 Ruby pressure uncertainty worked example

Construct a synthetic local calibration to show the calculation independently of the historical data. Choose reference pairs $(\Delta {\lambda }_i,p_i)=(1.00,2.69),(2.00,5.42),(3.00,8.09)$, with shifts in $\mathrm{nm}$ and pressures in $\mathrm{GPa}$. The reference zero is already subtracted. Treat the shifts as exact for this fit and give the three reference pressures equal weights. Minimizing the ordinary least squares objectiveand setting its derivative to zero givesThe regression residuals are $(-0.01,0.02,-0.01)\mathrm{GPa}$. Their weighted moment ${\sum }_i\Delta {\lambda }_i(p_i-a\Delta {\lambda }_i)$ vanishes, as required by the least-squares normal equations. An unknown reference offset requires fitting an intercept instead; forcing it to zero can bias the slope. If shift errors matter, this elementary fit is insufficient.

Use the local measurement modelHere $p$ denotes pressure relative to the reference, $a$ is calibrated sensitivity, and $k_T$ is a local wavelength sensitivity to temperature at fixed pressure. Choose $a=2.70\mathrm{GPa}\mathrm{n}{\mathrm{m}}^{-1}$, $k_T=0.010\mathrm{nm}{\mathrm{K}}^{-1}$ and a reading $\lambda -{\lambda }_0=2.00\mathrm{nm}$ at $T=T_0$. The inferred pressure is $5.40\mathrm{GPa}$. These coefficients and readings are stipulated, not measured ruby constants. With the same pressure, a $10\mathrm{K}$ increase in temperature would add $0.100\mathrm{nm}$ to the shift in this model; neglecting it would overestimate pressure by $0.270\mathrm{GPa}$.

Assign independent standard uncertainties $u_a=0.020\mathrm{GPa}\mathrm{n}{\mathrm{m}}^{-1}$, $u_{\lambda }=u_0=0.010\mathrm{nm}$ and $u_T=0.50\mathrm{K}$ for the temperature difference. Here $u_a$ is an independently assigned calibration uncertainty, not an estimate from the regression residuals above; treat $k_T$ as exact in this example. First-order propagation of uncertainty follows by differentiating the measurement model:The four standard contributions are $0.0400,0.0270,0.0270,0.0135\mathrm{GPa}$, giving $u_p=0.0569\mathrm{GPa}$. This is a standard uncertainty, not automatically a $95\%$ interval. Correlated reference and loaded wavelength readings require the corresponding covariance term; common wavelength-axis drift may cancel in their difference. Uncertain $k_T$, nonzero $T-T_0$, model inadequacy and unresolved deviatoric stress require a broader budget. Taking more optical readings reduces some random errors while leaving a common reference-scale error.