5.3 Least-Squares Adjustment of Level Nets, Triangulation, and Plane Traverses

Least-Squares Adjustment of Level Nets, Triangulation, and Plane Traverses

Modern geodetic data processing relies on rigorous matrix formulation of least squares to adjust complex 1D elevation networks, 2D triangulation/trilateration control figures, and plane traverses. This section details parametric matrix techniques ($A^T P A X = A^T P L$), non-linear model linearization, error ellipse computation, and residual quality control tests required for GELE mastery.

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1. Parametric (Observation Equations) vs. Correlative (Condition Equations) Methods

Least-squares adjustments are executed through two primary formulation approaches:

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2. Rigorous Matrix Formulation of Parametric Least Squares

For a system of $n$ observations and $u$ unknown parameters ($n > u$), the linear observation model in matrix notation is:

$$V = A X - L$$

where:

• $V_{n \times 1}$: Vector of observation residuals ($[v_1, v_2, \dots, v_n]^T$).
• $A_{n \times u}$: Design Matrix containing partial derivatives of observation equations with respect to unknown parameters ($\partial f_i / \partial x_j$).
• $X_{u \times 1}$: Vector of unknown parameter corrections ($[\delta x_1, \delta x_2, \dots, \delta x_u]^T$).
• $L_{n \times 1}$: Vector of misclosures ($L = L_{observed} - L_{computed}^0$).
• $P_{n \times n}$: Weight Matrix (diagonal matrix $P = \text{diag}(p_1, p_2, \dots, p_n)$ for independent observations).

2.1 Matrix Derivation of Normal Equations

Minimizing the weighted residual sum $V^T P V = (A X - L)^T P (A X - L)$:

$$V^T P V = X^T A^T P A X - 2 X^T A^T P L + L^T P L$$

Taking partial derivatives with respect to $X$ and setting to zero:

$$\frac{\partial(V^T P V)}{\partial X} = 2 A^T P A X - 2 A^T P L = 0 \implies (A^T P A) X = A^T P L$$

Solving for parameter corrections $X$:

$$X = (A^T P A)^{-1} A^T P L$$

2.2 Residuals and Variance-Covariance Matrices

• Adjusted Residuals Vector: $V = A X - L$
• A Posteriori Reference Variance ($\hat{\sigma}_0^2$): $$\hat{\sigma}_0^2 = \frac{V^T P V}{n - u} = \frac{V^T P V}{r}$$
• Parameter Variance-Covariance Matrix ($\Sigma_X$): $$\Sigma_X = \hat{\sigma}_0^2 (A^T P A)^{-1} = \begin{bmatrix} \sigma_{x_1}^2 & \sigma_{x_1 x_2} & \dots \\ \sigma_{x_2 x_1} & \sigma_{x_2}^2 & \dots \\ \vdots & \vdots & \ddots \end{bmatrix}$$
• Standard errors of adjusted parameters: $\sigma_{x_j} = \sqrt{\Sigma_{X, jj}}$.

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3. 1D Level Network Adjustment: Step-by-Step Worked Matrix Example

Network Geometry

Consider a level net with two fixed control benchmarks BM-1 ($Elev = 10.000\text{ m}$) and BM-2 ($Elev = 25.000\text{ m}$) and two unknown junction points A ($H_A$) and B ($H_B$). Four elevation differences are measured:

Initial approximate elevations: $H_A^0 = 15.100\text{ m}$, $H_B^0 = 24.900\text{ m}$.

Step 1: Form Observation Equations ($v_i = a_{i1} \delta H_A + a_{i2} \delta H_B - l_i$)

• Line 1: $H_A - 10.000 = 5.100 + v_1 \implies \delta H_A - (5.100 - 5.100) = \delta H_A - 0.000$
• Line 2: $H_B - H_A = 9.800 + v_2 \implies \delta H_B - \delta H_A - (9.800 - (24.900 - 15.100)) = -\delta H_A + \delta H_B - 0.000$
• Line 3: $25.000 - H_B = 0.200 + v_3 \implies -\delta H_B - (0.200 - (25.000 - 24.900)) = -\delta H_B - 0.100$
• Line 4: $H_B - 10.000 = 15.000 + v_4 \implies \delta H_B - (15.000 - 14.900) = \delta H_B - 0.100$

Step 2: Assemble Matrices

$$A = \begin{bmatrix} 1 & 0 \\ -1 & 1 \\ 0 & -1 \\ 0 & 1 \end{bmatrix}, \quad P = \begin{bmatrix} 1.0 & 0 & 0 & 0 \\ 0 & 0.5 & 0 & 0 \\ 0 & 0 & 1.0 & 0 \\ 0 & 0 & 0 & 0.5 \end{bmatrix}, \quad L = \begin{bmatrix} 0.000 \\ 0.000 \\ 0.100 \\ 0.100 \end{bmatrix}$$

Step 3: Compute Normal Matrix $N = A^T P A$

$$A^T P = \begin{bmatrix} 1 & -0.5 & 0 & 0 \\ 0 & 0.5 & -1.0 & 0.5 \end{bmatrix}$$ $$N = A^T P A = \begin{bmatrix} (1 + 0.5) & (-0.5) \\ (-0.5) & (0.5 + 1.0 + 0.5) \end{bmatrix} = \begin{bmatrix} 1.5 & -0.5 \\ -0.5 & 2.0 \end{bmatrix}$$

Step 4: Compute Right-Hand Side Vector $A^T P L$

$$A^T P L = \begin{bmatrix} 1(0) - 0.5(0) + 0(0.1) + 0(0.1) \\ 0(0) + 0.5(0) - 1.0(0.1) + 0.5(0.1) \end{bmatrix} = \begin{bmatrix} 0.000 \\ -0.050 \end{bmatrix}$$

Step 5: Invert Normal Matrix $N^{-1}$

$$\det(N) = (1.5)(2.0) - (-0.5)(-0.5) = 3.0 - 0.25 = 2.75$$ $$N^{-1} = \frac{1}{2.75} \begin{bmatrix} 2.0 & 0.5 \\ 0.5 & 1.5 \end{bmatrix} = \begin{bmatrix} 0.72727 & 0.18182 \\ 0.18182 & 0.54545 \end{bmatrix}$$

Step 6: Solve for Parameter Corrections $X = N^{-1} A^T P L$

$$X = \begin{bmatrix} 0.72727 & 0.18182 \\ 0.18182 & 0.54545 \end{bmatrix} \begin{bmatrix} 0.000 \\ -0.050 \end{bmatrix} = \begin{bmatrix} -0.00909\text{ m} \\ -0.02727\text{ m} \end{bmatrix}$$

Step 7: Calculate Final Adjusted Elevations

• $\hat{H}_A = H_A^0 + \delta H_A = 15.100 - 0.0091 = 15.0909\text{ m}$
• $\hat{H}_B = H_B^0 + \delta H_B = 24.900 - 0.0273 = 24.8727\text{ m}$

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4. Linearization of 2D Triangulation and Trilateration Equations

In 2D horizontal networks, distance and angle observation equations are non-linear functions of station coordinates $(X_A, Y_A, X_B, Y_B)$. They must be linearized using first-order Taylor series expansion around initial approximate coordinates $(X^0, Y^0)$.

Distance Observation Equation Linearization

$$S_{AB} = \sqrt{(X_B - X_A)^2 + (Y_B - Y_A)^2}$$ $$\delta S_{AB} = -\cos(\alpha_{AB}) \, \delta X_A - \sin(\alpha_{AB}) \, \delta Y_A + \cos(\alpha_{AB}) \, \delta X_B + \sin(\alpha_{AB}) \, \delta Y_B$$

where $\alpha_{AB}$ is the grid azimuth from station A to B.

Azimuth Observation Equation Linearization

$$\alpha_{AB} = \arctan\left(\frac{X_B - X_A}{Y_B - Y_A}\right)$$ $$d\alpha_{AB} = \frac{\cos(\alpha_{AB})}{S_{AB}} \, \delta X_A - \frac{\sin(\alpha_{AB})}{S_{AB}} \, \delta Y_A - \frac{\cos(\alpha_{AB})}{S_{AB}} \, \delta X_B + \frac{\sin(\alpha_{AB})}{S_{AB}} \, \delta Y_B$$

The adjustment is iterated until parameter update vector $\|X\| < \epsilon$ (e.g. $< 0.1\text{ mm}$).

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5. Plane Traverse Adjustment: Least Squares vs. Compass Rule

For decades, boundary surveys in the Philippines were adjusted using the empirical Compass (Bowditch) Rule. However, modern regulations under DENR Administrative Orders specify Least Squares for control networks.

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6. Point Error Ellipses and Residual Analysis

6.1 Point Error Ellipses (Confidence Ellipses)

For a 2D station $i$, the coordinate covariance matrix is extracted from $\Sigma_X$:

$$\Sigma_{XY} = \begin{bmatrix} \sigma_x^2 & \sigma_{xy} \\ \sigma_{xy} & \sigma_y^2 \end{bmatrix}$$

The geometric parameters of the standard ($39.4\%$ 2D probability) error ellipse are:

1. Orientation angle of semi-major axis ($\theta$):

   $$\tan(2\theta) = \frac{2 \sigma_{xy}}{\sigma_x^2 - \sigma_y^2}$$

2. Semi-major axis ($a$) and Semi-minor axis ($b$):

   $$a = \sqrt{ \frac{1}{2} \left( \sigma_x^2 + \sigma_y^2 + \sqrt{(\sigma_x^2 - \sigma_y^2)^2 + 4 \sigma_{xy}^2} \right) }$$ $$b = \sqrt{ \frac{1}{2} \left( \sigma_x^2 + \sigma_y^2 - \sqrt{(\sigma_x^2 - \sigma_y^2)^2 + 4 \sigma_{xy}^2} \right) }$$

To scale to a $95\%$ confidence ellipse, multiply $a$ and $b$ by $K = \sqrt{\chi_{2, 0.05}^2} = \sqrt{5.991} \approx 2.4477$.

                 Y (North)
                     ^       / Semi-Major Axis (a)
                     |      / 
                     |     /---. 
                     |   .'  |  `.
                     |  /    |    \
                     | |     +-----|---> X (East)
                     |  \\   /     /
                     |   `.  \\  .'  
                     |     `---' \\ Semi-Minor Axis (b)
                     |            \\ Angle theta
                     +----------------------->

6.2 Residual Analysis and Outlier Detection

To detect blunders in large observations sets, Standardized Residuals ($v_i^*$) are evaluated:

$$v_i^* = \frac{v_i}{\sigma_{v_i}} = \frac{v_i}{\hat{\sigma}_0 \sqrt{ (P^{-1} - A (A^T P A)^{-1} A^T)_{ii} }}$$

If $|v_i^*| > 3.0$, the observation is flagged as a statistical outlier (blunder) at the $99.7\%$ confidence level and subjected to rejection or re-measurement.