Least-squares adjustment is a model for the solution of an overdetermined system of equations based on the principle of least squares of observation residuals. It is used extensively in the disciplines of surveying, geodesy, and photogrammetry—the field of geomatics, collectively.
Formulation There are three forms of least squares adjustment: parametric, conditional, and combined:
In parametric adjustment, one can find an observation equation h(X) = Y relating observations Y explicitly in terms of parameters X (leading to the A-model below). In conditional adjustment, there exists a condition equation which is g(Y) = 0 involving only observations Y (leading to the B-model below) — with no parameters X at all. Finally, in a combined adjustment, both parameters X and observations Y are involved implicitly in a mixed-model equation f(X, Y) = 0. Clearly, parametric and conditional adjustments correspond to the more general combined case when f(X,Y) = h(X) - Y and f(X, Y) = g(Y), respectively. Yet the special cases warrant simpler solutions, as detailed below. Often in the literature, Y may be denoted L.
Solution The equalities above only hold for the estimated parameters X ^ {\displaystyle {\hat {X}}} and observations Y ^ {\displaystyle {\hat {Y}}} , thus f ( X ^ , Y ^ ) = 0 {\displaystyle f\left({\hat {X}},{\hat {Y}}\right)=0} . In contrast, measured observations Y ~ {\displaystyle {\tilde {Y}}} and approximate parameters X ~ {\displaystyle {\tilde {X}}} produce a nonzero misclosure:
w ~ = f ( X ~ , Y ~ ) . {\displaystyle {\tilde {w}}=f\left({\tilde {X}},{\tilde {Y}}\right).}
One can proceed to Taylor series expansion of the equations, which results in the Jacobians or design matrices: the first one,
A = ∂ f / ∂ X ; {\displaystyle A=\partial {f}/\partial {X};}
and the second one,
B = ∂ f / ∂ Y . {\displaystyle B=\partial {f}/\partial {Y}.}
The linearized model then reads:
w ~ + A x ^ + B y ^ = 0 , {\displaystyle {\tilde {w}}+A{\hat {x}}+B{\hat {y}}=0,}
where x ^ = X ^ − X ~ {\displaystyle {\hat {x}}={\hat {X}}-{\tilde {X}}} are estimated parameter corrections to the a priori values, and y ^ = Y ^ − Y ~ {\displaystyle {\hat {y}}={\hat {Y}}-{\tilde {Y}}} are post-fit observation residuals. In the parametric adjustment, the second design matrix is an identity, B=-I, and the misclosure vector can be interpreted as the pre-fit residuals, y ~ = w ~ = h ( X ~ ) − Y ~ {\displaystyle {\tilde {y}}={\tilde {w}}=h({\tilde {X}})-{\tilde {Y}}} , so the system simplifies to:
A x ^ = y ^ − y ~ , {\displaystyle A{\hat {x}}={\hat {y}}-{\tilde {y}},}
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