# Table 2 Basic transformation models to derive an IOT ($${\mathbf{S}}$$ and $${\mathbf{E}}$$ or $${\mathbf{B}}$$ and $${\mathbf{h}}$$) from a SUT ($${\mathbf{V}}$$ and $${\mathbf{U}}$$).
Model Formulas for $${\mathbf{S}}$$ or $${\mathbf{B}}$$ Formulas for $${\mathbf{E}}$$ Formulas for $${\mathbf{h}}$$
A (PTA-p*p) $${\mathbf{S}}^{{\left( {\text{A}} \right)}} = {\mathbf{UV}}^{{ - {\text{T}}}} {\hat{\mathbf{q}}}$$ $${\mathbf{E}}^{{\left( {\text{A}} \right)}} = {\mathbf{WV}}^{{ - {\text{T}}}} {\hat{\mathbf{q}}}$$
B (ITA-p*p) $${\mathbf{S}}^{{\left( {\text{B}} \right)}} = {\mathbf{U}}{\hat{\mathbf{g}}}^{ - 1} {\mathbf{V}}$$ $${\mathbf{E}}^{{\left( {\text{B}} \right)}} = {\mathbf{W}}{\hat{\mathbf{g}}}^{ - 1} {\mathbf{V}}$$
C (ISA-i*i) $${\mathbf{B}}^{{\left( {\text{C}} \right)}} = {\hat{\mathbf{g}}}{\mathbf{V}}^{{ - {\text{T}}}} {\mathbf{U}}$$ $${\mathbf{h}}^{{\left( {\text{C}} \right)}} = {\hat{\mathbf{g}}}{\mathbf{V}}^{{ - {\text{T}}}} {\mathbf{d}}$$
D (PSA-i*i) $${\mathbf{B}}^{{\left( {\text{D}} \right)}} = {\mathbf{V}}{\hat{\mathbf{q}}}^{ - 1} {\mathbf{U}}$$ $${\mathbf{h}}^{{\left( {\text{D}} \right)}} = {\mathbf{V}}{\hat{\mathbf{q}}}^{ - 1} {\mathbf{d}}$$
1. The superscript $$- 1$$ indicates inversion; the superscript $${-}{\text{T}}$$ indicates transposition and inversion; the hat ($${\hat{\mathbf{x}}}$$) on top of a vector indicates diagonalization into a square matrix