= Additional measurements and anatomical identifiability
\b[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 when
$$
F(\theta_1)=F(\theta_2)\ \Longrightarrow\ S(\theta_1)=S(\theta_2)
$$
throughout 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>:
$$
\boxed{S=2A+HP=\frac{2M}{\rho H}+HP.}
$$
For stipulated $H=1.5\,\mathrm m$, $M=60\,\mathrm{kg}$, $\rho=1000\,\mathrm{kg/m^3}$ 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.<Guide to the expression of uncertainty in measurement by JCGM>[§4.1.2: measurement models, corrections and incomplete 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.
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