseudo-estimator (e.g., Särndal and others 1992:205). 

In general, the method is object-oriented in that it disassembles a complex 

pseudo-estimator into small components that are recursively combined into the 

fi nal Taylor series function.

For example, consider the product estimator for missing data on p. 77. Equation 

94 defi nes the pseudo-estimator of the population total for biomass accretion on 

private lands that cannot be measured directly because of denied access, which is 

repeated here:

( ) ( ) »

1⁄4

o «

¬

a = f

f

f f A

t

t 2 θ 1

 (94)

The recursive process combines the objects in the complex Taylor series approximation in the following sequence:
84 USDA Forest Service RMRS-GTR-239. 2010.

1. The estimated area of “denied access” in the entire population (t

1

) is estimated 

with the linear transformation in Eq. 92, resulting in an indicator vector of 

zeros and ones.

2. The total biomass accretion for the “access granted” segment of the population 

(t

2

)f is estimated with the linear transformation in Eq. 93, resulting in an 

indicator vector of zeros and ones.

3. The census area imperfectly classifi ed as “forest” cover (Af) is computed with 

Eq. 57.

4. These two row vectors from steps 1 and 2 are stacked to form a 2-by-(qM=1 +

qM=2) indicator matrix.

5. This indicator matrix is multiplied by the 1-by-2 vector containing estimates of 

(t

1

) and (t

2

)f from steps 1 and 2.

6. The resulting 1-by-(qM=1 + qM=2) vector is multiplied by the scalar inverse of the 

remotely sensed census of forest cover in the target population from step 3.

The fi nal 1-by-(qM=1 + qM=2) vector from step 6 is available for the linear Taylor 

series approximation for the variance of the pseudo-estimator in Eq. 94. This recursive process is continued in Eq. 99 to accommodate an even more complicated 

pseudo-estimator for the entire target population. Although not fully shown, the 

complex ratio pseudo-estimator in Eq. 101 is similarly developed as a recursive 

process. These procedures are readily applied to non-linear transformations of 

large numbers of elements in a vector of estimated population parameters and its 

estimated covariance matrix. The following is an example.

Returning to the example of total wood volume per unit forest area, assume 

that total wood volume in the vector estimate is distributed among multiple elements that segregate wood volume by tree species groups, and total forest area 

in the vector estimate is distributed among multiple elements that segregate 

forest area by forest type groups. Given a q-by-1 vector estimate of t, the pseudoestimator is simply the [sum of all estimated volume over all tree species groups = 

t

1

] divided by the [sum of all area estimates for each forest type group = t

2

]:

[ ] [ ]

( ) [ ]

( )

( ) [ ]

( )

[ ] °

°

°

°

 ̄

°

°

°

°

®

­

≠

° ̄

°

®

­

=

= =

° ̄

°

®

­

=

= =

 ̧

1

·  ̈

©

§ =

ˆ 0

0 otherwise

is an estimated forest area of ˆ 1if element 

element of

0 otherwise

1if element of ˆ is an estimated volume

element of

where

ˆ

ˆ ˆ

2

2

2

2

1

1

1

2

1

h t

h

h t

h

h

h t

h

h t

h t

i

i i

i

i

i i

i

th

th

th

th

θ

 (104)

Both h1

 and h2 in Eq. 104 are 1-by-q row vectors. The Taylor-series approximation 

of this pseudo-estimator is derived from Eq. 89 as:
USDA Forest Service RMRS-GTR-239. 2010. 85

( ) [ ]V( ) [ ]t = a[ ] V( )t a′

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

≈

=

ˆ ˆ ˆ ˆ ˆ ˆ

M 2

2

1

1 2

q

q

a

a

a

V a a a  θ  (105)

where

[ ] ( )

[ ] ( )

°

°

°

 ̄

°

°

°

®

­

 ̧

 ̧

1

·

 ̈

 ̈

©

§−

 ̧

1

·  ̈

©

§

= ∂

∂ =

0 otherwise

if ˆ is an estimated forest area ˆ

ˆ

if ˆ is an estimated estimated volume ˆ

2

2

i a

i a

a

a i

i

i

i t

h t

t

h t

θ θ

With modest algebra, the 1-by-q row vector a in Eq. 105 may be expressed as:

[ ] ( ) 1 2

2

ˆ ˆ 1 h h

h t

a  ̧ −θ 1

·  ̈

©

§ = (106)

The simple expression in Eq. 106 yields a valid Taylor-series variance approximation for any scalar statistic that is the ratio of two scalar linear tra nsformations 

of t, i.e., (h1t)/ (h2t), where the elements of h1 and h2 may be any real number such 

that h2t≠0, even if the same element of t has a non-zero coeffi cient in both the 

numerator and denominator of this ratio statistic.

The following derivation of Eq. 106 is offered as another example of a simplifi ed approach to Taylor-series approximations. First, collapse the q-by-1 vector 

estimate of t into a 2-by-1 vector, where the fi rst vector element is the sum of all 

elements in the t vector that are in the numerator of the pseudo-estimator, and 

the second vector element is the sum of all elements in the t vector that are in the 

denominator of the pseudo-estimator.

[ ]t

h

h ˆ ˆ

ˆ , where

2

1

2

1

2

1 »

1⁄4

o «

¬

a =

»

»

1⁄4

o

«

«

¬

a  ̧

1

·  ̈

©

§ = t

t

t

t θ (107)

Then, treat the resulting 2-by-1 vector in Eq. 107 as the ratio of two scalar estimates in Eqs. 101 and 102 and Eq. 5.6.10 in Särndal and others (1992). The 

outcome is simply:
86 USDA Forest Service RMRS-GTR-239. 2010.

[ ] [ ]

[ ] [ ]

[ ] ( ) 1 2

2

2

1

2

2

1

2 2

ˆ ˆ 1

ˆ 1 ˆ 1

ˆ

ˆ

ˆ 1

h h

h t

h

h

h t

h

h

h t h t

a

θ

θ

θ

 ̧ − 1

·  ̈

©

§ =

»

1⁄4

o «

¬

a  ̧ − 1

·  ̈

©

§ =

»

1⁄4

o «

¬

a

»

1⁄4

o «

¬

a  ̧

1

·  ̈

©

§−  ̧

1

·  ̈

©

§ =

 (108)

which validates Eq. 106.

The Taylor-series approximation may be stacked to recursively construct more 

complex pseudo-estimators (Wolter 2007:244). For example, the estimated difference between total volume per unit area of forest type A compared to forest 

type B is:

( )

( )

( )

( )  ̧

 ̧

1

·  ̈

 ̈

©

§ − ̧

 ̧

1

·  ̈

 ̈

©

§ =

 ̧

1

·  ̈

©

§  ̧

1

·  ̈

©

§ =

= −

h t

h t

h t

h t

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ - ˆˆ

ˆ ˆ ˆ

2

1

2

1

2 B

1

2 A

1

B

B

A

A

A B

t

t

t

t

θ θ θ

 (109)

The corresponding fi rst-order Taylor-series linear approximation of the variance 

for the scalar difference in Eq. 109 is easily derived with the help of E q. 106 from 

the q-by-1 vector estimate of t and its estimated q-by-q covariance matrix with 

the 1-by-q vector a as:

( )= a [ ] V( )t

ˆ a′ ˆ ˆ V θ (110)

where

[ ] ( ) ( )

( ) ( )

( )

( )

( )

( )

( ) [ ] () () ( ) [ ] () ()  ̧

 ̧

1

·

 ̈

 ̈

©

§ − »

1⁄4

o «

¬

a − ̧

 ̧

1

·

 ̈

 ̈

©

§ − »

1⁄4

o «

¬

a =

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

−

−

= −

B B B

B

A A A

A

B

B

A

A

B

B

B

A

A

A

1 2

2

1 2

2

2

1

2

1

2 2

2 2

ˆ ˆ ˆ 1 ˆ 1

ˆ

ˆ

ˆ

1 0 0

0 0 ˆ

ˆ

ˆ

1

1 1

h h

h t

h h

h t

h

h

h

h

h t h t

h t h t

a

θ θ

θ

θ

Equation 110 may be rewritten as:
USDA Forest Service RMRS-GTR-239. 2010. 87

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

d V d

d

h

h

h

h

V t

h

h

h

h

d

′

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

=

′

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

 ̈

 ̈

©

§ ′

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

=

B

B

A

A

B

B

A

A

B

B

A

A

t

t

t

t

V

2

1

2

1

2

1

2

1

2

1

2

1

ˆ

ˆ

ˆ

ˆ

ˆ θ

ˆ ˆ

 (111)

where

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) »

1⁄4

o «

¬

a  ̧

1

·  ̈

©

− §  ̧

1

·  ̈

©

− § =

»

1⁄4

o «

¬

a − − =

2 B

1

A 2 2 2

1

2 2

2 2 2 2

ˆ

ˆ

ˆ

1

ˆ

1

ˆ

ˆ

ˆ

1

ˆ

1

ˆ

ˆ

ˆ

1

ˆ

ˆ

ˆ

1

t

t

t t t

t

t t A A B B

B

B

A B

A

A h t h t h t h t

d θ θ

Equation 111 agrees with Wolter’s (2007:Eq. 6.9.3) results. Again, the covariance 

matrix may be the fi nal results from the sequential application of RRE in any of 

the complex designs above. The estimators in Eqs. 109 and 110 might serve in 

a test-of-hypothesis regarding signifi cant differences in volume per unit area of 

forest type A compared to forest type B.

While the procedures above require tedious application of simple matrix algebra, the tedium would be far more onerous with scalar algebra conducted on 

the full non-linear transformation, especially if applied directly to the covariance 

matrix with double summations. The scalar operators of addition, subtraction, 

multiplication, and inverse are considered above. Power functions are not, but 

these should not be too daunting in a recursive process. Furthermore, except for 

division, the second-order terms of the Taylor series are zero for these operators. Therefore, this recursive process could be used to generate more simply the 

second-order Taylor series approximation for any scalar pseudo-estimator that 

uses these four operators.

A Model-Based Estimator for Small-Domain Predictions

The next example uses a moderately intensive sample of relatively “lessexpensive” fi eld measurements of tree volume and a sparse sub-sample of 

“more-expensive” volume measurements. The “less-expensive” protocol includes 

systematically biased measurement errors, while the “more-expensive” protocol 

has unbiased measurement errors. Model-based predictions are applied to small 

domains, many of which do not have suffi cient sample sizes for reliable direct 

estimates with the unbiased, “more-expensive” measurements of individual tree 

volumes.

Suppose a three-phase sampling design (e.g., Eqs. 44 to 51) employs inexpensive airborne photo-interpretation techniques (protocol M = 1), including 

photogrammetric estimates of stand volume; FIA fi eld data (protocol M = 2), including relatively “less-expensive” measurements of tree diameter at breast height 

(DBH) and regional tree-volume equations, and a very precise “more-expensive” 

fi eld protocol (M = 3), including upper stem diameter measurements made with 

ground-based laser instruments. Protocol M = 1 is applied to a very large Phase 1 
88 USDA Forest Service RMRS-GTR-239. 2010.

sample (s = 1) of photo-interpreted sampling units. Protocols M = 1 and M = 2 are 

applied to all points in the moderate-intensity Phase 2 sample (s = 2). Protocols 

M = 1, M = 2, and M = 3 are applied to all points in the sparse Phase 3 sample 

(s = 3). The outcome is the three-phase vector estimate similar to Eq. 50.

The “more-expensive” M = 3 fi eld protocol very accurately measures upper 

stem diameters and total tree heights of individual standing trees (e.g., Williams 

and others 1999). These physical measurements are inputs to a sophisticated stem 

profi le model (e.g., Flewelling and others 2000) that provides accurate and unbiased estimates of wood volume for individual standing trees. Because these 

measurements are very expensive, they are only available for the sparse sample of 

Phase 3 (s = 3) plots, although correlations among variables measured at Phase 1 

and Phase 2 improve overall precision in the three-phase sampling design.

As in Eq. 20, the vector of remotely sensed variables for each point i in the 

Phase 1 sample is composed of:

[ ]

( )

( )

( )

( )

( ) [ ] [ ]

( )

¦∈ =

=

=

= =

=

=

=

=

= =

»

»

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

«

«

¬

a

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

¬

a

= 1

M 1

1

1 M 1

M 1

M 1

M 1

1 M 1

M 1 ˆ

j s

i

s

s

U i

q D

i D

D

i

n

A

I

I

I

D

t y

y

y

y

y

y 



 (112)

where

( )

( ) other remotely sensed variables

0 otherwise

1if sample point isin domain 

remotely sensed variables, for domain estimates

M 1

M 1

=

 ̄

®

­ =

=

=

=

U

i

D

i

I

y

y

The vector of remotely sensed and fi eld variables for each point i in the Phase 2 

sample is defi ned in Eq. 113 as:

( )

( )

( )

( )

( ) [ ]

( )

¦∈ = =

=

=

= =

=

=

=

=

=

=

=

»

1⁄4

o «

¬

a =

»

»

»

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

«

«

«

¬

a

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

¬

a

= »

1⁄4

o «

¬

a

2 M 2

M 1

2

2 M 1,2

M 2

M 2

M 2

1 M 2

M 1

M 2

M 1 ˆ

j s s i

s

U i

q D

i D

D

i n

A

I

I

I

D

y

y t

y

y

y

y

y

y

y





 (113)

where

( )

( ) other "less- expensive"field variables, for population estimates

0 otherwise

1if sample point isin domain 

"less- expensive"field variables, for domain estimates

M 2

M 2

=

 ̄

®

­ =

=

=

=

U

i

D

i

I

y

y
USDA Forest Service RMRS-GTR-239. 2010. 89

Finally, the vector of remotely sensed, “less-expensive” fi eld variables, and 

“more-expensive” fi eld variables for each point i in the Phase 3 sample is defi ned 

in Eq. 114 as:

( )

( )

( )

( )

( ) [ ]

( )

¦∈ =

=

=

=

=

= =

=

=

=

=

=

=

=

=

=

»

»

»

1⁄4

o

«

«

«

¬

a

=

»

»

»

»

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

«

«

«

«

¬

a

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

¬

a

=

»

»

»

1⁄4

o

«

«

«

¬

a

3

M 3

M 2

M 1

3

3 M 1,2,3

M 3

M 3

M 3

1 M 3

M 2

M 1

M 3

M 2

M 1

ˆ

j s

i

s

s

U i

q D

i D

D

i

n

A

I

I

I

D

y

y

y

t

y

y

y

y

y

y

y

y

y





 (114)

where

( )

( ) other "more - expensive"field variables, for population estimates

0 otherwise

1if sample point isin domain 

"more - expensive"field variables, for domains estimates

M 3

M 3

=

 ̄

®

­ =

=

=

=

U

i

D

i

I

y

y

RRE is sequentially used to combine the independent estimates from the three 

phases of sampling, as in Eqs. 44 to 51.

Defi ne the scalar t

1

 as the population total for tree volume measured with the 

M = 3 protocol, a scalar estimate for which is extracted from the fi nal RRE vector 

estimate (e.g., Eq. 50) with the 1-by-qM=3 indicator matrix h1 in Eq. 115:

[ ]

1,2,3 M 3

M 2

M 1

1 1

ˆ

ˆ

ˆ

ˆ

= =

=

=

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

=

s

t

t

t

t

0 0 h (115)

The M = 2 fi eld protocol includes “less-expensive” estimates of individual tree 

volumes that use measurements of DBH, ocular estimates of tree heights, and 

regional volume equati ons that were fi t to off-frame convenience sample of historical data. Hypothetically, assume this M = 2 protocol produces biased estimates 

of individual tree volume at both the population and small-domain levels. In these 

situations, biases of 10 percent or more are possible, especially for domains that 

are small relative to the geographic scope of the database that was used to fi t the 

regional volume equation. The vector estimate from the Phase 2 sample includes 

separate elements for the total volume in each of qD domains, such as counties 

(Särndal and others 1992:220, 304, 397), as measured with the M = 2 protocol. 

The sampling error at Phase 2 is suffi ciently small to make reliable tree volume 

estimates for each of qD small domains. However, these estimates are biased because the “less-expensive” tree volume measurements (M = 2) are biased.
90 USDA Forest Service RMRS-GTR-239. 2010.

The vector partition with estimated total tree volumes for each of the q

D domains, which are measured with the M = 2 protocol, is isolated from the overall 

RRE vector estimate (Eq. 50) with the q

D-byqM=2 indicator matrix 

H

3

:

[ ]

1,2,3

M3

M2

M1

33

ˆˆˆ

ˆ

=

===

»»»»1⁄4o

««««¬a

=

s

ttt

t0H0 (116)

In this example, all elements in the conformable indicator matrix H

3

 equal zero 

except for each of the q

D elements that corresponds to the volume estimate made 

with the M = 2 protocol, in which case the element of H

3

 equals 1. Assume for 

notational convenience that these q

D elements are contiguous in the vector estimate; thus, the elements equal to 1 in indicator matrix H3

 are an identity matrix 

partition (I) in Eq. 116.

Defi ne the sum of all elements in the t

3

 vector as the sum of all estimated domain volumes with the M = 2 protocol as t2:

[ ]

[ ] ¦( ) [ ] = = =

=

===

=

»»»»1⁄4o««««¬a

==

q

D

d

s

d

s

t

t

1

M2

1,2,3

1,2,3

M3

M2

M1

2

23

ˆ

ˆˆˆ

ˆ

11

ˆ

ttt

0h0

t

 (117)

The scalar t2 in Eq. 117 equals the total tree volume in the entire population as 

measured with the biased M = 2 protocol. This scalar estimate is isolated from 

RRE vector estimate ( E q. 50) with the 1-byqM=3 indicator matrix h

2

. All elements 

of the conformable row vector h

2

 in Eq. 117 equal zero except for any of the q

D

elements corresponding to estimated total tree volume made with the M = 2 protocol for a domain; each of these latter elements equals 1 in h2

. For convenience 

of notation, these q

D elements are contiguous in RRE vector estimate (Eq. 50).

The objective is an unbiased, model-based prediction of the total volume for 

each of the q

D small domains using the biased estimates of total volume, which are 

based on the “less-expensive” M = 2 protocol. The model assumes that the bias 

of the M = 2 protocol relative to the M = 3 protocol, which is observed directly 

from the Phase 3 sample (s = 3), is constant for all domains. The resulting vector 

pseudo-estimator is defi ned for each of the q

D domains as:

»»»1⁄4o«««¬a

 ̧1·

 ̈©§

=

»1⁄4o«¬a

 ̧1·

 ̈©§

==

»»»1⁄4o

«««¬a

====

D

dq

D

d

dq

d

tt

t

t

t

t

ˆˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆˆ

1

2

1

3

2

1

1

șt

θ θ

 (118)

This model in Eq. 118 is similar to a synthetic estimator for small domains (Särndal 

and others 1992:410), which often provides exceptionally precise estimates, but it 

can induce consid erable estimation bias if the model is inaccurate. The latter is a 

substantial risk because there is often no empirical evidence to judge the accuracy 

of the model.
USDA Forest Service RMRS-GTR-239. 2010. 91

The two scalar estimates (t

1

 and t

2

) and the qD-by-1 column vector estimate (t3) 

in Eq. 118 equal the linear matrix function (Eqs. 115 to 117):

»

»

1⁄4

o

«

«

¬

a

′

′ ′

»

»

»

1⁄4

o

«

«

«

¬

a

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

=

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

=

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

= =

=

= =

=

=

h 0 0

0 h H

t

t

V

H 0

h 0

0 h

t

V

t

t

t

0 H 0

0 h 0

0 0 h

t

1

2 3

M 3 1,2,3

M 2

3

2

1

3

2

1

1,2,3 M 3

M 2

M 1

3

2

1

3

2

1

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

s

s

t

t

t

t

 (119)

The model in Eq. 118 is the basis for a non-linear pseudo-estimator of total volume in each of the qD domains (i.e., elements of qD-by-1 column vector θ) based 

on the “more-expensive” M = 3 tree measurement protocol.

Row i for the vector model in Eq. 118 is:

 ̧

 ̧

1

·

 ̈

 ̈

©

§

+ + + + + + = ̧

 ̧

1

·

 ̈

 ̈

©

§ = = = = =

= =

=

d d i d j d qD

d i d i

d i t t t t

t

t

t

t

t1

2

1

1    θ (120)

The partial fi rst derivatives for the function in Eq. 120, evaluated at the estimated 

values, are:

2

2

1

2

2

1

2

1

2

1 1

1

ˆ

ˆ

ˆ

ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

t

t t

t t

t

t

t t

t

t

t t t

t

t

t t t

d i

j i

j

j i

j

d i

d j

d i

i i

d i

d i

d i

d i

=

≠ ≠

=

=

=

=

=

=

=

− =

=  ̧

 ̧

1

·

 ̈

 ̈

©

§

∂

∂

= −

=  ̧

 ̧

1

·

 ̈

 ̈

©

§

∂

∂

=

=  ̧

 ̧

1

·

 ̈

 ̈

©

§

∂

∂

θ

θ

θ

 (121)

From Eqs. 89, 90, and 121, the fi rst-order Taylor-series approximation of the nonlinear pseudo-estimator in Eq. 120 is:
92 USDA Forest Service RMRS-GTR-239. 2010.

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

»

»

»

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

«

«

«

¬

a

−

−

−

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−  ̧

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·

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·

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§ − + − −  ̧

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1

·

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§ − + −  ̧

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1

·

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©

§ ≈ +

= =

= =

= =

= =

=

= =

= =

=

= =

=

= =

= =

=

≠

=

= =

=

= =

= =

= =

¦

¦

D D

D

D

d q d q

d i d i

d d

d i d i

d i

d i d i

d i d i

d i

d i d i

q

j

d j d j

d i d i

d i

q

j i

j

d j d j

d i

d i d i

d i d i

d i d i

t t

t t

t t

t t

t t

t

t

t

t t

t

t

t t

t

t

t t

t

t t

t t

t

t

t t

t

t

t t

t

t t

t t

t

t

t t

t

t t

t t

t

t t

t

t

t t

t

t

ˆ

ˆ

ˆ

ˆ

ˆ

0 ˆ

ˆ 0 ˆ

ˆ ˆ

ˆ

ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ ˆˆ ˆ

ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ ˆˆ ˆ

ˆ ˆ

ˆ ˆ ˆ ˆ

ˆ ˆ

ˆ

ˆ ˆ ˆ

ˆ ˆ

1 1

2 2

1 1

2

1

2

2

1

2

2

1

2 2 2

2

1

1 1

2

2

1

1

2

2

1

1 1

2

1

2

2

1

2

2

1

2

1

1 1

2



θ   

θ

θ

θ θ

 (122)

Expanding Eqs. 118 to 122 for each domain into the 1-by-qD ve ctor of modelbased estimates of total volume θD for each of the qD domains:

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

−

−

−

≈ +

3 3

2 2

1 1

ˆ

ˆ

ˆ

ˆ

t t

ș ș A t t

t t

D D (123)

where A in Eq. 123 is the qD-by-(qD + 2) mat rix of coeffi cients from the Taylorseries approximation and the defi nitions in Eq. 119:

»

1⁄4

o «

¬

a − =

»

»

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

«

«

¬

a

−

−

−

=

= =

= =

= =

t t I

A

2

1

2 3

2

1

3

2

2

1

2

2

1 1

2

1

2

1

2

2

1 1

2

1

2

1

2

2

1 1

2

1

ˆ

ˆ ˆ ˆˆ ˆ ˆ

1

ˆ

ˆ 0 0 ˆ

ˆ ˆ

ˆ

ˆ

0 ˆ

ˆ 0 ˆ

ˆ ˆ

ˆ

ˆ

0 0 ˆ

ˆ

ˆ

ˆ ˆ

ˆ

ˆ

t

t

t

t

t

t

t

t

t t

t

t

t

t

t

t t

t

t

t

t

t

t t

t

t

d d

d d

d d



     





 (124)

Equations 123 and 124 yield an approximate estimat or (Eq. 125) of the covariance matrix for the vector pseudo-estimator of unbiased volume estimates for 

each domain (D = d) in the population :
USDA Forest Service RMRS-GTR-239. 2010. 93

[ ]

A

h 0 0

0 h H

t

t

V

H 0

h 0

0 h

A

A

t

V ș A V

′

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

©

§

»

»

1⁄4

o

«

«

¬

a

′

′ ′

»

»

1⁄4

o

«

«

¬

a

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

≈

′

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

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 ̈

 ̈

©

§

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

≈

= =

=

=

=

1

2 3

M 3 1,2,3

M 2

3

2

1

1,2,3 3

2

1

1,2,3

ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ ˆ ˆ

s

s

D s t

t

 (125)

Straightforward, though tedious, matrix algebra demonstrates that the population estimate, and the sum of all elements in the Taylor-series approximation of 

its covariance matrix, for the sum of model-based predicted volumes over all 

domains estimates equals the direct estimate of the population volume with the 

M = 3 protocol:

( ) A 1

t

1A V

1 ș

′ ′

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

 ̈

 ̈

©

§

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

=

=

=1,2,3 3

2

1

1

1

ˆ

ˆ

ˆ

ˆ ˆ ˆ

ˆ ˆ

s

t

t

V t

t

Therefore, the vector of model-based volume estimates θD for domain D, which 

uses the estimators in Eqs. 118 and 125, is consistent at the population level with 

the corresponding direct estimate (t

1

), which uses the unbiased M = 3 measurement protocol. The biases in model-based, domain-level variance and covariance 

estimates, which are caused by the remainders from the fi rst-order Taylor-series 

approximation, sum to zero when domain-level estimates are summed into population-level estimates. This supports generic inference (Opsomer and others 

2007). However, this analysis does not demonstrate that the model-based estimator is unbiased at the domain level.

Finally, the covariance matrix for the model-based volume estimates, which 

is defi ned from Eqs. 124 and 125, and the population estimates for the remaining 

variables measured with the M = 2 protocol equal:

»

»

»

»

1⁄4

o

«

«

«

«

¬

a

′ ′

′ ′ + ′ −

»

1⁄4

o «

¬

a

»

»

1⁄4

o

«

«

¬

a

+ = − »

1⁄4

o «

¬

a

= =

=

=

=

1 3

2

3

2

1

2 2 3

2

1

M 3 1,2,3

M 2

3 1

2

3

2

1

2 3 2

2

1

1,2,3

M 2

ˆ ˆ

1

ˆ

ˆ ˆ ˆ

ˆ

ˆ

ˆ ˆ ˆ ˆ

1

ˆ

ˆ ˆ ˆ

ˆ ˆ

ˆ ˆ

0 h t

I h t H

t

t

V t h H t h

I 0

ș

t V

t

t

t

t

t

t t

t

t

t

D s s

 (126)

In this way, the model-based domain estimates (θD) may be merged with the direct 

estimates (t

M=2) for subsequent analyses and further application of other linear 

or non-linear pseudo-estimators (e.g., volume per forested acre with the M = 3 

protocol for each domain).

This approach might be useful in other FIA applications. For example, estimates of wood volume cull are needed. Cull is defi ned by Helms (1998) as “parts 

of logs deducted ... from gross timber volume because of defects. ... in a standing 

(live) tree, the cull expressed as a percent of the tree’s gross volume is termed the 

cull factor.” Direct measurement of the cull factor often requires destructive sampling, which is not conducted on a permanent FIA sample plot. Rather, estimates 
94 USDA Forest Service RMRS-GTR-239. 2010.

of the cull factor are made external to the FIA sampling frame, and prediction 

equations are constructed to estimate the cull factor for on-frame sample trees 

based on the standard FIA non-destructive fi eld protocol. If the destructive measurements come from a probability sample, then the methods in this section are 

relevant. These methods do not produce a PSU weight for individual sample trees 

from the Phase 2 FIA sample. This is an advantage because it avoids the curious 

situation where the exact same cull factor is used for a large number of sample 

trees. This does not necessarily produce more accurate population estimates, but 

it does avoid anomalous results in the FIA database.

Second-Order Taylor-Series Approximation to Assess Bias

The covariance matrix for non-linear pseudo-estimators can be biased by 

the fi rst-order Taylor-series approximation because the second-order and higher 

terms in the expansion are treated as zeros (Ahmed and Al-Khasawneh 2005). In 

other words, the remainder R in Eq. 91 is assumed to be negligibly small, even 

though this assumption is rarely tested. The reason for ignoring the second-order 

terms is that the derivation of the covariance matrix estimator is perceived as too 

daunting, and the magnitude of the Taylor-series remainder beyond the fi rst-order 

approximation is perceived to be small with no practical signifi cance. Both perceptions might be inaccurate. This section explores a type of recursive method 

that simplifi es derivation of the second-order Taylor-series approximation to assess the degree of bias in a fi rst-order approximation. If the bias is unacceptably 

large, then the same methods yield the less biased second-order estimator for the 

covariance matrix.

For example, continue the relatively complex case of model-based, smalldomain volume estimation in Eqs. 115 to 125. Using the methods of Deutsch 

(1965:Chapter 6) and Särndal and others (1992:173-174, 205-207, 499), the second-order Taylor-series approximation is:
USDA Forest Service RMRS-GTR-239. 2010. 95

( )

»

»

»

»

»

»

»

»

»

»

»

»

»

»

»

»

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»

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1⁄4

o

«

«

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+

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+ ∂

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+ ̧

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§

∂

∂

+

≈ +

¦ ¦ ¦¦ ¦

¦ ¦ ¦

¦ ¦ ¦

¦ ¦ ¦

≠

=

≠

≠

=

≠

≠

=

= = =

=

= =

= =

≠

=

≠

=

= =

=

=

=

≠

=

≠

≠

=

= = =

=

= =

= =

≠

= =

=

=

=

≠

=

= =

=

=

=

≠

=

≠

=

= =

=

=

=

=

≠

=

= =

=

=

=

≠

=

≠

=

= =

=

=

=

=

D D D D D

D D

D

D D D

D D D

q

j i

j

q

m j

m i m

q

k m

k j

k

d j d m d k

d i

d j d m

d j d m

q

j i

j

q

k j

k

d j d k

d i

d j

d j

q

m i m

q

k m

k i

k

d i d m d k

d i

d i d m

d i d m

q

k i

d i d k

d i

d i

d i

q

k i

k

d i d k

d i

d i

d i

q

j i

j

q

k j

k

d j d k

d i

d j

d j

d i

q

k i

k

d i d k

d i

d i

d i

q

j i

j

q

k j

k

d j d k

d i

d j

d j

d i

i i

t t t

t t

t t

d d t t

t t

t

d

t t t

t t

t t

d d

t t

t t

t

d

t t

t t

t t

d d t t

t t

t t

d d

t

t

t

t

d

t t

t t

t

d t t

t t

t

d

t

t

t

t

d

f

1 1 1

1

2

1 1

1

2

2

2

1 1

1

1 2

2

2

2

1

1

1

2

1

1 1

1

1

2

1

2

2 1

1

2

2

1

1

1

1 1

1

2

1

1

1

2

1

ˆ θ θ

 (127)

where

d m d m d m

d k d k d k

d j d j d j

d i d i d i

d t t

d t t

d t t

d t t

d t t

= = =

= = =

= = =

= = =

= −

= −

= −

= −

= −

ˆ

ˆ

ˆ

ˆ

ˆ 1 1 1
96 USDA Forest Service RMRS-GTR-239. 2010.

are evaluated at

d m d m

d k d k

d j d j

d i d i

t t

t t

t t

t t

t t

= =

= =

= =

= =

=

=

=

=

=

ˆ

ˆ

ˆ

ˆ

ˆ 1 1

The second-order partial derivatives in Eq. 127, analogous to Eq. 121, are:

2

2

1

2

2

2

2

1

2

2

2

1

1 1 1

2

2 2 1 1 1

2

ˆ

ˆ ˆ

ˆ

ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ 1 ˆ

1

ˆ

ˆ

t

t t

t t

t

t

t t

t t

t t t t

t

t t

t t

t t t t

t

t

t

t t

t t t t

d i

d i d i

d i

d i

d i

m i d m m i d m

d i d i d i

d i d m

d i

d i d jd i d j

d i

d j

d i

d i d i

d i

d i

=

= =

=

=

=

≠

= ≠=

= = =

= =

=

≠

= ≠=

=

=

=

= =

=

=

=

=  ̧

 ̧

1

·

 ̈

 ̈

©

§

∂

∂

=

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

©

§

=

 ̧ =  ̧

1

·

 ̈

 ̈

©

§

∂ ∂

∂

− =

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

©

§

=

 ̧ =  ̧

1

·

 ̈

 ̈

©

§

∂ ∂

∂

 ̧

 ̧

1

·

 ̈

 ̈

©

§ = −

 ̧

 ̧

1

·

 ̈

 ̈

©

§

=

 ̧ =  ̧

1

·

 ̈

 ̈

©

§

∂ ∂

∂

θ

θ

θ

θ

 (128)

where the remaining partial derivatives from Eq. 127 equal zero and do not appear 

in Eq. 128. Combining terms in Eqs. 127 and 128, the terms that are part of the 

second-order Taylor-series expansion are:
USDA Forest Service RMRS-GTR-239. 2010. 97

( ) ( ) ( ) ( )

( )( ) ( )( )

( )( )

( ) ( ) ( ) ( )

( )( ) ( )( )

( )( ) »

»

»

»

1⁄4

o

«

«

«

«

¬

a

+ − −

− − + − − + −

+

» +

1⁄4

o «

¬

a ≈ + − − − + −

»

»

»

»

»

1⁄4

o

«

«

«

«

«

¬

a

+ − −

+ ̧

 ̧

1

·

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©

§

+ − −  ̧

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1

·

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©

§ − − −

+

+

»

»

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o

«

«

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a −  ̧

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1

·

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§ » +

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o «

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·

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= =

=

= =

=

= =

= =

=

= = = =

=

= =

=

=

= =

= =

= =

¦

¦

2 2 2

2

1

1 1

2

2 1 1 2 2

2

1

2

1

2 2 2

2

1

1 1

2

1

2

2

1

2

1 1

1

2

2

1

1 1

2

1

2 2

2

1

1 1

2

ˆ ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ

1 ˆ ˆ ˆ

ˆ ˆ

2

1

ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ

ˆ

1 ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

2

1

ˆ ˆ

ˆ

ˆ

ˆ ˆ ˆ ˆ

ˆ ˆ

t t t t

t

t t

t t t t

t

t t t t

t

t t

t t

t

t

t t

t

t t

t t

t

t f

t t t t

t

t t

t

t t t t

t

t t

t t t t

t t

t

t d

t

t t

t t

t

t f

d i d i

d i

d i d i

d i

d i d i

d i d i

i

q

j

d i d i d j d j

d i

d i d i

q

j

d i

d j d j

d i d i

d i d i

i i

D

D

θ

θ θ

 

 (129)

As in Eqs. 123 to 125, Eq. 129 may be expressed in matrix format for domains 

i = 1,...,qD as:

( )( )

( )( )

( )( )

[ ]

»

1⁄4

o «

¬

a

»

1⁄4

o «

¬

a − »

1⁄4

o «

¬

a − =

′

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

 ̧

1

·

 ̈

 ̈

 ̈

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 ̈

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 ̈

 ̈

©

§

»

»

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

«

«

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a

≈

»

»

»

»

»

»

»

»

1⁄4

o

«

«

«

«

«

«

«

«

¬

a

»

»

»

1⁄4

o

«

«

«

¬

a

− −

− −

− −

»

»

»

1⁄4

o

«

«

«

¬

a

−

−

−

≈ +

=

A t t I I t I

A

t

t

t

V ș A V

t t

t t

t t

ș ș A

2

2 3

2

1

2

2

1

2

1

2 3

2

1

3

2

1 3

1 2

2 3

3

2

1

1,2,3

1 1 3 3

1 1 2 2

2 2 3 3

3 3

2 2

1 1

2ˆ

1 ˆ

2ˆ

ˆ

2ˆ

ˆ

ˆ

ˆ ˆ ˆˆ ˆ ˆ

1

ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ

ˆ

ˆ

ˆ

t t

t

t

t

t

t

t

t

t

t

t t

t

t

t

t t

t t t t

t t

t t

t t

D s

D D

 (130)

An empirical estimate of the covariance matrix in Eq. 130 is possible from 

the sample by adding the appropriate product transformations to form the second 

partition (Ahmed and Al-Khasawneh 2005). In this example, let the V subscript 

denote a volume measurement applicable to either the “less-expensive” M = 2 

protocol or the “more-expensive” M = 3 protocol, with the domain indicator variable Ii

 defi ned in Eqs. 113 and 114. Equation 131 augments the Phase 2 (s = 2) 

estimator (Eq. 113) and Phase 3 (s = 3) estimator (Eq. 114), with the additional 

variables necessary to apply the second-order Taylor-series linear approximation 

in Eq. 130:
98 USDA Forest Service RMRS-GTR-239. 2010.

( )( )

( )( )

()()

( ) ( )( ) ( )

¦ ¦∈ =

= =

=

=

∈ = =

= =

= =

= =

=

=

=

=

=

=

»

»

»

»

»

»

»

»

»

1⁄4

o

«

«

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a

=

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1⁄4

o

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a

=

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1⁄4

o

«

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¬

a

2

M 2 M 2

M 2

M 1

2 2

M 2 M 2

M 2 M 2

M 2 1 M 2

M 2

M 1

2

2 2 3

M 2

M 1

0

0

ˆ ˆ

ˆ

ˆ

i s

i

V i V

i s s

i V q V

V i V

V V

s

s y I y

n

A

y I y

y I y

y I y

n

A

t

D









y

y

y

y

t

t

t

and

( )( )

( )( )

()()

( )( )

( )( )

( )( )

()()

( )

( )( )

( )( )

( )( )

( )

¦ ¦∈ =

= =

= =

=

= =

=

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∈ = =

= =

= =

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= =

=

= =

= =

= =

=

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1⁄4

o

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a

=

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a

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1⁄4

o

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«

«

«

«

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¬

a

3

M 3 M 2

M 3 M 2

M 3

M 2 M 2

M 2

M 1

3 3

M 3 M 2

M 3 M 2

M 3 1 M 2

M 3 M 2

M 3

M 2 M 2

M 2 M 2

M 2 1 M 2

M 2

M 1

3

1 3 3

1 2

M 3

2 3

M 2

M 1

0

0

0

0

ˆ ˆ

ˆ ˆ

ˆ

ˆ ˆ

ˆ

ˆ

i s

i

V i V

V V

V i V

i s s

i V q V

V i V

V V

V V

V q V

V i V

V V

s

s

y I y

y y

y I y

n

A

y I y

y I y

y I y

y y

y I y

y I y

y I y

n

A

t

t t

t

D

D

















y

y

y

y

y

y

t

t

t

t

t

 (131)

80%

100%

60%

Accuracy

40%

0%

20%

Prevalence of category in population

0% 20% 40% 60% 80% 100%

None

Minimal

Moderate

Substantial

Excellent

Figure 2. Accuracy required from a 

remotely sensed, binary variable for 

gains in precision from stratifi cation 

(Czaplewski and Patterson 2003). 

Gains use the metric of design effect 

(Särndal and others 1992:54) with a 

design effect of 0.25 for “Excellent,” 

0.50 for “Substantial,” 0.67 for 

“Moderate,” and 0.75 for “Minimal.” 

The required accuracy is highest 

for common categories and lowest 

for rare categories. For example, 

a “Substantial” gain in precision 

requires an accuracy of 77 percent 

if the prevalence of a category is 

20 percent of the population, but it 

requires 94 percent accuracy if the 

prevalence is 80 percent (Czaplewski 

and Patterson 2003:Table 2).
USDA Forest Service RMRS-GTR-239. 2010. 99

Therefore, the variances of quadratic terms in Eq. 130, such as 2 3 ˆ ˆ ˆ ⎡ ⎤ t

⎣ ⎦ V t , may 

be estimated from the realized sample with Eq. 131. However, this may lead to 

multivariate estimators of high dimension, which can cause numerical problems. 

Therefore, second-order Taylor-series approximations might be more suitable to 

occasional assessments of the accuracy of fi rst-order Taylor-series approximations rather than daily application in production settings.

Recall that the fi rst major partition of the matrices in Eq. 130 is from the fi rstorder Taylor-series approximation (Eqs. 120 to 125), and the second partition is 

from the second order approximation (Eqs. 127 to 129). If the covariance matrix 

for the model-based estimates V[θD] with both fi rst- and second-order partitions 

(Eqs. 130 and 131) is approximately the same magnitude (e.g., trace of the covariance matrix) as the covariance matrix without the second-order partitions 

(Eq. 125), then there is evidence that the bias from the fi rst-order approximation 

is relatively small, and more confi dence may be placed in the simpler fi rst-order 

approximation. Otherwise, the more complex approximation in Eq. 130 is available in empirical applications, assuming that the extra (2qD + 1) dimensions of the 

estimated vector and its covariance matrix allow numerically feasible and stable 

results. Equation 130 may provide alternative insights into the magnitude of the 

bias in the fi rst-order Taylor-series approximation of the covariance matrix if realistic assumptions are possible for the relative magnitudes of the second-order 

terms. Of course, all of this assumes that the third- and higher-order terms in the 

Taylor-series are negligibly small.

In addition to the Taylor-series method, Wolter (2007:Chapter 9) evaluates 

alternative estimators of the covariance matrix, including the random group 

method, the balanced half sample method (pseudo-replication), the jackknife 

method, the method of generalized variance functions, and the bootstrap method. 

All these methods generally have identical asymptotic properties, so the concern 

is with smaller sample sizes. No one estimator is the most accurate or best overall. 

However, Wolter (2007:363-365) concludes that the

Taylor-series method is good, perhaps best in some circumstances, 

in terms of the (mean squared error) and bias criteria, but (other 

methods) are preferable form the point of view of confi dence 

interval coverage probabilities. ... The (Taylor-series) method can 

generally accommodate any survey estimator (that is a non-linear 

transformation of the population vector estimate), which includes 

most statistics used in survey sampling practice. It may be diffi cult to 

apply for very complex (transformations), but such statistics do not 

often occur in practice. ... The (Taylor-series) method can deal with 

any sample design for which an estimator or the covariance matrix 

can be given.

Accuracy and Registration of Remotely Sensed Data

The success of the statistical estimators presented above depends on the 

strength of the associations among the remotely sensed auxiliary variables and 

the study variables. Czaplewski and Patterson (2001, 2003) investigated the gains 

in statistical effi ciency as a function of classifi cation accuracy for categorical variables that are used for stratifi cation (Figure 2). These guidelines are extrapolated 

through heuristics to RRE and model-based estimators.

The required accuracy is highest for common categories and lowest for rare 

categories. For example, a substantial gain in statistical effi ciency requires an accuracy of 77 percent if the prevalence of a category is 20 percent of the population, 

but 94 percent accuracy is required if the prevalence is 80 percent (Czaplewski 
100 USDA Forest Service RMRS-GTR-239. 2010.

and Patterson 2003:Table 2). In detailed classifi cation systems with many different categories, most categories are relatively rare (e.g., prevalence less than 10 

percent).

Because RRE can accommodate categorical variables that are not mutually 

exclusive and classifi cation systems that are not exhaustive or complete, classifi -

cation algorithms for remotely sensed data may be separately optimized for each 

category. For example, a separate dichotomous thematic map may be optimized 

for each individual category of land cover (e.g., predominantly deciduous forest, 

predominantly coniferous forest, and mixed forest), which can increase statistical 

effi ciency with remotely sensed data (King 2002). Certain spectral conditions, 

such as cloud shadows, need not be classifi ed into any category of land cover. This 

would avoid dilution of accurate predictions with inaccurate predictions.

Czaplewski and Patterson (2001) consider the effects of misclassifi cation error caused by changes in land cover and land use between the acquisition date of 

the remotely sensed data and the measurement date for the fi eld data. The loss 

in statistical effi ciency is rapid as the agreement deteriorates (Czaplewski and 

Patterson 2001, 2003; Fattorini and others 2004), regardless of the cause, be it 

either misclassifi cation error or change in land cover. This is especially true in 

dynamic landscapes, which are the domains that often have the greatest demand 

for current monitoring data (Smith 2002; USDA Forest Service 2007).

It is feasible to frequently monitor major changes in land cover with spaceborne sensor data (e.g., Healey and others 2005; Sader and others 2005), and these 

measurements might substantially improve statistical reliability (e.g., Czaplewski 

1999). One might use the post-stratifi cation estimator, but this uses crossclassifi cation of annual, remotely sensed data with relatively rare land cover changes, along with remotely sensed data on more static forest conditions. Extensive 

cross-classifi cation produces many small strata with few PSUs, especially with 

annual panel designs. Hence, FIA uses post-stratifi cation solely in the context of 

the temporally indifferent estimator (Patterson and Reams 2005), which is not 

amiable to use of remotely sensed estimates of changes in forest cover. Therefore, 

the post-stratifi cation estimator constrains the potential for using relatively inexpensive time-series of spaceborne, remotely sensed data to improve statistical 

reliability. For example, Van Deusen (2005) recommends that a single thematic 

map is best with post-stratifi cation and time-invariant panel estimators. However, 

RRE can combine remotely sensed data with panel data from the same year without cross-classifi cation over multiple years. Hence, RRE is more amenable to true 

annual estimation in panel designs that have a time-series of remotely sensed data. 

In this sense, RRE is much like regression estimation, in which categorical auxiliary data are treated as dichotomous predictor variables rather than strata. RRE 

does use linear models that might unintentionally introduce model bias. Future 

research should be designed to evaluate the magnitude of the potential bias and 

investigate diagnostic tools that detect model bias.

For univariate continuous variables, Särndal and others (1992:250) demonstrate that, under certain assumptions, the correlation between a univariate 

predictor variable and a univariate population parameter must be 0.5 or higher for 

the ratio estimator to be more effi cient than the Horvitz and Thompson estimator. 

With very high correlations, e.g., 0.95, the gain in effi ciency with the difference 

estimator is eight-fold (Särndal and others 1992:224). However, similar guidance 

for multivariate predictors with multiple responses is less well known, and further 

research is required in the context of FIA applications.

Misregistration between sites measured in the fi eld and measured with remote 

sensing will degrade associations among remotely sensed and fi eld data, especially in heterogeneous, fi ne-grained landscapes (McRoberts 2010). Halme and 

Tomppo (2001), Czaplewski (2005), Magnussen (2007), and Nelson and others 

(2009) offer methods to improve registration accuracy between Landsat pixel data 

and fi eld plots.
USDA Forest Service RMRS-GTR-239. 2010. 101

Discussion

This report presents the static Kalman fi lter and multivariate pseudo-estimators 

as a fully integrated and consistent approach to statistical estimation in the FIA 

program. The Kalman fi lter is a sophist icated mathematical tool for performing 

estimation and prediction when data are observed sequentially. In typical applications of the dynamic Kalman fi lter, this sequence generally follows a time-series. 

However, in the case of RRE, which is equivalent to the static Kalman fi lter, the 

sequence is defi ned as levels in a hierarchy of data collection. At its roots, Kalman 

fi ltering is closely related to Best Linear Unbiased Prediction (BLUP), hence it 

relies on correlations between variables observed at the different levels to perform predictions. Because the variances and covariances are not observed, they 

are replaced by estimated moments, so that the resulting estimation approach is 

Empirical BLUP (EBLUP).

Calibration estimators, which include generalized regression estimators 

(GREG) as a special case, are widely used in large government sample surveys to gain effi ciency and reduce nonresponse bias (Deville and Särndal 1992; 

Lundström and Särndal 1999). However, Lundström and Särndal caution

In some surveys, there is an abundance of auxiliary information, so 

that a selection of the most relevant variables would have to precede 

the start of the calibration process. We do not necessarily recommend 

that the totality of the information be used. To blindly add auxiliary 

information, over and beyond a set of crucial variables, might do 

more harm than good. These problems are indicated in Nascimento 

Silva and Skinner (1997) and Lundström (1997). The selection of an 

‘optimal’ set of auxiliary variables is thus not a trivial problem, and 

will in many cases require the judgment of an experienced survey 

statistician.

Calibration uses a sub-optimal transformation of multivariate auxiliary information that is invariant for each study variable. Certain auxiliary variables might 

be well correlated with some study variables but poorly correlated with other 

study variables. In order to follow the advice of Lundström and Särndal fully, the 

set of core auxiliary variables must be well correlated with each study variable. 

This requires prudent intervention by an experienced survey statistician.

In contrast, RRE automatically produces optimal linear weights tailored to 

each survey variable. If an auxiliary variable is not well correlated with a specifi c study variable, then it receives little weight. If the same auxiliary variable 

is highly correlated with a different study variable, then it is heavily weighted. 

In the worst case, if a certain auxiliary variable is not well correlated with any 

study variable, it never receives much weight. This integral weighting is achieved 

through the optimal properties of the RRE, without routine maintenance by an 

experienced survey statistician. Compared to calibration estimators, RRE is expected to be more robust in the routine production of FIA sample survey statistics, 

especially if correlations signifi cantly vary over time between FIA study variables 

and different remotely sensed auxiliary variables.

The statistical estimation methods given in this report accommodate a wide 

range of estimation scenarios, including multiphase and multistage designs, calibration, nonlinear estimation, and small-area estimation. All of these scenarios use 

a fully integrated paradigm, which is particularly attractive for a large, multifaceted sample survey. Furthermore, the Kalman fi lter is a mature fi eld of statistics, 

and RRE is a special case of the Kalman fi lter. Solutions to associated numerical 

hazards are well developed. In addition, the Kalman fi lter is a well-established 

time-series estimator. The combination of the static Kalman fi lter (i.e., RRE) for 

remotely sensed data with the dynamic Kalman fi lter for the time-series of FIA 
102 USDA Forest Service RMRS-GTR-239. 2010.

panel data is a unifi ed and consistent solution to a complex sample survey design 

in FIA.

Limitations of the post-stratifi cation estimator

RRE has potential advantages compared to post-stratifi cation used by FIA. 

RRE correctly accommodates support-regions for sample points (e.g., FIA plots) 

that straddle strata boundaries (e.g., remotely sensed stand conditions and geopolitical units, such as counties and ownerships) because RRE does not require 

independent strata to utilize categorical census variables. Similar to a linear calibration estimator or generalized regression estimator (Zhang 2000), RRE treats 

these variables as dichotomous auxiliary variables. This is similar to regression 

estimation, in which categorical auxiliary data are treated as dichotomous predictor variables rather than strata.

RRE might be less vulnerable to sampling zeros (e.g., Agresti 1984:54), empty 

sample post-strata (Zhang 2000), and small, within-stratum sample sizes. These 

problems can be addressed through merging strata with small sample sizes into 

similar strata with larger sample sizes. However, this not only reduces statistical 

effi ciency, it is also inappropriate for inference unless the stratum means are expected to be equal (Cochran 1977:134; Särndal and others 1992:411; Knottnerus 

2003:169). In a citation to Jagers (1986), Zhang (2000) observes that “poststratifi ed estimation that ignores the empty sample post-strata is downward biased 

for non-negative (study-variables).” In contrast, RRE can use the full sample size 

of annual fi eld plots for a sampled population (e.g., a state with 1000 FIA sample 

points measured each year) in order to utilize auxiliary data, rather than partition 

the full sample into numerous small and pseudo-independent strata (e.g., counties 

separated into major land ownership categories and remotely sensed categories, 

with as few as four sample points per stratum). In this sense, RRE is similar to 

a linear calibration estimator or generalized regression estimator (Zhang 2000).

RRE can separately use many different census variables (e.g., separate classifi cations of land cover, land use, forest type, stand development, and multiple 

sources, such as several independent land cover maps produced by different agencies or programs, possibly with different legends). Furthermore, there is no need 

to segment continuous variables (e.g., predicted biomass) into ordinal categories, 

as in stratifi cation. Moreover, there is no need to cross-classify all such categorical variables into a single, detailed classifi cation system, which is necessary to 

apply post-stratifi cation. This also reduces problems with small sample sizes in 

v ery detailed strata. More important, it opens the door to more complete use of 

available remotely sensed auxiliary variables.

Problems associated with the post-stratifi cation estimator are avoided with 

RRE. This is also true with GREG and calibration estimators, which are much 

less constrained than the post-stratifi cation estimator. With RRE, GREG, and 

calibration estimators, the strata used with post-stratifi cation estimator are simply 

replaced with dichotomous categorical predictor variables. The amount and detail 

of the auxiliary variables is adjusted relative to the available sample size with any 

of these estimators. Compared to the calibration estimator, RRE and the regression estimators are less vulnerable to poorly correlated auxiliary variables. The 

RRE has advantages over all these alternatives. RRE can combine data from many 

different sources and at different scales within a prediction-based framework. In 

doing so, a number of previously disparate estimation problems are unifi ed, such 

as small-area estimation, domain estimation, and calibration (Opsomer, personal 

communication). This unifi ed approach harmonizes the broad array of estimates 

produced by FIA.

Multivariate estimates

The vast majority of statistical literature in sample surveys deals with estimates of univariate population parameters. However, large government statistical 

programs, like FIA, estimate thousands of variables from the same sample. This 
USDA Forest Service RMRS-GTR-239. 2010. 103

has been successfully accomplished with repetitive application of univariate 

estimators (e.g., Scott and others 2005). However, simultaneous estimation of 

multiple population variables is necessary to estimate the covariance matrix for 

multivariate estimates. This covariance matrix is an integral component to the 

pseudo-estimators in Eqs. 89 to 131 above, which are used to estimate covariance 

matrices for rates, ratios, proportions, and products. They also serve at the core 

of variance estimators that accommodate missing data and small-area estimation 

with certain model-based approaches. They are even capable of variance estimation in the context of standard linear programming applications from operations 

research (Knottnerus 2003:354). Therefore, RRE is relevant to core needs in programs like FIA.

Independence among sampling phases

RRE assumes that all phases in a multiphase design are mutually independent. 

However, often times one phase is a sub-sample of another phase. Recall that the 

estimators assume an infi nite population, and the joint inclusion probabilities are 

zero among any units in the realized sample. Even if the population is assumed to 

be fi nite, the joint inclusion probabilities are very small whenever the sampling 

fraction is small, in which case multiple phases may be considered “almost-independent” (Knottnerus 2003:258, 262). This condition is true for FIA. If the fi nite 

population model is used with a large sampling fraction, then Eq. 9 may be used 

with the non-null cross-covariance matrix C between different phases, an estimator for which is given by Knottnerus (2003:316, 375, 389). Similar methods 

(Knottnerus 2003:392) might be used to combine different annual panels in FIA.

Risk management with complex sampling designs

The traditional FIA system uses a very simple sampling design: simple systematic sampling with equal selection probabilities and the post-stratifi cation 

estimator. However, more complex sampling designs are needed to fully accommodate multiple remote sensing technologies. Databases need embellishment. 

New estimators must be assimilated into the information management system. 

This complexity is accompanied by risk, but risks can be managed.

In general, RRE equals the design-based estimator X t

ˆ plus an adjustment 

(e.g., Eqs. 2, 12, 22, and 132). The adjustment vector is the matrix-weighted (K) 

difference between the more precise ancillary vector estimate Z t

ˆ of a partition of 

the population vector and the relatively less precise estimate of a larger partition 

of the population vector, or the full population vector, from a more detailed stage 

or phase . In a recursive application of RRE, multiple adjustment vectors 

are generated. These may be grouped into a single term. The following is a generic representation of RRE after the j

th recursion:

where

( ) () ( ) () ( ) ()

( )

( ) () ( ) ( )

RRE RRE 2 2 RRE 1 1 1 2 11

adj

adj RRE RRE 1 o , 1, 1

ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ

e ˆ ˆ ˆˆ and 

X jZ i Z Z X j jj

X j

iZ i X j i i

i jj

−

− = −

= + − ++ − + − a o a oa o « » ¬ 1⁄4 ¬ 1⁄4¬ 1⁄4

= +

=− = a o ¦ ¬ 1⁄4

t t K t H t K t H t K t Ht

t t

t K t Ht t t 



 (132)

Therefore, RRE can be readily separated into two additive terms: (1) the 

traditional FIA design-based vector estimate for a single panel without 

post-stratifi cation, and (2) a vector of adjustments (t

adj)j

 that improve statis tical 

effi ciency through the recursive fi ltering of all auxiliary information. Even the 
104 USDA Forest Service RMRS-GTR-239. 2010.

RRE and calibration weights in Eq. 43 at the PSU level are conveniently factored 

into the sum of two terms: the simple, design-based π expanded value (i.e., a

k

 in 

Särndal and others 1992:43), and a term that fully captures the residual difference 

between the simple π estimator and the more effi cient RRE estimator. This structure might reduce the complexities of temporally dynamic state-space estimates 

or plot expansion factors. If there are concerns with the reliability of FIA estimates that use auxiliary, remotely sensed information, then the adjustment term 

may be simply omitted, leaving only a design-based π panel estimate based on the 

simple random sample of FIA plots in that panel.

Furthermore, (t

adj

)j equals the residual difference between the simple FIA 

panel estimate and RRE that contains all auxiliary information. If the size of (tadj)j

 is large relative to its covariance matrix, then there is empirical evidence of 

an anomaly that requires investigation. This form of quality assurance should be 

an integral part of the FIA data processing procedures.

Non-sampling errors are always a concern in any large government survey. 

They include measurement and numerical round-off errors and mistakes in data 

transcription, computer programming, and sampling frame construction. Nonsampling errors can introduce unrecognized bias into estimators of population 

totals and variance estimators. FIA already uses extensive edits to detect certain 

measurement and transcription errors. FIA also conducts thorough tests of computer programs and algorithms. Analyses of residuals that are discussed above can 

further reduce risk of more subtle biases for some variables (Francisco 2003:71; 

Estevao and Särndal 2006). For example, Knottnerus (2003:255) uses a runs test 

that might be applied to residuals for study variables that are sorted on their corresponding residuals for auxiliary variables.

Design-based and design-consistent estimators are the least risky. These 

include estimators previously used by FIA for decadal periodic surveys and 

estimators for a single panel of annual FIA data1. RREs in Eqs. 21 to 70 are 

design-consistent, similar to ratio and regression estimators. These introduce little 

incremental risk if implementation and auxiliary estimates are reliable. Residual 

analyses further mitigate any remaining risk.

The small-area, missing data, and prediction estimators in Eqs. 82 to 131 are 

model-based methods. They assure internal consistency to maintain credibility. 

However, these are more risky because they depend upon the reliability of the 

model (Schreuder and others 1993). Furthermore, they are used to compensate for 

small sample sizes, which limits the power of residual analyses to detect model 

prediction bias and non-sampling errors. These latter risks are unavoidable. They 

are part of the cost of maximizing the utility of expensive fi eld data to estimate 

small segments of the sampled population. These risks should be clearly communicated to users of such estimates.

FIA already improves data quality through external reviews of new statistical 

reports by State Forest Service agencies and other external users. This requires 

routine intervention by FIA staff. RRE might increase the amount of quality assurance procedures conducted by FIA staff. Some of this could require the expertise 

a senior statistician. These procedures include residual analyses, diagnosis of the 

cause for suspicious residuals, and elimination of the source of any anomalies. 

Some unexpectedly large residuals might be caused by bias in model predictions. 

Other suspicious residuals might be caused by non-random errors, such as systematic errors in computer programs or data collection. Residual analyses could 

be automated and made more user friendly for FIA staff. 1 Scott and others (2005) treat segments of a single FIA fi eld plot that contains multiple strati

fi cation conditions as 

independent in the variance estimator for a single panel. They acknowledge that this will bias the variance estimator, but 

they assume the bias is inconsequential. This assumption might be considered model-based. FIA estimators, which use 

the moving average or temporally indifferent estimator (Patterson and Reams 2005) to combine multiple panels, might 

also be considered model-based. They are unbiased for the model that assumes a static population among panels. FIA 

uses volume equations that predict tree volume based on measurements of DBH and tree height. These models are fi t to 

historical, off-frame, purposive samples of trees. Therefore, FIA volume estimates are model-based.
USDA Forest Service RMRS-GTR-239. 2010. 105

Concerns may arise about the data quality of remotely sensed auxiliary information. If left unattended, measurement bias in auxiliary data can propagate into 

bias in population estimates. Coulston (2008) provides an example that involves 

FIA’s Phase 1 photo-interpretation. Other concerns may arise during analysis 

of RRE residuals. In the worst case, RRE auxiliary adjustments, i.e., the 

term in [ ] from Eq. 132, may be omitted. This leaves the purely design-based estimate, i.e., , which is the least risky. This retreat from the RRE 

estimator obviates risks to data quality caused by anomalies with the remotely 

sensed or other geospatial sources of auxiliary data, although it also foregoes any 

gains in effi ciency through RRE. The sequential, object-oriented nature of RRE 

readily permits retention of reliable auxiliary data sets and omission of suspect 

sets.

Seminal Literature

The RRE estimators used in this report are special cases of the Kalman fi lter, which originated and matured within the discipline of aerospace engineering 

during the 1960s and 1970s. The original applications were primarily for astronavigation in the Apollo lunar mission and the Mariner interplanetary space 

probes. The astrionics of that era used 8-bit digital processors, which are primitive 

by today’s 32- and 64-bit standards. Astronautical navigation systems assimilate 

time-series of imperfect predictions from linearized astrophysical models plus 

“noisy” measurements from diverse sensor arrays. Aerospace engineers developed model-based statistical estimators to process this multivariate time-series 

of stochastic inputs to produce estimates of spacecraft location and trajectory. 

Mission success demanded effi cient and unbiased statistical estimators with accurate covariance matrices from the model-based approach. The consequences of 

failure were unpleasant, such as reports on the evening news and Congressional 

inquiries, or truly tragic in the case of manned space exploration.

Statisticians addressed these challenges through theoretical and applied 

techniques, including the Kalman fi lter and numerical analysis. One comprehensive and rigorous treatise is Peter Maybeck’s (1979) book, Stochastic Models, 

Estimation, and Control. Maybeck covers relevant mathematical statistics, probability theory, linear time-series models, and numerically robust variations of the 

Kalman fi lter.

Another contemporaneous source is Gerald J. Bierman’s (1977) book, 

Factorization Methods for Discrete Sequential Estimation. Bierman provides 

a concise description of the mathematical statistics that relates the Kalman fi lter to linear models that are more familiar to statisticians who are not involved 

with aerospace engineering. More importantly, Bierman thoroughly covers the 

problem of divergence, solutions to which are critically important to successful 

applications of the Kalman fi lter. Aerospace engineers use the term divergence 

when their navigation system misdirects a missile, spacecraft, or aircraft to miss 

the target. Statisticians consider the same phenomenon as an unrecognized bias 

in a model-based estimator. The source of the bias is often misspecifi cation of 

the model, and Bierman covers embellishments of the Kalman fi lter that estimate 

biases caused by inaccurate or incomplete models or parameters. Divergence may 

also be caused by numerical errors, and Bierman develops numerically robust 

versions of the Kalman fi lter that remain important in applications to spacecraft 

navigation (e.g., Garcia-Yarnoz and others 2006).

Andrew Jazwinski’s (1970) treatise, Stochastic Processes and Filtering Theory,

is another key reference. He tends to cover the detailed mathematic statistics in 

more depth than Bierman and Maybeck, but the latter authors provide a more useful perspective for applications.

None of these three seminal references, which are in Academic Press’s series 

on Mathematics in Science and Engineering, seem to be well recognized in the 

sample survey literature. Anyone interested in applications of RRE or the Kalman 
106 USDA Forest Service RMRS-GTR-239. 2010.

fi lter to complex sample surveys would be well served by thoroughly studying 

these references.

More current references are available. Kalman Filtering: Theory and Practice 

Using MATLAB by Grewal and Andrews (2001) is a comprehensive and concise 

reference. It includes computer code for numerous algorithms. The 2001 comprehensive treatise by Yaakov Bar-Shalom, Xiao-Rong Li, and Thiagalingam 

Kirubarajan covers material relevant to sample survey estimation with RRE, despite its application-oriented title, Estimation with Applications to Tracking and 

Navigation: Theory Algorithms and Software. Charles K. Chui and Guanrong 

Chen’s (2009) heavily cited treatise Kalman Filtering with Real-Time Applications

is an excellent source for theoretical and applied details.

Knottnerus’ (2003) book, Sample Survey Theory: Some Pythagorean 

Perspectives, makes the fi rst direct connection between the static Kalman fi lter 

and Pythagorean regression. He uses the latter as the basis for his standard sampling model as a unifying approach to estimation for complex sample surveys. 

Knottnerus (2003:50) shows how the recursive methods from the Kalman fi lter 

can use “additional information ... (to improve) an initial estimate or, in general, how the estimator from the previous step can be improved by using new 

information.” Knottnerus also incorporates auxiliary information as constraints. 

However, Knottnerus does not use the state space model, which I frequently use 

to further simply complex estimators, and Knottnerus does not emphasize importance of numerical risks and residual analysis.

Finally, many of Knottnerus’ sample survey concepts are derived from Särndal 

and others’ (1992) book, Model Assisted Survey Sampling. While the authors use 

fi nite population sampling theory in their book, it remains infl uential to this report, in which the infi nite population perspective is used.

Future Research and Development

The estimators introduced above are merely examples of many types of complex sampling designs that may be accommodated with sequential application of 

RREs and subsequent non-linear, multivariate pseudo-estimators. These exampl es 

represent basic modules that may be assembled in various ways to incorporate 

complex sample survey data generated by diverse remote sensing technologies 

and administrative census statistics. Some of the more obvious opportunities will 

be touched upon here. These may be the subject of future research.

The two-phase design with cluster plots (Eqs. 62 to 70) may be expanded to 

include remotely sensed predictions of continuous and categorical variables from 

full-coverage, wall-to-wall Landsat data (i.e., M = 0 and s = U). Multivariate 

predictions for an individual pixel, or a small window of pixels covering a supportregion (Czaplewski 2005; McRoberts 2010), can be registered to the location of 

the FIA plot that is also measured with the photo-interpretation protocol (M = 1). 

Census statistics from all pixels could improve estimated population totals from 

the photo-interpretation and FIA fi eld protocols.

Summary statistics from a larger support-region of contiguous Landsat pixels 

can be matched with summary statistics from a photo-interpreted cluster plot. 

Then, census statistics from Landsat could improve estimated population totals 

for variables photo-interpreted over the cluster plots, which, in turn, could improve estimated population totals from the FIA fi eld protocol. Likewise, remotely 

sensed measurements and predictions from coarser-resolution satellites, such as 

MODIS, could be assimilated at the scale of a larger support-region, analogous 

to a cluster plot that is also measured with a sample or local census of photointerpreted observations. A similar approach might be best for spaceborne LiDAR 

data if they can be registered to a relatively large cluster plot (e.g., 25 to 1000 

acres) with suffi cient accuracy, but those data cannot be adequately registered to 
USDA Forest Service RMRS-GTR-239. 2010. 107

the relatively small support-regions (e.g., 0.05 to 1 acre) for photo-interpretation 

or FIA fi eld measurements.

A Phase 2 sub-sample of airborne LiDAR data could be registered to photointerpreted data and FIA fi eld data, and a much larger Phase 1 sample of airborne 

LiDAR data could be used to improve estimates of forest biomass. Or airborne 

LiDAR could be acquired for a relatively large support-region (cluster plot), including the FIA plot as a sub-set, and used like the photo-interpreted data for a 

cluster plot in Eqs. 57 to 61. Other design modules might include collocating 

samples of spaceborne and airborne LiDAR data, which might be facilitated by 

collocated, high-resolution aerial photography that support other remote sensing 

protocols. The model-based methods in Eqs. 115 to 125 can use the association 

between spaceborne and airborne LiDAR data at the population level to improve 

precision of estimates for small domains.

Knottnerus (2003:325) uses RRE to assure logical correspondence among 

study variables. For example, the difference in the areas of forest cover at two different times should agree with a separate estimate of the annual net rate of change 

in forest extent. Mandallaz (2008:188) calls this time additivity. Knottnerus and 

van Duin (2006) cover recursive restriction methods to assure additivity among 

complex statistical tables. Another example is the difference between estimates 

of timber removals from FIA fi eld plots and estimates of timber consumption 

from the independent FIA Timber Products Output (TPO) survey (Smith 2002). 

The state vector in a Kalman fi lter might be confi gured to achieve these types 

of compatibilities. This will likely involve non-null cross-covariances C among 

estimates of the state vector and a separate measurement vector, which may be 

accommodated with the gain matrix K in Eq. 9. The variance estimate for external 

estimates, such as TPO, might be problematic; sub-optimal methods might be appropriate (e.g., Knottnerus 2003:334).

FIA fi eld data may be missing because of denied access to private lands or unsafe fi eld conditions. These areas are in the target population, but not the sampled 

population. In addition, an FIA fi eld plot might not exist in certain known, rare, 

analytically important conditions, such as the area of a county with state-managed public lands, or lands enrolled in the USDA Conservation Reserve Program. 

Such conditions are part of the sampled population, but they represent sampling 

zeros. These anomalies might be addressed through purely model-based pseudoestimators for missing fi eld data (e.g., Eqs. 91 to 99). Estimates for missing-data 

domains might be improved with photo-interpretation of high-resolution aerial 

photography that covers FIA plots that are not measured because of denied access 

or hazardous fi eld conditions. It might be possible to join together one or more 

model-based estimators for missing data and the design-based estimator for the 

remainder of the target population. Other complex estimators might be possible 

through other methods developed in this paper and elsewhere (e.g., Särndal and 

Lundström 2008). Further study should assess feasibility.

There are different approaches to small-area estimation. Inverse calibration in 

Eqs. 82 to 86 and RE in Eq. 88 are examples of different alternatives. While the 

literature in small-area estimation is large, very little of it covers opportunities 

offered by the multivariate vector response in RRE. Further research is needed, 

which might lead to improvements in internal consistency and variance estimators 

that combine design-based estimates for the sampled population and model-based 

estimates for the target population that cannot be sampled.

In all cases, it is important to use numerically stable methods with RRE. The 

special case of RRE as a calibration estimator is considered by Czaplewski (in 

prep.[b]). Bierman’s (1977) factorization methods use and preserve triangular 

square roots of the covariance matrices, which would reduce storage requirements 

and numerical errors in a database. Bierman’s UD decomposition is a modifi ed 

form of the Gram-Schmidt orthogonalization. These methods generally replace 

matrix inverses with element-by-element scalar division. A matrix language is not 

necessary. Numerical studies are required to understand better the dimensions of 
108 USDA Forest Service RMRS-GTR-239. 2010.

the matrix equations that can be reliably processed with a digital computer. Future 

development should consider implementation of these powerful algorithms in 

database software, such as OracleTM, which should facilitate transfer of related 

research to the FIA production environment.

Robust RRE requires an accurate estimate of the covariance matrix. However, 

the sample covariance matrix is not necessarily an accurate estimator whenever 

the number of variables is large relative to the sample size (Ledoit and Wolf 2004). 

Dimensions for the auxiliary variables increase multiplicatively with cross-classifi cation of polychotomous categorical variables. However, cross-classifi cation 

might not substantively increase statistical effi ciency with remotely sensed, 

auxiliary data (Czaplewski in prep.[b]). Furthermore, covariance shrinking can 

improve reliability of Kalman fi ltering with high-dimension state vectors (Furrer 

and Bengtsson 2007) and other applications that face similar challenges (e.g., 

Schafer and Strimmer 2005; Bickel and Levina 2008). Ledoit and Wolf (2003) 

provide a relatively straightforward algorithm. Furthermore, orthogonalization 

and factorization methods (e.g., Bierman 1997) might mitigate impacts of estimation error with these covariance matrices. Monitoring of residuals might detect 

consequences of imprecise estimates of the covariance matrix. Further research 

can reveal the best compromise between reliability of RRE and the detail of remotely sensed, auxiliary data and FIA study variables that are incorporated into 

the state vector.

Prudent applications of RRE strongly depend upon analyses of recursive residuals, which are called the innovation sequence in the Kalman fi lter literature. 

Additional research is required for residual analyses in the context of the recursive static Kalman fi lter as applied to complex sample surveys, such as FIA. This 

research might be facilitated with the robust methods with census data and other numerically stable methods. These methods sequentially apply orthogonally 

transformed auxiliary variables one at a time. Each has a corresponding variance 

estimate, which can standardize the scalar residual difference between the auxiliary measurement datum and its estimate from the state vector. Standardized 

residuals that are unexpectedly large offer a powerful diagnostic. The auxiliary 

variable associated with a large residual may be omitted from the fi nal RRE. 

Further study is needed to determine the order in which scalar auxiliary variables 

enter the RRE. There might be opportunities to use a stepwise approach to this 

order to improve estimates of FIA study variables and avoid over-zealous application of remotely sensed auxiliary variables (Eq. 38).

FIA statistical reports can be more conveniently produced with a vector of 

unique RRE expansion values for each individual PSU (see “Optimal PSU 

Expansion Valuesfor Each FIA Plot”). The section “Robust Numerics” also introduces practical methods to mitigate numerical errors inherent with RE. In order 

to ensure numerical stability of matrix inversions, these methods decompose the 

problematic covariance matrix for the estimates of the auxiliary data. The result 

is an upper triangular and diagonal matrix. This structure differs from the assumptions used for the expansion values in Eqs. 76 and 81. Further research is 

needed to express the results from robust RRE as plot expansion values. In addition, it would be convenient from the analysis perspective to express results from 

pseudo-estimators as expansion values (see “A Model-Based Estimator for SmallDomain Predictions”). This too requires further research.

As an alternative, Sõstra and Traat (2009) use Knottnerus’ RRE to derive 

simpler scalar weights that depend solely on the auxiliary variables; the same 

scalar weight applies to all study variables measured at a sampled PSU. These 

weights are sub-optimal, meaning they do not replicate the minimum variance linear estimator. However, scalar weights are more compatible with the current FIA 

database, which uses scalar expansion factors for area and volume. Additional 

research would evaluate the compromise between statistical effi ciency and database convenience.
USDA Forest Service RMRS-GTR-239. 2010. 109

Further research is needed to develop fully the equivalency of the RRE with 

complex sample survey estimators that are available in the statistical literature 

(e.g., Eqs. 39 to 43). This will better defi ne the statistical properties of the RRE in 

the context of sample survey estimators, and it might suggest algebraic identities 

that allow the RRE to be better expressed as analytical sampling weights (e.g., Eq. 

43). The view of complex sample survey estimation from the perspective of the 

RRE and the Kalman fi lter might pave the way for rapid development of relatively 

simple survey estimators.

RRE applies the static Kalman fi lter to complex sampling designs for data 

gathered at one point in time. The Kalman fi lter is most often applied to dynamic 

systems that change over time. Therefore, the Kalman fi lter provides a unifi ed 

approach to processing complex sample survey panel data. Future research will 

study the Kalman fi lter as a recursive, object-oriented approach to assimilating the 

time-series of FIA fi eld data, both the Phase 2 and Phase 3 sub-samples for forest health monitoring, with multiple asynchronous time-series of remotely sensed 

data from different sensors.

Future research should focus on the object-oriented approaches of recursive 

estimation for population parameters and recursive Taylor series approximations 

for variance estimation. Not only will these approaches simplify complex estimation problems, they might also facilitate adaptation of estimators, software, and 

databases to future remote sensing and other technologies, while minimizing impact on database maintenance and development.

Conclusions

By an Act of Congress (Gillespie 1999), FIA began the transition from a stateby-state, decadal, periodic survey to an annual panel survey in all states. The 

transition is nearly complete. The only limitation is a modest increase in annual 

funding. The current statistical methods used with FIA panel data are essentially 

identical to those used for the FIA period surveys in the 1950s to 1990s. The 

most current data for PSUs from different panels, which are measured in different years, are treated as though they were measured at the same instant, hence the 

term temporally indifferent estimation (Patterson and Reams 2005). Effi ciency 

is improved with post-stratifi cation based on remotely sensed classifi cations of 

forest and non-forest cover at one point in time. The next era of FIA will build 

on this foundation by more fully incorporating different types of remotely sensed 

data (USDA Forest Service 2007). In anticipation of this corporate strategy, reliable and practical statistical methods must be developed and thoroughly tested in 

the near future so that they can be confi dently incorporated into FIA production 

procedures.

This report is merely an initial step needed to satisfy this requirement. It represents one possible blueprint for a consistent and unifi ed statistical engine within 

the FIA production system for national statistics on forest ecosystems. It portrays a design intended to be a reliable, adaptable, cohesive, and comprehensive 

mathematical system for improving FIA estimates with diverse, remote sensing 

technologies. The same mathematical paradigm is directly extendable to time-series of FIA panel data. Future research must rigorously test this integrated design. 

There are viable alternatives under development (e.g., Roesch 1999, 2008). Future 

research will need to compare alternatives to the current FIA methods.

Use of complex, remotely sensed data requires a more complex statistical 

system. Introduction of additional statistical complexity into an already complex 

production system incurs some risk. Comprehensive pilot tests are critical before 

implementation is attempted. Numerous pilot studies have already investigated 

innovative uses of remotely sensed data relevant to FIA (Czaplewski in prep.[b]). 

These have involved challenges in acquisition and use of remotely sensed data. 

Database management has also presented demanding challenges. Development 
110 USDA Forest Service RMRS-GTR-239. 2010.

of the necessary statistical estimators has been especially elusive in these pilot 

studies because statistical complexities are inseparably associated with the technologies and logistics of remote sensing. No amount of innovation in remote 

sensing has full value to FIA statistical products until statistically combined with fi eld data to form better population estimates. The statistical estimators introduced 

above appear capable of accommodating many types of remotely sensed data. 

However, they must be thoroughly tested with realistic numerical datasets and 

published in a journal with suffi cient statistical expertise in the peer-review and 

editorial processes.

Post-stratifi cation is a common and simple approach in large government survey programs of natural resources, such as FIA (Mandallaz 2008). Its results are 

conveniently captured in the large FIA database (McRoberts and others 2004) 

with relative ease, and it poses no extraordinary risk. However, its potential for 

improving population estimates with remotely sensed data is limited. RRE is more 

complex, but it can improve accuracy and effi ciency by assimilating larger quantities of auxiliary remotely sensed information. This might reduce the sample size 

needed to achieve a minimum accuracy criterion. A decision to convert from poststratifi cation to RRE depends on the compromise among implementation costs, 

risks, effi ciency, accuracy, and long-term cost savings. The decision is reversible. 

If the costs of implementation and risk management with RRE are unacceptable 

in FIA, then RRE might remain an attractive choice for special analytical studies 

whenever extra accuracy merits extra effort.

RRE offers opportunities to improve detailed sample survey products with remotely sensed and other geospatial auxiliary data known as complex auxiliary 

information (Estevao and Särndal 2006). However, judicious applications require 

attention to numerical problems and monitoring for other anomalies. Otherwise, 

results are not necessarily reliable. These problems are not well recognized in 

the sample survey literature, which might explain the paucity of survey applications that use RRE, especially given widespread popularity of the closely related 

Kalman fi lter in engineering, econometrics, atmospheric sciences, and physical 

oceanography.

Given existing and emerging remote sensing technologies, each with their own 

unique capabilities and limitations, and the inherent complexity of large statistical 

monitoring programs with diverse objectives, design modules may be added and 

removed in the FIA sampling and estimation processes. New types of relatively 

inexpensive, remotely sensed measurements can be used to improve precision 

and effi ciency of estimates made with FIA fi eld data. The value of all these complex design modules depends on strong empirical correlations and associations 

between measurements made with different protocols, different support-regions 

(e.g., cluster plots), the incremental cost of adding these modules, and support 

from a fl exible information management system that can readily process and distribute statistical estimates.

RRE is a powerful solution to diffi cult sample survey problems. It separates a 

large complex sampling design into small modules, each of which is more easily 

addressed. These modules are sequentially reassembled into more effi cient and 

accurate population estimates. RRE offers the opportunity to consider complex 

sampling designs that would otherwise be too intimidating. The dynamic Kalman fi lter further extends closely related opportunities with the time-series of FIA 

panel data. Särndal (2007) hypothesizes that calibration estimation might offer 

the potential to “generalize earlier theories or approaches.” Based on the insights 

offered in this report, RRE might provide an even more complete foundation to 

make further advances in complex sample survey estimation.
USDA Forest Service RMRS-GTR-239. 2010. 111

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