Least-squares adjustment terminology: the equations obtained by multiplying each observation equation by the coefficient of its unknowns and summing are called what?

Introduction / Context:
In surveying and geodesy, least-squares adjustment is used to estimate unknowns (e.g., coordinates) from redundant observations while minimizing the sum of squared residuals. Understanding the formation of the core equations is fundamental for reliable network adjustment.

Given Data / Assumptions:

• A linearized observation model relating observations to unknown parameters.
• Matrix form: v = A x - l, with A as coefficient matrix.
• Weighted least-squares objective: minimize vᵀ W v.

Concept / Approach:
Setting the derivative of the objective with respect to the unknowns to zero yields the normal equations: (Aᵀ W A) x = Aᵀ W l. Conceptually, this is equivalent to multiplying each observation equation by its coefficients and summing to collect terms in the unknowns.

Step-by-Step Solution:
Start from observation equations relating measurements to unknowns.Form the least-squares objective and differentiate with respect to unknowns.Set derivatives to zero to obtain the normal equations.Recognize that the described algebraic process corresponds to constructing the normal equations.

Verification / Alternative check:
Textbook derivations show equivalence between matrix normal equations and coefficient-weighted summations.

Why Other Options Are Wrong:

• Observation equations: original measurement relationships.
• Conditional equations: constraints independent of observations (e.g., geometric conditions).
• Residual equation/None: do not describe the forming step.

Common Pitfalls:
Confusing conditional constraints with observation models; ignoring weighting in forming normal equations.

Final Answer:
Normal equation