The State Vector Formulation for Complex Sample Surveys: Forest Inventory and Analysis (FIA) Applications

Gemini Deep Research 
Editted by Raymond L Czaplewski 
October 3, 2025

Introduction: 
This review considers two monographs from the perspective of the USDA Forest Inventory and Analysis Program (FIA):
Czaplewski, Raymond L. Ph D. "Complex sample survey estimation in static state-space." Gen. Tech. Rep. RMRS-GTR-239. Fort Collins, CO: US Department of Agriculture, Forest Service, Rocky Mountain Research Station. 124 p. 239 (2010).
Czaplewski, Raymond L. "Generalized Multivariate Difference Estimator (GMDe): The Recursive Algorithm." (2023): 242.
The 2010 publication relates to the Recursive Restriction Estimator (RRE), originally published by Knottnerus6 in 2003, is a predecessor to the 2023 publication Generalized Multivariate Difference Estimator (GMDE).

Part I: Foundational Concepts and the Core Methodology
1.1 Introduction: The Estimation Challenge in Modern Forest Inventories
FIA is mandated to deliver current, consistent, and credible information regarding the status, condition, and trends of forests across the United States.1 Historically, this mission was accomplished through periodic, state-by-state field inventories. However, legislative mandates and evolving user needs have pushed the FIA towards a nationally consistent annual survey system, which requires more dynamic and efficient estimation procedures.1 A central component of the FIA's strategic plan is the increased integration of data from a diverse and growing array of remote sensing technologies, including satellite-based sensors like Landsat and MODIS, airborne platforms such as LiDAR, and high-resolution aerial photography.1
This integration presents a significant statistical challenge. The various remote sensing platforms differ in their spatial and temporal resolutions, measurement protocols, and error characteristics, necessitating complex sampling units and multi-layered sampling designs.1 For instance, while lower-resolution satellite data may provide full "wall-to-wall" coverage suitable for thematic mapping, higher-resolution data from sources like IKONOS or LiDAR, are often too costly for complete coverage and must be incorporated through sampling.1 Accomplishing the FIA's strategic objectives—such as enhancing support for the National Forest System, developing rapid assessment capabilities, and monitoring wildlife habitat—requires a statistical framework capable of cohesively and optimally combining these disparate population estimates with the foundational annual field plot data.1
The monograph "Complex Sample Survey Estimation in Static State-Space" by Czaplewski (2010) argues that traditional estimation methods are often insufficient for this task and proposes a unified suite of methods centered on the Recursive Restriction Estimator (RRE), which is a predecessor to the Generalized Multivariate Difference Estimator (GMDE).1 GMDe/RRE uses design-consistent estimates for a "state-vector" and “measurement‑vector” that contain all sufficient statistics for the sampled population.1 The core premise is that GMDe/RRE can accommodate complexity by decomposing a large estimation problem into a sequence of simpler, more manageable estimators.1 This approach is not an incremental extension of existing survey methods but rather a new concept, bring in a cohesive and highly developed methodology from the engineering and econometrics literature: the Kalman filter.1 GMDe/RRE is, in fact, a special case of the Kalman filter, and by reframing the survey estimation problem in this way, GMDe/RRE seeks to leverage decades of theory and practice to solve the pressing challenges of modern environmental monitoring.
1.2 The State-Space Paradigm for Survey Estimation
A theoretical foundation of GMDe/RRE is the state-space model, a concept originating in control engineering for describing stochastic processes.1 This paradigm offers a powerful and flexible way to structure complex estimation problems.
Defining the "State-Vector"
The central concept in the state-space model is the "state," which is defined as the minimum set of quantitative information needed to completely describe the system at a given time and predict its future behavior.1 In the context of a static sample survey, the "state" of the system is not the condition of an individual sampling unit but rather a vector containing all relevant population parameters of interest. This vector, denoted as the state-vector, contains all sufficient statistics about the population, such as the total biomass, the total area of each forest type, or the prevalence of specific land use categories.1 By defining the problem at the population level, the state-space approach abstracts away from individual population elements and inclusion probabilities, which are instead handled within the design-based estimators that provide inputs to the model.1
The Static Kalman Filter
GMDe/RRE is identified as a special case of the Kalman filter, namely the static Kalman filter.1 The Kalman filter is a sequential estimator that operates in multivariate state-space. While the more common dynamic Kalman filter is used for time-series applications where the state evolves over time, the static version is applied here to estimate a fixed state-vector at a single point in time.1 Instead of processing a time-series of observations, the static Kalman filter sequentially processes information from different components of a complex sample survey, such as different sampling phases or stages.1
This sequential processing is enabled by a key feature of the Kalman filter: it is "memoryless".1 This property means that the estimate of the state-vector at any given step in the recursion fully contains all relevant statistical information from the components that have already been processed. It is not necessary to store or re-process the population estimates from previous steps, as their information has been optimally "filtered" into the current estimate and its associated variance-covariance matrix.1 Instead of solving one massive, potentially intractable matrix equation involving all population estimates simultaneously, GMDe/RRE solves a series of smaller, more manageable problems in a sequential fashion. This modularity makes the approach highly scalable and adaptable to the addition of new population estimates in an “object‑oriented” sense.
Relationship to Generalized Least Squares (GLS)
The 2010 RRE paper grounds this engineering-derived methodology in the familiar language of statistical regression theory by demonstrating its algebraic equivalence to optimal sample survey estimators.1 The 2010 monograph on FIA applies the work of Knottnerus (2003), who uses the concept of Pythagorean regression to demonstrate the intimate connection between the Kalman filter and the Generalized Least Squares (GLS) estimator.1 The “restriction estimator” (RE), which is the core building block of RRE, is shown to be algebraically identical to the minimum variance GLS regression estimator under specific conditions, such as in two-phase sampling with independent phases.1 This connection is critical, as it confirms that RRE and GMDe are not an ad hoc procedures but are members of the class of optimal linear unbiased estimators, sharing the same foundational principles as well-established methods in the sample survey literature.
1.3 Deconstructing the Recursive Restriction Estimator (RRE)
RRE and GMDe are multivariate composite estimators that provide a mathematically optimal method for combining two previously calculated vectors of population estimates, which need not have the same dimensions, into a single, more precise estimate.1
The Restriction Estimator (RE) as the Building Block
The RE operates on two primary inputs: A state-vector estimate, which represents the current best estimate of the population parameters for both auxiliary variables and study variables. This estimate has an associated covariance matrix, representing its uncertainty. A measurement vector estimate, which represents new information about a linear transformation of the state-vector. This estimate has its own covariance matrix, representing its uncertainty.1 The relationship between the true measurement vector  and the true state-vector is given by a linear model.1
Mathematical Formulation
The RE combines these two pieces of information using a set of three core equations derived from the static Kalman filter formulation.1
The Kalman Gain matrix is a matrix of optimal weights. It is a multivariate generalization of the scalar weight used in a simple univariate composite estimator. It balances the relative uncertainties of the state-vector estimate and the measurement-vector estimate. The state update equation computes the new, more precise estimate by correcting the prior state estimate with the "innovation" or residual—the difference between the new measurement and what was expected based on the prior state. This residual is weighted by the Kalman Gain.
The covariance update equation computes the covariance matrix for the new estimate. The uncertainty of the estimate is always reduced after incorporating the new measurement.
The resulting covariance matrix is smaller than the prior covariance matrix (in the positive semi-definite sense), quantifying the gain in precision.1
The "Recursive" Process in Practice
The power of RRE and GMDe comes from applying the recursive method. Knottnerus describes this as an "object-oriented" process where two "objects" are combined at a time.1 An object can be a vector of known constants from a census, a vector estimate from a single sample phase, or the vector output from a previous RE application.
For example, in a three-phase design, the process would be:
1.	Recursion 1: Combine a "Census Object" (e.g., Landsat data) with a "Phase 1 Object" (e.g., a large sample of photo-plots) using the RE. The output is a new, more precise "Composite Object 1" that now contains all the information from both sources.
2.	Recursion 2: Combine "Composite Object 1" with a "Phase 2 Object" (e.g., a small sample of field plots) using the RE. The output is the final estimate, which has optimally assimilated the information from all three sources.
This sequential process elegantly separates the complexity of the overall design into a series of manageable, pairwise steps, leveraging the memoryless property of the filter at each stage.1

Part II: Comparative Analysis and Practical Applications
2.1 An Advance Over Traditional Methods: GMDE/RRE vs. Post-Stratification
A central argument is that the GMDe/RRE framework offers significant advantages over the post-stratification estimator, which is the standard method used by the FIA to incorporate auxiliary information from remote sensing.1
Limitations of the Post-Stratification Estimator
The 2010 paper by Czaplewski provides a detailed critique of post-stratification, highlighting several fundamental limitations that become particularly acute in the context of the FIA's modern, panel-based design and its reliance on complex auxiliary data.1
●	The Problem of Small Strata: Post-stratification requires cross classifying all sources of categorical auxiliary data to form strata. When multiple sources are used (e.g., a remote sensing-based forest type map, a land ownership map, and county boundaries), the number of potential strata can become enormous. Given the limited sample size of an annual FIA panel, this leads to a proliferation of strata with very small or even zero sample sizes.1 This results in highly unstable stratum-level estimates and, consequently, inaccurate estimates of variance for the population totals.1
●	Sampling Zeros: A common issue with small strata is the "sampling zero," where a condition is known to exist in a stratum but, by chance, is not captured by any one of the few sample plots that fall within it. In this case, the post-stratified estimate for that condition in that stratum is zero, with an estimated sampling error of zero, which is misleading and contributes to an underestimation of overall variance.1
●	Straddling Strata Boundaries: FIA plots, which have a spatial extent, frequently straddle the boundaries of post-strata (e.g., a plot that is part forest and part non-forest). The methods used to handle these "mixed-condition" plots violate the core assumption of post-stratification that sampling errors among strata are independent, which can lead to biased or overly optimistic variance estimates.1
●	Information Loss: Post-stratification is inherently limited to categorical auxiliary variables. This forces continuous data, such as a satellite-derived prediction of biomass, to be discretized into ordinal categories (e.g., "low," "medium," "high" biomass). This categorization inevitably results in a loss of information and statistical efficiency.1
How GMDE/RRE Overcomes These Limitations
The GMDe/RRE framework is designed to resolve these issues by fundamentally changing how auxiliary information is incorporated into the estimation process. It moves from the logic of sample partitioning (stratification) to a logic of multivariate regression and calibration.1
●	No Strata Required: GMDE does not require the sample to be partitioned into strata. Instead, categorical auxiliary data are converted into a series of dichotomous (0/1) indicator variables, which are then included in the state-vector alongside continuous variables. This allows the estimator to use the full sample size for every estimate, completely avoiding the problem of small strata and sampling zeros.1
●	Handling High-Dimensional Auxiliary Data: Because GMDE/RRE does not require cross-classification, it can simultaneously accommodate a vast number of auxiliary variables from many different sources (e.g., multiple independent land cover maps, continuous biomass predictions, geopolitical boundaries). This opens the door to a much more complete and efficient use of all available remotely sensed information.1
●	Correctly Accommodating Complex Boundaries: The issue of plots straddling boundaries is naturally handled. A plot that is, for example, 40% "forest" and 60% "non-forest" according to a remote sensing map would simply have values of 0.4 and 0.6 for the corresponding auxiliary variables, which are then correctly processed by the regression-like structure of GMDe/RRE.1
●	Optimal, Data-Driven Weighting: Perhaps the most significant advance is the shift in responsibility for handling auxiliary information. Under post-stratification, the statistician must make numerous a priori decisions about which variables to use, how to categorize them, and how to merge strata—a process of manually curating and simplifying the information landscape. In contrast, GMDE/RRE uses the empirically estimated variance-covariance structure of the data to automatically calculate the optimal weighting matrix. This matrix provides a unique, optimal weight for each auxiliary variable as it pertains to each study variable. Auxiliary variables that are poorly correlated with a given study variable will receive little to no weight for that estimate, while strongly correlated variables will be heavily weighted. This data-driven, a posteriori determination of information value makes GMDe/RRE more robust and adaptable, particularly in a production environment where new remote sensing products may become available over time.1
The following table summarizes the key distinctions between the two frameworks.
Table 1: Comparison of Estimation Frameworks: Post-Stratification vs. GMDE/RRE
Feature	Post-Stratification	GMDE/RRE	
Handling of Auxiliary Data	Requires cross-classification of categorical variables into mutually exclusive strata. Continuous data must be categorized.	Treats auxiliary variables (both continuous and categorical) as predictors in a multivariate regression-like framework. No cross-classification needed.	
Sample Size Requirements	Partitions the total sample, leading to small or empty strata, which can cause unstable estimates and inaccurate variances.	Uses the full sample for every estimate, avoiding the problems associated with small sample sizes.	
Robustness to Complex Boundaries	Assumes independence between strata; violated when plots straddle boundaries, potentially biasing variance estimates.	Naturally handles "mixed-condition" plots by treating proportions as continuous auxiliary variables.	
Information Efficiency	Incurs information loss by forcing continuous variables into categories. Limited in the number of auxiliary variables that can be used.	Uses the full information content of continuous variables. Can accommodate a very high-dimensional vector of auxiliary information.	
Variance Estimation	Variance estimators can be biased or misleading due to small sample sizes, sampling zeros, and boundary issues.	Provides a design-consistent, optimal (EBLUE) variance-covariance matrix that correctly propagates uncertainty.	
Implementation Complexity	Conceptually simple, but requires complex and often subjective decisions about stratum definition and merging.	Mathematically more complex, but automates the optimal weighting of auxiliary information. Requires robust numerical algorithms.	

2.2 Applying GMDE/RRE to Complex FIA Sampling Design Options
The flexibility of the GMDe/RRE framework is best demonstrated by its application to the hypothetical sampling designs relevant to the FIA program. The 2010 paper provides several detailed examples, which serve as practical guides for possible implementation.1
Two-Phase Sampling with a Census at Phase 1
This scenario is highly relevant to the FIA, which commonly uses wall-to-wall remote sensing products (a "census") as auxiliary information for its field plot sample.1
●	Scenario: A census of auxiliary variables (protocol , e.g., Landsat pixel classifications) provides known population totals. A smaller, independent Phase 2 sample provides measurements for both the auxiliary variables and the study variables (protocol , e.g., FIA field data).1
●	GMDE/RRE Formulation: The state-vector is defined from the Phase 2 sample as . The measurement vector is the vector of known census totals, which has a covariance matrix of zero. The GMDe/RRE then acts as an optimal calibration estimator, constraining the final estimate for the auxiliary variables to exactly match the known census totals.1 
●	Numerically Stable Algorithm: A critical issue arises because the covariance matrix  can be singular or ill-conditioned, making its inverse infeasible. The 2023 paper presents a robust solution: instead of applying the entire vector of census constraints at once, the constraints are processed recursively, one scalar element at a time. This elegant algorithm replaces a single, potentially impossible matrix inversion with a sequence of simple scalar divisions, making the method numerically stable and highly practical for implementation in database software.1
Three-Phase Sampling
This design extends the two-phase model, for instance, to incorporate the sparse but highly detailed measurements from the FIA's Forest Health Monitoring (FHM) Phase 3 plots.1
●	Scenario: A census of remote sensing data (Phase 1, ), a large sample of field plots (Phase 2), and a small sub-sample with more expensive FHM measurements (Phase 3).
●	GMDE/RRE Formulation: GMDe/RRE is applied sequentially. First, the census and Phase 2 data are combined to produce an improved estimate for the Phase 2 variables. This improved estimate and its updated covariance matrix then become the "prior" estimate that is combined with the Phase 3 data in a second recursive step. This process ensures that the information from the large, inexpensive samples is efficiently propagated to improve the precision of the estimates from the smallest, most expensive sample.1 The paper also discusses a sub-optimal but potentially more robust alternative that reduces the dimensions of the matrices at each step by discarding variables that are not common to the two objects being combined, which can improve numerical stability at the cost of some statistical efficiency.1
Designs with Cluster Plots
Many FIA designs involve measurements that are not independent, such as when remote sensing data and field data are collected within the same cluster plot. The Nevada Photo-based Inventory Pilot (NPIP) is used as an example.1
●	Scenario: A sample of 50-acre photo-plots (a cluster of photo-interpreted points, protocol ) each contains a 1-acre field plot at its center (protocol ). Measurements are taken at both the center point and across the entire cluster.
●	GMDE/RRE Formulation: This design violates the assumption of independence between the different population estimates. The estimation errors for the cluster-mean photointerpretation and the center-point field measurements are correlated. This requires the use of the full GMDE/RRE formulation where the cross-covariance matrix, , is non-zero. The state-vector is defined using the measurements at the center point, , while the measurement vector is defined from the more precise cluster-mean, . The GMDe/RRE then optimally combines these correlated estimates to produce a final estimate that is more precise than either component alone.1
The following table provides a summary of these applications.
Table 2: Summary of GMDE/RRE Applications to FIA Sampling Designs
Sampling Design	State-Vector Definition	Measurement Vector Definition	Key Modeling Assumption	
Two-Phase with Census	Phase 2 sample estimate of auxiliary and study variables.	Known census totals for auxiliary variables.	Measurement error is zero. Phase 1 and 2 are independent.	
Three-Phase Sampling	Phase 3 sample estimate of all variables.	Composite estimate from combining Phase 1 and Phase 2.	All phases are independent at each step.	
Single-Phase with Cluster Plots	Estimate from center point of cluster	Estimate from the mean of the entire cluster	State and measurement are correlated.	
Two-Phase with Cluster Plots	Composite estimate from a small sample of fully measured clusters.	Composite estimate from a large sample of partially measured clusters.	A multi-step recursive process combining correlated and independent components.	

Part III: Advanced Applications and Implementation Considerations
The GMDe/RRE framework is not merely an estimator for population totals; it is a foundation upon which a wide range of advanced statistical products can be built. The final GMDE/RRE output—a vector of population estimates and its full variance-covariance matrix—enables the rigorous propagation of uncertainty through subsequent linear and non-linear transformations, known as pseudo-estimators.1
3.1 Extending the Framework: Linear and Non-Linear Pseudo-Estimators

Linear Transformations for Small-Area Estimation
A significant challenge for national surveys like the FIA is producing reliable estimates for small geographic areas or domains (e.g., counties or individual 6000-acre hexagons) that contain too few sample plots for direct, design-based estimation.1 GMDe/RRE output can be used to generate model-based synthetic estimates for these small areas through a linear transformation known as inverse calibration (or direct calibration).1
The model assumes that the relationship between remotely sensed predictions and true field conditions, as estimated by GMDe/RRE for the entire population, holds constant for all small areas within that population. For a given small area, the known census of remotely sensed pixel values is multiplied by a matrix of calibration coefficients derived from the population-level GMDE/RRE. A crucial property of this method is that the small-area estimates are consistent: their sum over all small areas is guaranteed to equal the design-consistent GMDE/RRE estimate for the total population, ensuring coherence between local and regional statistics.1
Non-Linear Transformations and Taylor-Series Approximations
Many important statistics are non-linear functions of population totals, such as ratios (e.g., volume per acre) or products. While computing the estimate itself is straightforward, estimating its variance is complex. The paper advocates for the use of multivariate Taylor-series approximations to linearize the non-linear function, allowing for the propagation of variance through the estimated covariance matrix from GMDe/RRE.1
●	Product Estimator for Missing Data: A practical example is provided for estimating biomass accretion on "denied access" lands where field crews cannot take measurements. The model estimates the total missing accretion as the product of two random variables derived from GMDe/RRE: [the estimated total area of denied access lands] and [the estimated biomass accretion per unit area on similar, accessible lands]. The paper shows that for a product, the first-order Taylor series is exact, providing an unbiased variance estimate for this model-based imputation.1
●	Ratio Estimator: The common problem of estimating a ratio, such as total wood volume divided by total forest area, is also detailed. Both the numerator and the denominator are random variables from GMDe/RRE state-vector. The variance of their ratio is approximated using the first-order Taylor series.1
●	Recursive Simplification of Taylor Series: The paper presents an elegant, object-oriented approach to deriving these potentially daunting multivariate Taylor-series approximations. A complex non-linear function is broken down into a sequence of simple, fundamental operators (addition, subtraction, multiplication, division). The Taylor-series approximation is derived for each simple operator and then recursively combined, greatly simplifying the overall derivation.1
3.2 Practical Implementation: Numerical Stability and Expansion Values
The successful implementation of an advanced statistical framework like GMDE/RRE in a large production environment depends on addressing two critical practical issues: numerical robustness and compatibility with existing database structures.
The Critical Issue of Numerical Robustness
The Kalman filter is notoriously vulnerable to numerical round-off errors, a problem known as divergence in the engineering literature.1 This instability is especially severe when dealing with highly precise auxiliary data (where the measurement covariance matrix approaches zero) or when the state-vector is high-dimensional with highly co-dependent (collinear) variables. In such cases, the matrix inversions required to calculate the Kalman Gain can become infeasible, or accumulated round-off errors can lead to unreliable or nonsensical results.1
GMDe/RRE adopts solutions developed over decades in the aerospace engineering community, which are standard practice there but novel in survey sampling. These include factorization methods, such as U-D factorization or other square-root filters. These algorithms do not work with the covariance matrix directly but rather with its matrix square root (or factors thereof). This approach is numerically more stable and can double the effective precision of the computations. Furthermore, these methods often replace large matrix inversions with a sequence of numerically stable scalar operations, making them more robust and easier to implement.1
Deriving Optimal PSU-Level "Expansion Values"
A major practical hurdle for adopting a complex multivariate estimator within the FIA is its incompatibility with the program's traditional data structures, which are built around a simple, scalar "expansion factor" for each plot.1 The GMDe/RRE produces a final vector and covariance matrix, not a simple set of plot weights.
To bridge this gap between advanced theory and operational reality, the 2010 RRE paper presents a novel derivation of PSU-level expansion values. It demonstrates that the final GMDE/RRE vector estimate can be expressed as the sum of PSU-specific vectors. This vector is a function of the plot's own measurements and the optimal GMDE/RRE weights.1
Furthermore, the 2010 RRE paper shows that the final GMDE/RRE covariance matrix can also be disaggregated to the PSU level. It can be computed as the sum of outer products of residual vectors derived from its corresponding expansion vector. This development is a critical breakthrough for implementation. It allows the full, complex result of GMDe/RRE to be stored and processed within a database using a structure analogous to traditional expansion factors. By storing a vector of values for each plot, the familiar summation process can be preserved, dramatically lowering the barrier to adoption and allowing users to compute estimates and variances for any sub-population by simply summing the appropriate plot-level vectors.1 This solves the "last mile" problem of integrating a sophisticated estimator into a large-scale production system.
Part IV: Synthesis with the Generalized Multivariate Difference Estimator (GMDe/RRE)
4.1 Defining the Generalized Multivariate Difference Estimator (GMDe)
Based on available literature, the Generalized Multivariate Difference estimator (GMDe) is a simple, robust, model-based method designed for complex, high-dimensional, and often longitudinal sample surveys.2 It is a multivariate generalization of the classic difference estimator first published by Hansen, Hurwitz, and Madow in 1953 and is positioned as a powerful alternative to generalized regression (GREG) and calibration estimators.2
The GMDe's key characteristic is its hybrid structure, which separates the estimation process into two distinct components 3:
1.	Design-Based Component: The process begins with a standard design-based estimator (e.g., the multivariate Horvitz-Thompson estimator) applied to the probability sample. This step produces an initial vector of population estimates for both the study variables of interest and any auxiliary variables, along with the full variance-covariance matrix for this vector.
2.	Model-Based Component: A linear "adjustment" is then applied to the initial estimates of the study variables. This adjustment is calculated as the product of an "optimal" coefficient matrix and the vector of residuals for the auxiliary variables (e.g., the difference between an estimate from a large Phase 1 sample and a small Phase 2 sample). The coefficient matrix is derived using a multivariate minimum variance criterion, effectively finding the optimal weight for each auxiliary variable's contribution to each study variable.3
Crucially, GMDe is explicitly described in the literature as being "closely related to the multivariate composite estimator and the Kalman filter update," the very same theoretical foundations upon which RRE is built.2
4.2 A Unified Framework: Connecting GMDE and GMDe
RRE and GMDe are not competing methodologies but are better understood as two complementary perspectives on the same fundamental statistical principle: optimal linear estimation via a multivariate composite estimator, which is mathematically embodied in the Kalman filter update equation. This is precisely the structure of the RE state update equation (Eq. 10 from the paper) and matches the description of GMDe's model-based adjustment component.1 For a simple two-phase design, RRE formulation presented in the Czaplewski 2010 paper is algebraically identical to GMDe. The GMDe takes the design-based estimate from Phase 2 as its initial estimate and adjusts it using the auxiliary information from Phase 1; this perfectly mirrors GMDe process.1
The apparent difference between the two frameworks lies more in their framing and emphasis than in their mathematical substance.
●	GMDE as a Constructive Process: The 2023 paper presents GMDe as a constructive, sequential process. It emphasizes the flow of information through a complex system, showing how to build a final, optimal estimate from the ground up by recursively integrating different population estimates (phases, stages, censuses). The narrative is about managing complexity through recursion.
●	GMDe as an Adjustment Process: The literature on GMDe frames it as an adjustment process. It assumes one begins with a single, comprehensive design-based estimate that already incorporates the full complexity of the survey design, and then applies a final model-based adjustment to achieve optimality and enhance robustness (e.g., by mitigating outliers or imposing constraints).2
This distinction suggests a powerful pathway for integration rather than a choice between methods. Both RRE and GMDe, with their object-oriented, recursive approach, provide a systematic and numerically robust methodology for assimilation of the initial "design-based component" as input to RRE or GMDe, especially in surveys with many population estimates. Once GMDe process has produced its final vector estimate and covariance matrix, one could then apply GMDe framework to perform final, robust adjustments. For example, GMDe literature places a strong emphasis on modifying the optimal coefficient matrix to impose inequality constraints or mitigate the influence of outliers—features that are mentioned but less central in the 2010 RRE paper.2
Therefore, a unified workflow could be envisioned:
1.	Use the recursive, object-oriented GMDe/RRE process to manage the complexity of integrating multiple, heterogeneous population estimates into a single, design-consistent multivariate estimate and its associated covariance matrix.
2.	Use GMDe/RRE framework to apply final, robust adjustments, ensuring coherence, imposing constraints, and mitigating the influence of anomalous data points.
In this way, RRS and GMDe can be seen as two sides of the same coin, offering a complete and powerful toolkit for modern complex survey estimation.

Part V: FIA Applications
The GMDe/RRE framework is a flexible solution for integrating the diverse data sources available to FIA. 
5.1 Two-Phase Sampling with a Census
This common FIA scenario involves using wall-to-wall remote sensing data (a "census" of pixels, e.g., from Landsat) as auxiliary information for the sample of field plots.1
The RRE functions as an optimal calibration estimator. It uses the known population totals from the remote sensing census as constraints, adjusting the estimates from the field plot sample to be perfectly consistent with the census data. This process improves the precision of estimates for the primary study variables (e.g., forest volume, biomass) measured in the field. The paper also details a numerically stable algorithm that processes these census constraints sequentially, one at a time, which avoids the need for a single, large, and potentially infeasible matrix inversion.1
5.2 Three-Phase Sampling 
This design is directly relevant to the FIA's Forest Health Monitoring (FHM) program, which consists of a census of remote sensing data (Phase 1), a large sample of standard field plots (Phase 2), and a smaller sub-sample of plots with more intensive and expensive health measurements (Phase 3).1 The GMDe/RRE is applied recursively. First, it combines the census and Phase 2 data to produce an improved estimate for the Phase 2 variables. This improved estimate and its updated covariance matrix then serve as the input for a second step, where they are combined with the sparse Phase 3 data. This sequential process efficiently propagates the statistical strength from the large, inexpensive samples to improve the precision of estimates from the smallest, most expensive sample.1
5.3	 Designs with Cluster Plots
Many FIA designs involve correlated measurements, such as when remote sensing data and field data are collected within the same plot. The FIA's Nevada Photo-based Inventory Pilot (NPIP), which uses 50-acre photo-plots surrounding a central 1-acre field plot, is a key example.1 In this case, the estimation errors from the photo-interpretation (of the large cluster) and the field measurements (at the center) are correlated. The RRE framework accommodates this by using a non-zero cross-covariance matrix to optimally combine the two correlated data sources, resulting in a final estimate that is more precise than either component alone.1
5.4	Complex Multi-Source Designs
GMDe/RRE can support a highly complex design that combines a census of satellite data, a large independent sample of cluster photo-plots, and a small sample of cluster plots that also contain field measurements.1 This scenario demonstrates the full power of the sequential, "object‑oriented" approach. GMDe/RRE breaks the complex problem into a series of manageable steps, such as first combining the census with the large photo-plot sample, and then combining that result with the fully-measured plot sample. This illustrates how GMDe/RRE can provide a simple and elegant solution to otherwise intractable designs.1
5.5 Advanced Model-Based Applications (Pseudo-Estimators)
The RRE framework serves as a foundation for producing a variety of advanced statistical products from the final RRE output (the vector of estimates and its full covariance matrix).1
Small-Area Estimation
To produce reliable estimates for small domains like counties, which have too few sample plots for direct estimation, GMDe/RRE can be used to create model-based estimates. The model assumes the relationships between remote sensing and field data estimated for the whole population also apply to the small areas. These small-area estimates are consistent, meaning they are guaranteed to sum to the RRE total for the entire population.1
Missing Data Imputation
For situations like "denied access" lands where field measurements are impossible, GMDe/RRE can be used in a product estimator (e.g., [estimated area of denied access] x [estimated biomass per area on similar lands]). The full covariance matrix from the RRE allows for a rigorous variance estimate for this model-based imputation.1
Calibration for Measurement Bias
In a design with both inexpensive, potentially biased measurements (e.g., standard volume equations) and expensive, unbiased measurements (e.g., from upper-stem measurements on a sub-sample), GMDe/RRE can first optimally combine all data. The resulting estimates are then used to create a calibration model that corrects the biased measurements in small domains, ensuring consistency with the unbiased population total.1
Conclusions
The statistical framework detailed in Czaplewski (2010) and reformulated with the Generalized Multivariate Difference Estimator (GMDE), represents a significant and cohesive advancement in the theory and practice of complex sample survey estimation. By reframing the estimation problem within the state-space paradigm of the static Kalman filter, GMDe/RRE provides a unified and mathematically rigorous solution to the challenges posed by the integration of diverse population estimates, particularly the multi-resolution remote sensing data that are central to the future of the FIA program.
GMDe/RRE has several key advantages of over traditional methods like post-stratification. By treating auxiliary information within a regression-like framework rather than through sample partitioning, GMDe avoids ubiquitous problems of small strata and sampling zeros, uses the full information content of continuous auxiliary data, and correctly handles complex boundary conditions. Its data-driven, optimal weighting of auxiliary information makes it a more robust and adaptable tool for a production environment.
Furthermore, the GMDe/RRE framework extends far beyond the estimation of population totals. Its output—a final estimate vector and its full covariance matrix—serves as a foundation for a wide array of advanced pseudo‑estimators, including consistent small-area estimates and variance estimates for non-linear transformations, all within a single, coherent system. The development of methods to disaggregate the final GMDe/RRE results into PSU-level expansion values represents a critical innovation, bridging the gap between advanced statistical theory and the practical implementation needs of a large-scale survey program.
The principles underlying GMDe/RRE are part of a broader class of modern, robust estimators. The RRS and GMDe share a common theoretical heritage in the Kalman filter update and the multivariate composite estimator. GMDe/RRE provides a powerful engine for constructing an optimal estimate from complex, multi-source data, and offering a framework for applying final, robust adjustments to that estimate. GMDe/RRE offers a powerful, flexible, and unified solution to many of the most difficult problems in modern sample survey estimation. The adoption of such a framework would equip programs like the FIA with the statistical machinery needed to fully assimilate the wealth of information available from new technologies and to meet the growing demand for more detailed, timely, and credible environmental monitoring.
Works cited
1.	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. 
Czaplewski, Raymond L. "Generalized Multivariate Difference Estimator (GMDe): The Recursive Algorithm (rGMDe)." (2023).
2.	GMDe Summary - Environmetrika, accessed October 2, 2025, https://environmetrika.org/GMDe-summary
3.	GMDe | Environmetrika, accessed October 2, 2025, https://environmetrika.org/GMDe
4.	Generalized Multivariate Difference Estimator (GMDe) - ResearchGate, accessed October 2, 2025, https://www.researchgate.net/profile/Raymond-Czaplewski/publication/353411938_Generalized_Multivariate_Difference_Estimator_GMDe/links/610944bf1e95fe241aa8983b/Generalized-Multivariate-Difference-Estimator-GMDe.pdf
5.	Knottnerus, Paul. "General Restriction Estimator in Multisurvey Sampling." In Sample Survey Theory: Some Pythagorean Perspectives, pp. 325-390. New York, NY: Springer New York, 2003.