Recovery-correction uncertainty quantifies uncertainty in an amount inferred through a recovery correction. If $\hat{q}=y/r$ uses measured recovered amount $y$ and recovery estimate $r>0$, both inputs contribute. The uncertainty of the corrected result is not obtained merely by copying the uncertainty attached to $y$.

The first-order sensitivity coefficients are $\partial \hat{q}/\partial y=1/r$ and $\partial \hat{q}/\partial r=-y/r^2$. Combining them givesThis is the linear propagation of uncertainty law with correlated inputs.^[ref] With positive nominal inputs and correlation coefficient $\rho$, the relative form isRecovery-estimate covariance can matter because an amount and its estimated recovery may share measurements or common response changes.

For original values $y=6\mathrm{nmol}$, $r=0.60$, $u_y=0.30\mathrm{nmol}$ and $u_r=0.06$, the corrected amount is $10\mathrm{nmol}$. Independence gives relative variance $0.05^2+0.10^2=0.0125$ and standard uncertainty about $1.12\mathrm{nmol}$. With $\rho =0.8$, the variance becomes $0.0045$ and uncertainty about $0.67\mathrm{nmol}$. Correlation cannot be chosen to improve the answer; it must describe the actual joint measurements.

The formula is an approximation near the nominal values. A recovery near zero or with large relative uncertainty can make a quotient strongly asymmetric and poorly represented by a symmetric standard uncertainty. Model inadequacy—such as applying a recovery from an unrepresentative sample matrix—is a separate issue that careful numerical propagation alone does not resolve.

A nonnegative real number is a real number $x\ge 0$. The set of all such numbers is ${\mathbb{R}}_{\ge 0}=[0,\infty )$, which includes zero as well as the positive real numbers. The word nonnegative is therefore weaker than strictly positive.

This set is closed under addition and multiplication, but negating a nonzero member leaves it. Every real square and every absolute value is nonnegative. A variance can equal zero, as it does for a constant random variable; calling every variance positive would incorrectly exclude that legitimate case. Nonnegative constraints similarly allow zero as a feasible boundary value in mathematical optimization.

A negative real number is a real number $x<0$, lying to the left of zero on the number line. Equivalently, $x=-a$ for a positive real number $a$. Examples include $-3$, $-1/2$ and $-\sqrt{2}$; zero is excluded.

The sum of two negative real numbers is negative, while their product is positive. Multiplication by a negative real number reverses an inequality: if $x<y$ and $c<0$, then $cx>cy$. These are consequences of compatibility between the total order and the arithmetic of an ordered field, rather than independent rules about written minus signs.

A shared-excitation resistance ratio measurement compares two electrical resistances by passing the same electric current through them and comparing their electric potential differences. It is the circuit realization of a ratiometric measurement: the unknown excitation becomes a common multiplicative input.

For a sensor $R_T$ in series with a reference resistor $R_{\mathrm{ref}}$, ideal high-input-impedance sensing givesThe electric current must be nonzero and shared at the time relevant to each sample. One implementation feeds $V_T$ to an analog-to-digital converter input and $V_{\mathrm{ref}}$ to its external reference pins; the conversion is then scaled by their ratio. The primary circuit guide shows this wiring and includes the programmed gain when translating a digital code to resistance.^[ref]

For an original check, choose $R_T=180\Omega$ and $R_{\mathrm{ref}}=1200\Omega$. Excitation at $0.4\mathrm{mA}$ gives $0.072\mathrm{V}$ and $0.480\mathrm{V}$; excitation at $1.2\mathrm{mA}$ gives $0.216\mathrm{V}$ and $1.440\mathrm{V}$. Both give $r=0.15$ and recover $180\Omega$. This calculation assumes unchanged resistances; the larger excitation dissipates nine times the sensor electric power, making sensor self-heating in a ratiometric resistance measurement an independent test.

A common supply is insufficient if different branch currents pass through the two resistances. With currents $I_T,I_{\mathrm{ref}}$,This is branch-current mismatch in a resistance ratio measurement. Sequential sampling of a changing excitation has the same problem unless the two paths retain the same effective excitation history. Ratiometric cancellation with matched signal and reference filters specifies the dynamic condition.

Wiring matters even with an ideal shared current source. A two-wire sensor reading includes lead electrical resistance; four-terminal sensing separates the current leads from high-impedance potential-sensing leads. The reference itself needs the same care if lead drops would otherwise enter its measured denominator.^[ref]

A bench check can replace the sensor with a stable known resistor, sweep excitation, and compare each measured potential with their ratio. A constant ratio accompanied by changing individual potentials tests shared excitation. It does not establish reference accuracy, eliminate sensor heating, or prove operation outside the converter's input and reference ranges.

Reference-resistor error in ratiometric resistance measurement is the error left when an instrument converts a measured resistance ratio using an inaccurate value for its reference resistor. Shared excitation removes one nuisance input; it does not establish the denominator's true electrical resistance.

Let $R_a$ be the actual reference resistor value and $R_c$ the value used in calculation. For an ideal simultaneous reading $r=R_T/R_a$, the inferred sensor resistance isWriting $R_a=R_c(1+e)$ gives the exact fractional error $-e/(1+e)$. The familiar $-e$ is its first-order approximation. If a deterministic bound is $|e|\le E<1$, the inferred fractional error lies inA symmetric reference tolerance therefore produces a slightly asymmetric resistance-error interval.

For an original example, suppose the instrument uses $R_c=2000\Omega$ while the actual reference resistor is $2001.6\Omega$. A true $150\Omega$ sensor is reported as $149.880096\dots \Omega$, an error near $-0.07994\%$. Multiplying the shared excitation by any common positive number leaves this error unchanged in the ideal model. Using a measured and corrected $R_c$ instead of its nominal marking can remove the known part, while the corrected value retains its own measurement uncertainty.

Temperature variation can produce the remaining error after measurement calibration. Locally, $e(T)\simeq e(T_0)+{\alpha }_R(T-T_0)$, where ${\alpha }_R$ is the temperature coefficient of resistance. For example, a $20$-parts-per-million-per-kelvin coefficient over $15\mathrm{K}$ contributes roughly $300$ parts per million; it survives excitation cancellation. The primary circuit guide explicitly requires reference accuracy and low drift, and separates these requirements from shared-current cancellation.^[ref]

For small random errors, applying the law of propagation of uncertainty to ${\hat{R}}_T=r{R}_c$ givesThe covariance term must follow how the reference value and reading were obtained; it cannot simply be assumed zero when inputs are shared.^[ref]

Ratiometric cancellation with matched signal and reference filters preserves a resistance ratio despite changing excitation when both paths apply the same dynamics to the shared electric current. Matching only the steady gain is insufficient during a transient.

Let constant sensor and reference resistances be $R_T,R_{\mathrm{ref}}$, and let both paths be linear time-invariant systems with impulse responses $h_T,h_{\mathrm{ref}}$. With consistent initial conditions, their filtered electric potential differences arewhere $\ast$ is convolution. If $h_T=h_{\mathrm{ref}}$, simultaneous sampling gives $y_T/y_{\mathrm{ref}}=R_T/R_{\mathrm{ref}}$ whenever the denominator is nonzero. Arbitrary unequal stored initial states need not cancel. Nor can a time-varying $R_T$ generally be pulled outside the convolution.

For unit steady gains, write $I(t)=I_0(1+\eta (t))$ after settling at $I_0$. Define $q_T=h_T\ast \eta$ and $q_{\mathrm{ref}}=h_{\mathrm{ref}}\ast \eta$. ThenThis displays exactly which excitation component survives. For small fluctuations the fractional residual is $q_T-q_{\mathrm{ref}}$, so both amplitude and timing differences between the transfer functions matter.

As an original transient example, step the excitation up by ten percent from a settled state. Two first-order low-pass filters with time constants ${\tau }_T=1\mathrm{ms}$ and ${\tau }_{\mathrm{ref}}=2\mathrm{ms}$ give, for $t\ge 0$,At $t=1\mathrm{ms}$ the reading is high by approximately $2.2962\%$, although both filters have the same final gain and the error tends to zero. Setting both time constants to $1\mathrm{ms}$ cancels this particular excitation step at every time. Sampling skew after matched ratiometric filters can still break simultaneous cancellation even with identical hardware.

The primary ADS1248 experiment used a resistance decade box, a $1\mathrm{mA}$ excitation and a $2\mathrm{k\Omega }$ reference. Reference-only filtering produced signal-dependent noise; matching the signal and reference paths substantially reduced that dependence. The report provides the circuit and noise-versus-input plots, rather than merely recommending a common nominal cutoff.^[ref]

Sensor self-heating in a ratiometric resistance measurement occurs when the sensing electric current changes the sensor's own temperature through Joule heating. A ratiometric measurement can remove the direct dependence of the electrical reading on that electric current while still reading the heated sensor correctly. For a resistance thermometer, that means the indicated temperature can differ from the unperturbed surroundings.

With actual sensor electrical resistance $R_T(I)$ and a stable reference resistor $R_{\mathrm{ref}}$ carrying the same electric current,The cancellation leaves the dependence of $R_T$ on $I$ intact. In a small, steady perturbation, a model with thermal resistance $R_{\mathrm{th}}$ gives $\Delta T\simeq R_{\mathrm{th}}{I}^2{R}_T(0)$ andwhere ${\alpha }_R$ is the temperature coefficient of resistance. These approximations require small heating and thermal conditions that remain comparable between readings. The BIPM guide distinguishes heat flow inside the probe from heat flow into its surroundings; changing the surrounding medium can change the effect.^[ref]

A practical diagnostic is a two-current extrapolation. Model $R_T(I)\simeq R_0+c{I}^2$ over the tested range. Measurements $R_1,R_2$ at distinct nonzero current magnitudes $I_1,I_2$ imply$R_0$ is an inferred zero-power electrical resistance, not an electrical reading taken with zero excitation. The guide describes this squared-current method and requires a stable surrounding temperature and settled readings.^[ref]

For an original numerical example, suppose a $1000\Omega$ reference resistor gives ratios $0.120006$ at $0.5\mathrm{mA}$ and $0.120024$ at $1.0\mathrm{mA}$. The inferred resistances are $120.006\Omega$ and $120.024\Omega$. Since $I_2^2=4I_1^2$, the estimate is $(4R_1-R_2)/3=120.000\Omega$. Both ratios canceled their excitation exactly, yet they disagree because the sensor changed.

A bath experiment should alternate the two excitation settings, allow thermal settling and monitor the bath temperature. Repeated low/high/low readings help reveal drift rather than attributing every difference to heating. Resistance thermometer self-heating gives the wider physical setting; reference-resistor error in ratiometric resistance measurement and mismatched filters are separate contributions.

Radiant exposure measures the radiant energy received per unit surface area during a specified interval. At a fixed surface point $\mathbf{x}$, it is the time integral of incident irradiance $E_{\mathrm{e}}$:The area belongs to the receiving surface: its orientation and location matter. An irradiance in $\mathrm{W}{\mathrm{m}}^{-2}$ specifies a rate, while radiant exposure in $\mathrm{J}{\mathrm{m}}^{-2}$ specifies the accumulated arrival of energy.^[ref]

For a uniform rectangular pulse, $H_{\mathrm{e}}=E_0\tau$, with constant irradiance $E_0$ and duration $\tau$. Stipulate one pulse at $20\mathrm{W}{\mathrm{m}}^{-2}$ for $1\mathrm{s}$, and another at $80\mathrm{W}{\mathrm{m}}^{-2}$ for $0.25\mathrm{s}$. Both deliver $20\mathrm{J}{\mathrm{m}}^{-2}$, but their peak irradiances differ by a factor of four. The two radiant exposures agree only after each complete pulse has arrived. A shorter observation window can intercept different fractions of them, and a nonlinear receiver can respond differently even to their complete pulses.

Integrating over a receiving patch gives its incident radiant energy:Here ${\overline{H}}_A$ is the spatially averaged radiant exposure, rather than necessarily the local maximum. Uniform illumination is the special case where this spatially averaged radiant exposure equals the local value everywhere on the patch. In the rectangular-pulse example, a uniformly illuminated $1\mathrm{c}{\mathrm{m}}^2=10^{-4}{\mathrm{m}}^2$ patch receives $20\times 10^{-4}=2\mathrm{mJ}$.

For an original spatial example, consider a circular Gaussian beam at a fixed transverse plane, with constant Gaussian beam radius $w$ throughout the pulse. Let $P(t)$ be its total radiant power and $U=\int P(t)dt$ its laser pulse energy. Integrating its transverse irradiance in time gives the radiant exposure of a Gaussian beam:The normalization follows by integrating $H(r)2\pi rdr$ from $0$ to $\infty$. The disk of radius $w$ contains only $U(1-e^{-2})$, so its actual spatially averaged radiant exposure is $U(1-e^{-2})/(\pi w^2)$, not $U/(\pi w^2)$. With stipulated $U=1\mu \mathrm{J}$ and $w=1\mathrm{mm}$, the peak is $0.6366\mathrm{J}{\mathrm{m}}^{-2}$, while the average inside that disk is $0.2752\mathrm{J}{\mathrm{m}}^{-2}$. State the Gaussian beam radius convention and the receiving area with any reported value.

An energy readout alone therefore does not specify a surface's radiant exposure. In an ideal linear model for pulse energy measurement with a photodiode, dark-subtracted photocurrent $i_{\mathrm{ph}}(t)=R{P}_{\det }(t)$ has constant spectral responsivity $R$ at the chosen wavelength. Integrating the electric current gives signal electric charge $Q_{\mathrm{sig}}=R{U}_{\det }$. If a known patch of area $A$ is uniformly illuminated and all of its incident energy is collected, thenAs a separate stipulated example, $Q_{\mathrm{sig}}=0.40\mathrm{nC}$ and $R=0.40\mathrm{A}{\mathrm{W}}^{-1}$ correspond to $U_{\det }=1.0\mathrm{nJ}$. Uniform illumination over $A=1.0\mathrm{m}{\mathrm{m}}^2$ gives $H_{\mathrm{e}}=1.0\mathrm{mJ}{\mathrm{m}}^{-2}$. Background subtraction, collection geometry, detector linearity and an integration window covering the full response are part of this measurement model. Continuous-wave responsivity alone does not establish linearity for a short pulse: silicon-photodiode experiments have observed pulse-dependent saturation.^[ref]

A historical NIST measurement calibration separates these operations experimentally. A modulator produces pulses from a continuous-wave laser; a calibrated beam splitter sends one beam to a slow thermal reference and another to the pulse detector. The reference records the accumulated energy, while the pulse repetition rate and injection duration determine the mean energy per pulse. The detector's integrating electronics provide a pulse-energy readout. Converting such a readout to radiant exposure additionally requires the illuminated area and its spatial distribution.^[ref]

The beam-use term fluence denotes the same time integral for a directed beam normal to the chosen surface. For light arriving from many directions, specify the angular geometry: planar irradiance and spherical irradiance are different quantities, so a value defined using one cannot simply be substituted for the other.^[ref]

Radiant energy is the amount of energy emitted, transferred or received as electromagnetic radiation. It is measured in joules. For the specified beam or collection of rays crossing a boundary,where radiant flux ${\Phi }_{\mathrm{e}}$ is radiation power. The receiving geometry and time interval are part of the statement; dividing by a receiving area produces radiant exposure, rather than another total energy.^[ref]

For an original passive-optic example, stipulate $5.0\mathrm{mJ}$ incident during an interval, with no remaining change of stored energy afterward. Suppose $3.0\mathrm{mJ}$ is transmitted, $1.25\mathrm{mJ}$ reflected and $0.75\mathrm{mJ}$ absorbed. Their sum is the incoming $5.0\mathrm{mJ}$. A detector collecting only the transmitted path receives $3.0\mathrm{mJ}$, even though the source supplied more. This radiometric energy balance specifies the collection boundary instead of equating every energy readout in the apparatus.

A thermal detector converts absorbed radiation into an observable temperature response; a calibrated model relates that response to received energy. A documented NIST arrangement uses a slow thermal reference and calibrated beam splitting to establish pulse-energy measurements.^[ref] Absorbed energy, incident radiant energy and energy in an uncollected branch must remain distinct in that calibration.

Two sources can deliver the same radiant energy with different radiant flux histories or spectra. Their total energies agree only for the specified integration and collection; this alone does not determine peak power, radiant exposure at a point or detector response.

Spherical irradiance counts incident radiance from all directions without selecting a receiving-plane normal. Using the projected-area convention specified in the CIE entry's note2,Here $d\Omega$ is an element of solid angle. An ideal small sphere of radius $a$ intercepts radiant flux ${\Phi }_{\mathrm{in}}=\pi a^2{E}_{\mathrm{e},\mathrm{o}}$. The denominator is its diametrical projected area, not its full surface area.^[ref]

For an original isotropic radiance field example, take $L_{\mathrm{e}}=L_0=2\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$ in every direction. Then $E_{\mathrm{e},\mathrm{o}}=4\pi L_0=8\pi \mathrm{W}{\mathrm{m}}^{-2}$. A one-sided plane instead receivesThe factor four follows from the different angular geometries; it is not a unit conversion. Averaging the sphere's intercepted power over its actual $4\pi a^2$ surface also gives $E_{\mathrm{e},\mathrm{o}}/4$, rather than the projected-area-normalized quantity above.

An all-direction receiver must have a characterized angular response. A single flat irradiance detector inherently samples a hemisphere with projected-area weighting, so rotating it or enclosing it in a diffuser does not supply an all-direction value without an appropriate response model and measurement calibration. Report the angular and area conventions with the result; equal units do not make irradiance and spherical irradiance interchangeable.

An integrating sphere is an optical cavity with diffusely reflecting interior walls and ports for illumination, samples or detectors. Repeated wall reflections mix the incident radiant flux, reducing sensitivity to where it first strikes. Baffles prevent a receiver from directly viewing an unwanted source or illuminated patch; port size and wall reflectance remain part of the response model.^[ref]

An original encounter model illustrates the effect of openings. Suppose a well-mixed reflection retains fraction $\rho$ and fraction $f$ of the reflected light escapes through ports before the next wall encounter. The integrating-sphere port-loss factor $1-f$ combines with wall reflectance to give a survival fraction $q=\rho (1-f)$. If the first wall encounter carries energy $U$, successive encounters carry $U,qU,q^{2}U,\dots$, so their summed incident energy is $U/(1-q)$.

With stipulated $\rho =0.98$ and $f=0.02$, this sum is $25.25U$. Enlarging the openings to $f=0.05$ reduces it to $14.49U$. These are repeated encounters with the same initially supplied energy, not optical gain: absorption and escape together still account for that original energy. The example counts incident wall encounters from the first one and does not assert a calibrated detector multiplier.

An actual sphere measurement compares its detector signal with a calibrated input or reference. Coating wavelength dependence, ports, sample substitution and temporal response can affect the conversion. The primary guide discusses reflectance/transmittance arrangements and their measurement geometry.^[ref] An integrating sphere is therefore useful apparatus, rather than a guarantee that every outgoing component has been measured without correction.