 Additional measurements and anatomical identifiability

Additional measurements and anatomical identifiability asks which observations distinguish the anatomical quantities that a measurement model aims to recover. More observations help only if they constrain information missing from the existing measurements. Repeating one height measurement and measuring an independent dimension are different operations.

Formally, let $F(\theta )$ denote the observations predicted from shape parameters $\theta$, and let $S(\theta )$ be the desired surface area. The area is identifiable from those observations precisely whenthroughout the allowed model class. This is quantity-specific identifiability: identifying a target need not require reconstructing every shape parameter. Measurement precision is a separate issue from this structural property.

Extend the original cuboid construction in non-identifiability of shape from height and mass. Fixed height $H$, mass $M$ and assumed uniform density $\rho$ determine $A=ab=M/(\rho H)$, but not the side lengths $a,b$. An additional horizontal perimeter measurement $P=2(a+b)$ determines the total surface area:For stipulated $H=1.5\mathrm{m}$, $M=60\mathrm{kg}$, $\rho =1000kg/{\mathrm{m}}^3$ and $P=1.0\mathrm{m}$, this gives $A=0.040{\mathrm{m}}^2$ and $S=1.58{\mathrm{m}}^2$. The horizontal sides solve the quadratic equation $z^2-(P/2)z+A=0$, giving $0.40$ and $0.10\mathrm{m}$. Their interchange leaves the same area. A square section of the same $A$ has $P=0.80\mathrm{m}$ and total area $1.28{\mathrm{m}}^2$; height and mass alone could not distinguish these cases.

The conclusion relies on the stated geometry and density. A circumference at one location does not generally determine the entire body surface area of an arbitrary person. Two shapes can share that circumference and differ elsewhere. Validation therefore needs independent geometric evidence and a justified model class, not simply a larger number of inputs. The GUM likewise requires relevant model quantities and corrections to be included rather than equating precise inputs with complete knowledge.^[ref]

Local sensitivity also deserves attention. In the parameterization $a=\sqrt{A}{e}^t$, $b=\sqrt{A}{e}^{-t}$, the perimeter is $P(t)=4\sqrt{A}\cosh t$. Its derivative vanishes at the square $t=0$, and $P(t)-P(0)=2\sqrt{A}{t}^2+O(t^4)$. Small departures from a square can therefore be difficult to resolve from a noisy perimeter even when exact observations determine the area. Structural identifiability and practical precision answer different questions.