Geometry, Identifiability and Estimability

Measurement geometry, rank, null spaces, observability, identifiability, parameter correlations, singular values, condition number, information matrices, covariance geometry, weak estimation directions, geometry evolution, excitation, and practical estimability.

4.3.16 Geometry, Identifiability and Estimability

Geometry, identifiability, and estimability describe whether the available measurements contain enough independent information to determine the states and parameters that an estimator is trying to solve for.

An estimator can be mathematically correct and numerically stable and still fail to determine some states if the measurement geometry is poor or if different parameters produce nearly indistinguishable effects.

The central idea is

\[ \boxed{ \text{Measurement Geometry} + \text{Dynamic Coupling} + \text{Parameter Sensitivities} \rightarrow \text{Estimability} } \]

This section focuses on:

These ideas are fundamental in orbit determination, navigation, calibration, parameter estimation, maneuver estimation, and general relative-navigation problems.

4.3.16.1 Why Geometry Matters

Suppose a measurement is

\[ \mathbf z = h(\mathbf x) + \mathbf v. \]

Its local sensitivity to the state is

\[ \boxed{ \mathbf H = \frac{\partial h}{\partial\mathbf x}. } \]

The matrix \mathbf H determines which combinations of the state are visible to the measurement.

If a particular state perturbation

\[ \delta\mathbf x \]

produces almost no change in the measurement,

\[ \boxed{ \mathbf H\,\delta\mathbf x \approx \mathbf0, } \]

then that direction is weakly observed.

Thus measurement accuracy alone is not enough.

\[ \boxed{ \text{Accurate Sensor} + \text{Poor Geometry} \rightarrow \text{Weak State Knowledge} } \]

4.3.16.2 Geometric Interpretation of a Measurement

Different measurements constrain different geometric directions.

For example, a range measurement

\[ \rho = \| \mathbf r_{\rm target} - \mathbf r_{\rm observer} \| \]

is primarily sensitive to displacement along the line of sight.

A LOS measurement is primarily sensitive to motion perpendicular to the LOS.

Therefore,

\[ \boxed{ \text{Range} \rightarrow \text{Radial Information} } \]

while

\[ \boxed{ \text{LOS} \rightarrow \text{Transverse Information}. } \]

Combining complementary measurement geometries improves estimability.

4.3.16.3 Measurement Geometry Matrix

For a set of linearized measurements,

\[ \delta\mathbf z = \mathbf H \delta\mathbf x + \mathbf v. \]

The matrix

\[ \boxed{ \mathbf H } \]

may be called a geometry matrix or measurement-sensitivity matrix in many estimation contexts.

Its rows describe measurement sensitivities.

Its columns describe how each state or parameter affects the measurements.

4.3.16.4 Geometry in Batch Orbit Determination

In batch OD, the estimated unknown may be the reference-epoch state.

Then the measurement sensitivity at epoch t_k is

\[ \boxed{ \mathbf G_k = \mathbf H_k \boldsymbol\Phi(t_k,t_0). } \]

Stacking all epochs gives

\[ \boxed{ \mathbf G = \begin{bmatrix} \mathbf G_1\\ \mathbf G_2\\ \vdots\\ \mathbf G_N \end{bmatrix}. } \]

This stacked matrix contains both

Therefore batch estimability depends on the entire observation arc, not only one instantaneous sensor configuration.

4.3.16.5 Geometry in Sequential Estimation

In sequential estimation, each individual measurement may constrain only part of the state.

But as the geometry evolves over time, information accumulates.

Thus,

\[ \boxed{ \text{Weak Instantaneous Geometry} + \text{Changing Geometry Over Time} \rightarrow \text{Strong Overall Estimability} } \]

This is especially important for angle-only or range-only navigation.

4.3.16.6 Rank of the Measurement Matrix

Consider

\[ \mathbf H \in \mathbb R^{m\times n}. \]

The maximum number of independently observable local state combinations is

\[ \boxed{ \operatorname{rank}(\mathbf H). } \]

If

\[ \operatorname{rank}(\mathbf H)=n, \]

the state is locally fully constrained by the current linearized measurement set.

If

\[ \operatorname{rank}(\mathbf H)

at least one state combination is unobservable instantaneously.

4.3.16.7 Rank Deficiency

Suppose

\[ \boxed{ \operatorname{rank}(\mathbf H)

Then there exists a nonzero vector

\[ \mathbf q \]

such that

\[ \boxed{ \mathbf H\mathbf q = \mathbf0. } \]

This means moving the state along direction \mathbf q does not change the measurement to first order.

That direction lies in the null space of \mathbf H.

4.3.16.8 Null Space

The null space is

\[ \boxed{ \mathcal N(\mathbf H) = \{ \mathbf q: \mathbf H\mathbf q=\mathbf0 \}. } \]

A state perturbation in this space cannot be detected by the linearized measurement model.

Therefore,

\[ \boxed{ \text{Null-Space Direction} \rightarrow \text{Unobservable State Combination}. } \]

4.3.16.9 Physical Meaning of a Null-Space Direction

Suppose a sensor measures only range.

Then a small displacement tangential to the range sphere may produce almost no first-order range change.

That tangential direction is weakly constrained by the instantaneous range measurement.

Thus a null-space vector is not merely a mathematical object; it identifies a physical state direction that the sensor cannot distinguish.

4.3.16.10 Left Null Space

The left null space contains vectors

\[ \mathbf y \]

such that

\[ \boxed{ \mathbf y^T\mathbf H = \mathbf0^T. } \]

This is associated with redundant combinations of measurement equations.

While the state null space is usually more important for estimability, the left null space can reveal redundancy within the measurement set.

4.3.16.11 Column Space

The column space of \mathbf H contains all first-order measurement changes that can be generated by state perturbations.

Thus,

\[ \boxed{ \mathbf H\delta\mathbf x } \]

lies in the column space of \mathbf H.

Measurement residual components outside this space cannot be explained by the estimated state correction alone.

4.3.16.12 Row Space

The row space describes the independent state combinations constrained by the measurements.

If many rows are nearly parallel, then the measurements contain significant redundancy.

Thus,

\[ \boxed{ \text{Many Measurements} \neq \text{Many Independent Constraints}. } \]

4.3.16.13 Identifiability

A parameter is identifiable when its effect on the measurements can be distinguished from the effects of other unknown parameters.

Suppose the unknown parameter vector is

\[ \boldsymbol\theta. \]

The local sensitivity is

\[ \boxed{ \mathbf G_\theta = \frac{\partial\mathbf z} {\partial\boldsymbol\theta}. } \]

If the columns of \mathbf G_\theta are linearly independent, the parameters are locally distinguishable.

4.3.16.14 Non-Identifiable Parameters

Suppose

\[ \boxed{ \mathbf g_1 = c\mathbf g_2. } \]

Then changes in \theta_1 and \theta_2 produce proportional measurement effects.

The estimator cannot uniquely determine them separately.

Only some combination may be identifiable.

Thus,

\[ \boxed{ \text{Separate Parameters} \neq \text{Separately Observable Effects}. } \]

4.3.16.15 Practical Identifiability

Exact linear dependence is not required for a practical problem.

If

\[ \mathbf g_1 \approx c\mathbf g_2, \]

the parameters may technically be identifiable but only very weakly.

This leads to

This is called practical identifiability or weak estimability.

4.3.16.16 Structural versus Practical Identifiability

Structural identifiability asks whether the parameter can theoretically be determined with perfect measurements.

Practical identifiability asks whether it can be determined accurately with realistic noise, geometry, and data duration.

Thus,

\[ \boxed{ \text{Structurally Identifiable} \not\Rightarrow \text{Practically Well Estimated}. } \]

4.3.16.17 Estimability

Estimability is broader than identifiability.

A state or parameter may be mathematically identifiable but poorly estimated because

Thus,

\[ \boxed{ \text{Estimability} = \text{Identifiability} + \text{Information Strength} + \text{Noise Level}. } \]

4.3.16.18 Observability versus Identifiability

The terms are closely related but often used differently.

Observability generally refers to whether system states can be reconstructed from outputs over time.

Identifiability usually refers to whether unknown parameters can be uniquely determined.

For an augmented estimator, both concepts become closely coupled.

\[ \boxed{ \text{State Observability} + \text{Parameter Identifiability} \rightarrow \text{Augmented-State Estimability}. } \]

4.3.16.19 Linear Observability Matrix

For a linear time-invariant system

\[ \dot{\mathbf x} = \mathbf A\mathbf x, \]

\[ \mathbf z = \mathbf H\mathbf x, \]

the observability matrix is

\[ \boxed{ \mathcal O = \begin{bmatrix} \mathbf H\\ \mathbf H\mathbf A\\ \mathbf H\mathbf A^2\\ \vdots\\ \mathbf H\mathbf A^{n-1} \end{bmatrix}. } \]

The system is observable if

\[ \boxed{ \operatorname{rank}(\mathcal O)=n. } \]

4.3.16.20 Why Dynamics Improve Observability

A state component not directly measured may still influence a measured state later through dynamics.

For example, velocity may not appear in a pure position measurement directly.

But velocity changes future position.

Thus,

\[ \boxed{ \text{Dynamic Coupling} \rightarrow \text{Indirect Observability}. } \]

4.3.16.21 Time-Varying Observability

Spacecraft systems are often time varying.

Then the instantaneous matrix \mathbf H_k alone does not determine observability.

The relevant information accumulates through the sequence

\[ \boxed{ \mathbf H_k \boldsymbol\Phi(t_k,t_0). } \]

Changing orbital geometry can therefore turn a weakly observed state into a well-estimated one.

4.3.16.22 Observability Gramian

For a continuous linear time-varying system, the observability Gramian over [t_0,t_f] may be written

\[ \boxed{ \mathbf W_o = \int_{t_0}^{t_f} \boldsymbol\Phi^T(t,t_0) \mathbf H^T(t) \mathbf H(t) \boldsymbol\Phi(t,t_0) \,dt. } \]

With measurement weighting,

\[ \boxed{ \mathbf W_o = \int \boldsymbol\Phi^T \mathbf H^T \mathbf R^{-1} \mathbf H \boldsymbol\Phi \,dt. } \]

4.3.16.23 Interpretation of the Observability Gramian

The eigenstructure of \mathbf W_o describes how strongly different state directions are observed over the interval.

Large eigenvalues:

\[ \boxed{ \text{Strongly Observed Directions} } \]

Small eigenvalues:

\[ \boxed{ \text{Weakly Observed Directions}. } \]

A zero eigenvalue indicates an unobservable direction.

4.3.16.24 Information Matrix

For discrete measurements, the information matrix is commonly

\[ \boxed{ \mathbf I = \mathbf G^T \mathbf R^{-1} \mathbf G. } \]

This matrix is closely related to the normal matrix in batch least squares.

It describes how much information the observations provide about the estimated variables.

4.3.16.25 Information Matrix and Covariance

For a locally linear Gaussian problem with no prior,

\[ \boxed{ \mathbf P \approx \mathbf I^{-1} } \]

when \mathbf I is nonsingular.

Thus,

\[ \boxed{ \text{Small Information Eigenvalue} \leftrightarrow \text{Large Covariance Eigenvalue}. } \]

Weak measurement geometry appears directly as large uncertainty.

4.3.16.26 Prior Information

With prior covariance

\[ \mathbf P_0, \]

the total information becomes

\[ \boxed{ \mathbf I_{\rm post} = \mathbf P_0^{-1} + \mathbf G^T \mathbf R^{-1} \mathbf G. } \]

Thus a prior can make an otherwise weakly observed parameter numerically well constrained.

But this does not mean the new measurements themselves observed it strongly.

4.3.16.27 Prior-Dominated Estimate

If

\[ \mathbf P_0^{-1} \gg \mathbf G^T\mathbf R^{-1}\mathbf G \]

in a parameter direction, then the posterior estimate is dominated by the prior.

Therefore,

\[ \boxed{ \text{Small Posterior Covariance} \not\Rightarrow \text{Strong Measurement Identifiability}. } \]

4.3.16.28 Singular Value Decomposition

A powerful way to study geometry is the singular value decomposition:

\[ \boxed{ \mathbf G = \mathbf U \boldsymbol\Sigma \mathbf V^T. } \]

The diagonal elements

\[ \sigma_1,\sigma_2,\ldots,\sigma_n \]

are the singular values.

4.3.16.29 Meaning of Singular Values

Large singular values correspond to state combinations that strongly affect the measurements.

Small singular values correspond to weakly observed combinations.

Thus,

\[ \boxed{ \sigma_{\rm large} \rightarrow \text{Strong Direction} } \]

and

\[ \boxed{ \sigma_{\rm small} \rightarrow \text{Weak Direction}. } \]

4.3.16.30 Right Singular Vectors

The columns of

\[ \mathbf V \]

identify state or parameter directions.

The right singular vector associated with the smallest singular value identifies the weakest estimation direction.

This is extremely useful because it shows which combination of unknowns is poorly constrained.

4.3.16.31 Left Singular Vectors

The columns of

\[ \mathbf U \]

describe corresponding directions in measurement space.

They show which combinations of residuals carry information about each singular direction.

4.3.16.32 Rank from SVD

A numerical rank can be determined by comparing singular values to a tolerance.

If

\[ \sigma_i \approx0, \]

the corresponding state direction is effectively unobservable.

Because numerical calculations never produce perfect zeros, the tolerance must be selected sensibly.

4.3.16.33 Condition Number

The condition number is

\[ \boxed{ \kappa(\mathbf G) = \frac{\sigma_{\max}} {\sigma_{\min}} } \]

for a full-rank matrix.

A large condition number means the estimation problem is sensitive to measurement noise and numerical errors.

4.3.16.34 Interpreting the Condition Number

If

\[ \kappa \approx1, \]

the measurement geometry is relatively balanced.

If

\[ \kappa\gg1, \]

some state combinations are much more weakly constrained than others.

Thus,

\[ \boxed{ \text{Large Condition Number} \rightarrow \text{Weak Geometric Direction}. } \]

4.3.16.35 Normal Matrix Conditioning

The normal matrix is

\[ \mathbf N = \mathbf G^T \mathbf W \mathbf G. \]

Its condition number approximately squares that of the appropriately weighted design matrix.

Therefore forming normal equations can worsen numerical conditioning.

This is one reason QR or SVD solutions can be preferable.

4.3.16.36 Parameter Correlation

For covariance matrix

\[ \mathbf P, \]

the correlation coefficient is

\[ \boxed{ \rho_{ij} = \frac{P_{ij}} {\sqrt{P_{ii}P_{jj}}}. } \]

It satisfies

\[ -1 \le \rho_{ij} \le 1. \]

4.3.16.37 Interpretation of High Correlation

If

\[ |\rho_{ij}|\approx1, \]

two estimated quantities are strongly coupled.

This means changes in one can be compensated by changes in the other with little impact on predicted measurements.

Thus,

\[ \boxed{ \text{High Correlation} \rightarrow \text{Weak Parameter Separation}. } \]

4.3.16.38 High Correlation Is Not Automatically Wrong

Strong parameter correlation does not necessarily indicate an incorrect estimator.

It may simply reflect the true information content of the measurements.

The correct response may be to

4.3.16.39 Typical Correlations in Orbit Estimation

Common examples include

\[ \boxed{ \text{Position} \leftrightarrow \text{Velocity} } \]

\[ \boxed{ C_D \leftrightarrow \text{Atmospheric Density} } \]

\[ \boxed{ C_R \leftrightarrow \text{Area-to-Mass Ratio} } \]

\[ \boxed{ \text{Range Bias} \leftrightarrow \text{Radial Position} } \]

\[ \boxed{ \text{Clock Bias} \leftrightarrow \text{Range}. } \]

4.3.16.40 Maneuver-Parameter Correlation

For maneuver estimation, common correlations include

\[ \boxed{ \text{Thrust Magnitude} \leftrightarrow \text{Burn Duration} } \]

\[ \boxed{ \text{Burn Time} \leftrightarrow \text{Initial State} } \]

\[ \boxed{ \text{Thrust Direction} \leftrightarrow \text{Attitude / Alignment}. } \]

These correlations directly affect how reliably individual maneuver parameters can be estimated.

4.3.16.41 Measurement Geometry and Correlation

Poor geometry often produces high state correlation.

For example, if measurements constrain mainly one spatial direction, uncertainty in the perpendicular directions remains large and may become strongly coupled.

Improving geometry often reduces correlation naturally.

4.3.16.42 Covariance Ellipsoid

The position covariance

\[ \mathbf P_{rr} \]

defines an uncertainty ellipsoid.

The eigenvalue decomposition is

\[ \boxed{ \mathbf P_{rr} = \mathbf V \boldsymbol\Lambda \mathbf V^T. } \]

The eigenvectors define ellipsoid axes.

The eigenvalues define uncertainty magnitude along those axes.

4.3.16.43 Weak Direction from Covariance

The largest eigenvalue of

\[ \mathbf P \]

corresponds to the largest uncertainty direction.

Thus,

\[ \boxed{ \lambda_{\max}(\mathbf P) \rightarrow \text{Weakest Estimated Direction}. } \]

This complements the SVD view, where the smallest information singular value identifies the weakest direction.

4.3.16.44 Information Ellipsoid versus Covariance Ellipsoid

Information and covariance are inversely related.

Strong information corresponds to small covariance.

Therefore,

\[ \boxed{ \text{Large Information Axis} \leftrightarrow \text{Small Covariance Axis}. } \]

Both representations describe the same geometry from opposite perspectives.

4.3.16.45 Measurement Geometry Example: Single Range

For a single range measurement,

\[ \rho = \|\mathbf r\|, \]

the Jacobian is

\[ \boxed{ \mathbf H_\rho = \hat{\mathbf r}^T. } \]

This is a 1\times3 matrix.

Its rank is at most 1.

Therefore a single range measurement cannot determine full 3-D position instantaneously.

4.3.16.46 Single LOS Measurement

A LOS measurement provides angular direction but no direct range.

Its differential sensitivity is approximately

\[ \boxed{ \frac{\partial\hat{\mathbf r}} {\partial\mathbf r} = \frac{1}{r} ( \mathbf I - \hat{\mathbf r}\hat{\mathbf r}^T ). } \]

This matrix removes the radial direction.

Thus LOS is weak to first order along the line of sight.

4.3.16.47 Range + LOS Geometry

Combining range and LOS gives complementary radial and transverse information.

Conceptually,

\[ \boxed{ \text{Range} + \text{LOS} \rightarrow \text{Complete Relative Position Geometry} } \]

subject to noise and calibration.

This is an important example of complementary sensor geometry.

4.3.16.48 Range-Rate Geometry

Range rate provides

\[ \boxed{ \dot\rho = \hat{\boldsymbol\rho}^T \mathbf v_{\rm rel}. } \]

Therefore it strongly constrains the velocity component along the line of sight.

Transverse velocity remains weak instantaneously.

4.3.16.49 Angles-Only Estimation

Angles-only measurements often have poor instantaneous range observability.

But as the observer and target move, parallax and dynamics provide range information over time.

Thus,

\[ \boxed{ \text{Angles Only} + \text{Geometry Evolution} + \text{Dynamics} \rightarrow \text{Range Estimability}. } \]

4.3.16.50 Poor Parallax Geometry

If observer motion is nearly along the LOS, the viewing direction changes very little.

Then range remains weakly determined.

Large transverse baseline usually improves parallax.

Therefore geometry, not simply observation duration, is critical.

4.3.16.51 Multi-Station Geometry

Measurements from multiple stations provide different LOS directions.

If the station geometry is well distributed,

\[ \boxed{ \text{Multiple Independent LOS Directions} \rightarrow \text{Improved Position Geometry}. } \]

If all stations lie in nearly the same direction relative to the spacecraft, improvement may be limited.

4.3.16.52 GNSS Geometry

GNSS pseudorange geometry is represented by rows containing satellite LOS vectors.

Good satellite distribution in the sky produces strong 3-D positioning geometry.

Poor clustering produces weak geometry.

This leads directly to the idea of dilution of precision.

4.3.16.53 Dilution of Precision

For GNSS-like geometry,

\[ \mathbf G \]

maps position and clock perturbations into pseudorange residuals.

The covariance due to unit measurement variance is approximately

\[ \boxed{ \mathbf Q = ( \mathbf G^T\mathbf G )^{-1}. } \]

DOP metrics summarize selected diagonal elements of this matrix.

4.3.16.54 Geometric Dilution of Precision

A common total geometry measure is

\[ \boxed{ GDOP = \sqrt{ Q_{xx} + Q_{yy} + Q_{zz} + Q_{tt} }. } \]

Lower GDOP generally indicates better geometry.

Higher GDOP indicates measurement noise is more strongly amplified into state uncertainty.

4.3.16.55 Position Dilution of Precision

Position DOP is

\[ \boxed{ PDOP = \sqrt{ Q_{xx} + Q_{yy} + Q_{zz} }. } \]

This isolates geometric amplification affecting 3-D position.

4.3.16.56 DOP Is a Geometry Metric

DOP does not directly contain sensor noise magnitude if the covariance has been normalized.

Actual position covariance scales with measurement uncertainty.

Thus,

\[ \boxed{ \text{Navigation Error} \approx \text{Measurement Error} \times \text{Geometry Amplification}. } \]

4.3.16.57 Geometry Evolution

A configuration that is poor at one time may become excellent later as orbital motion changes viewing geometry.

Therefore geometry should often be evaluated as a function of time:

\[ \boxed{ \mathbf G=\mathbf G(t). } \]

This can guide measurement scheduling and sensor selection.

4.3.16.58 Geometry Planning

If measurement opportunities can be scheduled, it may be better to select observations with diverse geometry rather than simply maximize observation count.

Thus,

\[ \boxed{ \text{Measurement Scheduling} \rightarrow \text{Information Optimization}. } \]

4.3.16.59 Fisher Information Matrix

The Fisher information matrix is closely related to sensitivity and covariance.

For Gaussian measurements,

\[ \boxed{ \mathbf F = \mathbf G^T \mathbf R^{-1} \mathbf G. } \]

Under suitable regularity conditions, it determines the local lower bound on parameter uncertainty.

4.3.16.60 Cramér-Rao Lower Bound

For an unbiased estimator,

\[ \boxed{ \operatorname{Cov}( \hat{\boldsymbol\theta} ) \succeq \mathbf F^{-1}. } \]

This is the Cramér-Rao lower bound.

It provides a theoretical best-case covariance under the assumed model.

4.3.16.61 Meaning of the CRLB

The CRLB answers:

Given this model, noise level, and measurement geometry, how accurately could an ideal unbiased estimator determine the parameters?

It therefore provides a valuable pre-estimation performance metric.

4.3.16.62 Sensitivity Magnitude

A parameter with very small sensitivity

\[ \left\| \frac{\partial\mathbf z}{\partial\theta} \right\| \ll1 \]

has little influence on the measurements.

Even if it is technically independent of other parameters, its uncertainty may remain large.

Thus identifiability alone does not imply strong estimability.

4.3.16.63 Sensitivity Scaling

Comparing raw Jacobian columns can be misleading if parameters have different units.

For example,

Scaled sensitivities may be defined as

\[ \boxed{ \tilde{\mathbf G} = \mathbf G \mathbf S } \]

where \mathbf S normalizes parameter scales.

4.3.16.64 Why Scaling Matters for SVD

Without scaling, a parameter with naturally large numerical units may dominate the singular-value analysis.

Proper scaling allows SVD and condition number to reflect physical estimability rather than arbitrary unit choice.

4.3.16.65 Dimensionless Sensitivity

One possible normalized sensitivity is

\[ \boxed{ S_{ij}^{*} = \frac{\theta_j} {z_i} \frac{\partial z_i} {\partial\theta_j}. } \]

This measures fractional output change relative to fractional parameter change.

Such normalization can be useful when comparing parameters of different physical units.

4.3.16.66 Weak Parameter Direction

Suppose the smallest right singular vector is

\[ \mathbf v_{\min}. \]

Then

\[ \boxed{ \delta\boldsymbol\theta = \alpha \mathbf v_{\min} } \]

changes the measurements only weakly.

This identifies the least distinguishable parameter combination.

4.3.16.67 Null-Space Parameter Combination

If

\[ \sigma_{\min}=0, \]

then

\[ \boxed{ \mathbf G \mathbf v_{\min} = \mathbf0. } \]

The corresponding parameter combination is completely unidentifiable under the linearized measurement model.

4.3.16.68 Near-Null Direction

If

\[ \sigma_{\min} \]

is small but nonzero, the direction is nearly unidentifiable.

Noise can cause very large uncertainty in that direction.

This is often more common in practical applications than exact rank deficiency.

4.3.16.69 Weak State Directions

In orbit estimation, weak directions may correspond to combinations such as

depending on the sensor geometry.

They are not fixed universally.

The weakest direction depends on the actual measurement set and dynamic evolution.

4.3.16.70 Weak Parameter Directions

For dynamic-parameter estimation, weak combinations may involve

These must be interpreted physically.

4.3.16.71 Improving Estimability by Removing Parameters

If two parameters are strongly correlated, estimating only one effective parameter can improve robustness.

For example, instead of estimating

\[ C_D,\quad A,\quad m \]

separately, one might estimate

\[ \boxed{ \frac{C_DA}{m}. } \]

This aligns parameterization with what the dynamics actually observe.

4.3.16.72 Composite Parameters

Suppose measurements depend only on

\[ \gamma = \theta_1\theta_2. \]

Then estimating \theta_1 and \theta_2 separately may be impossible.

Estimating

\[ \boxed{ \gamma } \]

directly is more appropriate.

The principle is

\[ \boxed{ \text{Estimate Observable Combinations} \text{ rather than arbitrary physical decompositions}. } \]

4.3.16.73 Improving Estimability with Priors

A prior can constrain a weak parameter:

\[ \boxed{ \theta \sim \mathcal N( \bar\theta, \sigma_\theta^2 ). } \]

This reduces posterior uncertainty.

However, the estimator should report that the solution is prior assisted.

4.3.16.74 Improving Estimability with Complementary Sensors

Suppose one sensor measures range and another measures LOS.

Their sensitivity directions are complementary.

Then combined information is

\[ \boxed{ \mathbf I_{\rm total} = \mathbf I_{\rm range} + \mathbf I_{\rm LOS}. } \]

This can significantly reduce weak directions.

4.3.16.75 Improving Estimability through Maneuvers

Intentional spacecraft motion can improve geometry.

A maneuver may change

Thus,

\[ \boxed{ \text{Control Maneuver} \rightarrow \text{Improved Estimation Geometry} } \]

in some mission designs.

4.3.16.76 Persistent Excitation

Parameters such as

may require sufficiently rich system motion.

If the system never excites a parameter, measurements may contain little information about it.

Therefore,

\[ \boxed{ \text{Parameter Estimation} \text{ requires informative excitation}. } \]

4.3.16.77 Calibration Geometry

A camera-alignment parameter may be weakly estimable if the target always remains near the same image location.

Changing target direction across the camera field can provide more independent information.

Thus calibration quality depends strongly on motion and viewing geometry.

4.3.16.78 Time-Window Selection

Estimability can depend on the selected data arc.

A short window may have poor rank.

A longer window may improve geometry.

But if parameters vary over time, using too long a window may violate the constant-parameter assumption.

Thus,

\[ \boxed{ \text{Data Arc} = \text{Information Gain} \leftrightarrow \text{Model Validity}. } \]

4.3.16.79 Sliding-Window Estimability

For time-varying systems, estimability may be evaluated over a moving data window.

At each epoch,

\[ \mathbf G_{\rm window}(t) \]

can be formed.

Monitoring its singular values reveals when the navigation geometry becomes weak.

4.3.16.80 Real-Time Geometry Monitoring

A real-time estimator can monitor metrics such as

This allows the system to detect periods of poor navigation geometry.

4.3.16.81 Geometry-Based Measurement Scheduling

If multiple candidate measurements are available, one can choose those expected to improve information most.

Potential objectives include

\[ \boxed{ \max \det(\mathbf I) } \]

or

\[ \boxed{ \min \operatorname{tr}(\mathbf P). } \]

These are examples of optimal experiment or sensor-scheduling criteria.

4.3.16.82 D-Optimality

D-optimal design seeks to maximize

\[ \boxed{ \det(\mathbf I). } \]

This approximately minimizes the volume of the covariance ellipsoid.

Thus it favors overall information content.

4.3.16.83 A-Optimality

A-optimal design minimizes

\[ \boxed{ \operatorname{tr}(\mathbf P). } \]

This minimizes the sum of parameter variances.

It emphasizes average uncertainty.

4.3.16.84 E-Optimality

E-optimal design seeks to improve the weakest direction.

One form maximizes the minimum eigenvalue:

\[ \boxed{ \max \lambda_{\min}(\mathbf I). } \]

This is useful when the goal is to avoid one extremely weak state direction.

4.3.16.85 Geometry and Sensor Placement

Sensor placement directly affects measurement sensitivity.

Examples include

Good geometry can improve estimation without changing sensor intrinsic precision.

4.3.16.86 Baseline

For triangulation, a larger baseline often improves depth sensitivity.

In simplified stereo geometry,

\[ \boxed{ Z \approx \frac{fB}{d}. } \]

Depth sensitivity improves with suitable baseline B, though practical limits arise from correspondence and field-of-view constraints.

4.3.16.87 Relative Navigation Geometry

In general relative navigation, estimability depends on the geometry between the observer and target.

Relevant measurements may include

The same rank, SVD, covariance, and information concepts apply.

This is a general principle and not specific to any particular RPO system.

4.3.16.88 Relative Range-Only Geometry

With only one relative-range measurement,

\[ \rho = \|\mathbf r_{\rm rel}\|, \]

the instantaneous direction of

\[ \mathbf r_{\rm rel} \]

is not determined.

Dynamic evolution and repeated measurements are required to recover additional state information.

4.3.16.89 Relative LOS-Only Geometry

LOS provides direction but not scale.

Thus absolute relative range is weak instantaneously.

Relative motion over time can provide scale through changing LOS geometry.

4.3.16.90 Relative Range + LOS

Combining

\[ \rho \]

and

\[ \hat{\boldsymbol\rho} \]

gives

\[ \boxed{ \mathbf r_{\rm rel} = \rho \hat{\boldsymbol\rho}. } \]

This greatly improves instantaneous relative-position geometry.

Relative velocity still generally requires either

4.3.16.91 Geometry and Maneuver Estimation

A maneuver parameter can only be estimated if its trajectory effect projects into measured directions.

For example, a small out-of-plane burn may be weakly estimable if all measurements are insensitive to cross-track motion.

Thus maneuver estimability is inherently geometry dependent.

4.3.16.92 Geometry and Drag Estimation

Drag mainly affects along-track orbital motion over time.

Measurements must be sensitive enough to the accumulated along-track trajectory change.

Short arcs or poor along-track sensitivity can make drag parameters difficult to estimate.

4.3.16.93 Geometry and SRP Estimation

SRP sensitivity depends on Sun-spacecraft geometry.

If the geometry remains nearly constant or other forces produce similar effects, C_R may be weakly estimated.

Changing solar geometry and eclipse transitions can provide additional information.

4.3.16.94 Parameter Correlation Matrix Visualization

A useful diagnostic is the correlation matrix

\[ \boxed{ \mathbf C_{ij} = \frac{P_{ij}} {\sqrt{P_{ii}P_{jj}}}. } \]

Visualizing \mathbf C can quickly reveal groups of strongly coupled parameters.

Blocks of correlations may indicate redundant parameterization.

4.3.16.95 Sensitivity-Matrix Visualization

Plotting columns of the sensitivity matrix over time can also reveal estimability.

If two columns are nearly identical throughout the arc, the corresponding parameters will be difficult to separate.

Thus a useful diagnostic is

\[ \boxed{ \mathbf g_i(t) \quad\text{versus}\quad \mathbf g_j(t). } \]

4.3.16.96 Common Mistakes

Common mistakes include:

  1. Assuming many measurements automatically imply good observability.
  2. Checking only matrix rank and ignoring weak singular values.
  3. Using unscaled parameters when computing condition number.
  4. Interpreting prior-constrained parameters as strongly observed.
  5. Ignoring null-space directions.
  6. Estimating too many nearly redundant parameters.
  7. Treating high parameter correlation as purely numerical rather than geometric.
  8. Ignoring changing geometry with time.
  9. Using only one instantaneous \mathbf H to assess a dynamic problem.
  10. Ignoring state-transition effects in batch estimability.
  11. Comparing Jacobian columns with incompatible units.
  12. Declaring full rank as proof of good practical estimability.
  13. Ignoring weak covariance eigen-directions.
  14. Improving numerical conditioning without addressing the underlying geometric weakness.
  15. Using a long data arc even when parameters vary over time.
  16. Using a short arc that contains insufficient excitation.
  17. Ignoring complementary sensor geometry.
  18. Failing to examine parameter correlations.
  19. Adding priors without reporting their influence.
  20. Treating low residual RMS as evidence of strong parameter identifiability.

4.3.16.97 Practical Geometry and Estimability Workflow

A disciplined workflow is

\[ \boxed{ \begin{array}{c} \text{Define State and Parameter Vector}\\ \downarrow\\ \text{Define Measurements and Geometry}\\ \downarrow\\ \text{Compute Measurement Jacobians}\\ \downarrow\\ \text{Propagate STM / Parameter Sensitivities}\\ \downarrow\\ \text{Build Stacked Sensitivity Matrix}\\ \downarrow\\ \text{Scale States and Parameters}\\ \downarrow\\ \text{Check Rank}\\ \downarrow\\ \text{Compute SVD}\\ \downarrow\\ \text{Inspect Singular Values}\\ \downarrow\\ \text{Identify Weak / Null Directions}\\ \downarrow\\ \text{Compute Information Matrix}\\ \downarrow\\ \text{Inspect Covariance and Correlations}\\ \downarrow\\ \text{Assess Geometry over Time}\\ \downarrow\\ \text{Modify Sensor Set / Arc / Parameterization if Needed}\\ \downarrow\\ \text{Validate with Simulation or Independent Data} \end{array} } \]

4.3.16.98 General Estimability Architecture

The overall structure can be represented as

\[ \boxed{ \begin{array}{c} \text{State / Parameters}\\ \boldsymbol\theta \\ \downarrow\\ \text{Dynamics}\\ \downarrow\\ \text{Measurement Sensitivities}\\ \mathbf G \\ \downarrow\\ \text{Rank / SVD / Information Matrix}\\ \downarrow\\ \text{Strong and Weak Directions}\\ \downarrow\\ \text{Covariance / Correlation}\\ \downarrow\\ \text{Estimability Assessment}\\ \downarrow\\ \text{Sensor / Geometry / Parameter-Set Design} \end{array} } \]

4.3.16.99 Core Equations

Linearized measurement model

\[ \boxed{ \delta\mathbf z = \mathbf H \delta\mathbf x + \mathbf v } \]

Reference-state sensitivity

\[ \boxed{ \mathbf G_k = \mathbf H_k \boldsymbol\Phi(t_k,t_0) } \]

Stacked design matrix

\[ \boxed{ \mathbf G = \begin{bmatrix} \mathbf G_1\\ \mathbf G_2\\ \vdots\\ \mathbf G_N \end{bmatrix} } \]

Rank criterion

\[ \boxed{ \operatorname{rank}(\mathbf G)=n } \]

for local full-rank estimability.

Null space

\[ \boxed{ \mathbf G\mathbf q = \mathbf0 } \]

SVD

\[ \boxed{ \mathbf G = \mathbf U \boldsymbol\Sigma \mathbf V^T } \]

Condition number

\[ \boxed{ \kappa(\mathbf G) = \frac{\sigma_{\max}} {\sigma_{\min}} } \]

Information matrix

\[ \boxed{ \mathbf I = \mathbf G^T \mathbf R^{-1} \mathbf G } \]

Approximate covariance

\[ \boxed{ \mathbf P \approx \mathbf I^{-1} } \]

for a full-rank local Gaussian problem.

Correlation coefficient

\[ \boxed{ \rho_{ij} = \frac{P_{ij}} {\sqrt{P_{ii}P_{jj}}} } \]

Observability Gramian

\[ \boxed{ \mathbf W_o = \int \boldsymbol\Phi^T \mathbf H^T \mathbf R^{-1} \mathbf H \boldsymbol\Phi \,dt } \]

4.3.16.100 Key Takeaways

Geometry, identifiability, and estimability determine whether a navigation or orbit-determination problem actually contains enough information to estimate its states and parameters.

The linearized relationship

\[ \boxed{ \delta\mathbf z = \mathbf G \delta\boldsymbol\theta + \mathbf v } \]

contains the essential information.

If

\[ \boxed{ \operatorname{rank}(\mathbf G)

some combination of states or parameters lies in the null space and cannot be identified.

However, full rank alone is not sufficient.

A full-rank system may still be poorly estimable if one singular value is extremely small.

Therefore,

\[ \boxed{ \text{Full Rank} \neq \text{Good Conditioning}. } \]

The SVD

\[ \boxed{ \mathbf G = \mathbf U \boldsymbol\Sigma \mathbf V^T } \]

provides one of the clearest tools for identifying strong and weak estimation directions.

The smallest singular value identifies the weakest direction, while

\[ \boxed{ \kappa = \frac{\sigma_{\max}} {\sigma_{\min}} } \]

quantifies conditioning.

The information matrix

\[ \boxed{ \mathbf I = \mathbf G^T \mathbf R^{-1} \mathbf G } \]

links measurement geometry directly to estimation uncertainty.

When invertible,

\[ \boxed{ \mathbf P \approx \mathbf I^{-1}. } \]

Large covariance eigenvalues therefore correspond to weak information directions.

Parameter correlation is another major diagnostic:

\[ \boxed{ \rho_{ij} = \frac{P_{ij}} {\sqrt{P_{ii}P_{jj}}}. } \]

When

\[ |\rho_{ij}|\approx1, \]

two estimated quantities are difficult to separate.

The most important practical principle is:

\[ \boxed{ \text{Good Estimation} \text{ requires not only good sensors, but good geometry.} } \]

Estimability can often be improved by

A well-designed estimation problem should therefore answer three questions:

\[ \boxed{ \begin{array}{c} \text{What can be observed?}\\ \text{How strongly can it be observed?}\\ \text{Which combinations remain weak or indistinguishable?} \end{array} } \]

Those questions form the foundation for robust geometry assessment, parameter selection, covariance interpretation, and reliable navigation-system design.

Next: 4.3.17 Numerical Conditioning and Robust Solution Methods

Geometry and estimability determine whether the measurements contain enough independent information. The next section explains how to solve that estimation problem robustly in finite-precision computation, including normal equations, QR, Cholesky, SVD, square-root methods, scaling, regularization, condition-number analysis, and rank-deficient solution methods.

4.3.17 Numerical Conditioning and Robust Solution Methods

Normal equations, QR, Cholesky, SVD, square-root least squares, condition number, scaling, regularization, and rank deficiency.

Continue to Numerical Conditioning and Robust Solution Methods →