Solving One Problem, Creating Another: Ratio Method in Two-Colour Pyrometry

The ratio method attempts to simultaneously determine a single temperature and emissivity ratio that satisfies the measurements from two detectors at two independent wavelengths. The ratio method is a technique used in two-colour (or two-wavelength) pyrometry. It helps estimate the true surface temperature of an object. It does this using two detectors that each measure radiation at a different wavelength.

The basic idea behind two-colour pyrometry is straightforward: In conventional non-contact temperature measurement, there are typically two unknowns—the object’s true temperature and its emissivity—but only one measurement from a single detector. The two-color method adds a second detector operating at a different wavelength, effectively providing a second measurement. At first glance, this appears to create a solvable system: two unknowns (temperature and emissivity) and two independent measurements. However, in reality, the system still involves three unknowns, because the emissivity can vary with wavelength. This means that the emissivity at one wavelength may differ from that at the second, so assumptions about the emissivity ratio must still be made to solve for the true surface temperature.

Background or optical path effects are also an important point in favour of ratio pyrometers. Suppose there is dust, steam, or a semi-transparent window in the optical path. In that case, a ratio pyrometer will only cancel out those losses if the attenuation is equal at both wavelengths. In many cases, broadband obstructions, like a uniformly dirty lens or smoke that affects both channels similarly, are indeed mitigated by ratio measurements. But if the transmittance of the path differs with wavelength, like non-grey attenuation, it effectively alters the measured emissivity ratio and introduces error, just like an emissivity change would. Even if the emissivity ratio is constant, unequal channel transmission can introduce errors similar to emissivity errors.

The assumption for a two-colour pyrometer is often that the emissivity is almost equal at both wavelengths. [math]λ_T[/math] and [math]λ_B[/math], so that [math]ε_1 (λ_T )= ε_2 (λ_B )[/math]. If [math]ε_1 (λ_T )[/math] does not equal [math]ε_1 (λ_B )[/math] Then, an adjustment must be made, which is typically called “Slope” or emissivity ratio in ratio pyrometers.

[math]ε_slope= (ε_1 (λ_T ))/(ε_2 (λ_B ) )≈1 [/math]

One way to express the ratio pyrometer analytically is as follows. From the definition of emissivity, which is defined as the factor between brightness and blackbody brightness at the same temperature. Inserting the Planck expression and noting that the [math]λ^5[/math] and [math]c_1[/math] terms cancel out, the following term, which includes the radiation constants [math]c_1[/math] and [math]c_2[/math].

[math]ε_slope= (ε_1 (λ_T ))/(ε_2 (λ_B ) )=(M_(λ_T ) (λ_T,T_T ))/(M_(λ_T)^° (λ_T,T) )∙(M_(λ_B)^° (λ_B,T))/(M_(λ_B ) (λ_B,T_B ) )=⋯=(e^(c_2/(λ_T ∙T_T ))-1)/(e^(c_2/(λ_T ∙T))-1)∙(e^(c_2/(λ_B ∙T))-1)/(e^(c_2/(λ_B ∙T_B ))-1)[/math]

Taking the ratio of the two emissivity at the two wavelengths results in the following relationship between the two equivalent brightness temperatures [math]T_T[/math],[math]T_B[/math] at both wavelengths and the actual surface temperature [math]𝑇[/math].

There is no analytic solution to the equation, so the solution must be found numerically by iteration. Therefore, the ratio temperature is introduced, which is calculated from the two detector readings. It’s the temperature of a blackbody that would emit the same radiation ratio at the two wavelengths.

[math]1/T_R = Λ (1/(λ_T T_T )- 1/(λ_B T_B ))[/math]

Using Wien’s approximation and the information on the relationship between the two emissivities, a simpler formula is derived.

[math]1/T=1/T_R + Λ/c_2 ln⁡(ε_slope )[/math]

Although Wien’s law introduces some error at lower temperatures or longer wavelengths, since it neglects the factor in Planck’s law, the following equation can be derived. Still, it is often quite accurate in the high-temperature, short-wavelength regime where two-colour pyrometers operate.

The effective wavelength Λ is hereby introduced, depending on the two wavelengths used. It is not a physical wavelength but a mathematical construct that appears in the formula for determining the surface temperature.

[math]Λ= (λ_B λ_T )/(λ_T-λ_B )[/math]

With this set of equations, the temperature can be calculated by measuring the brightness temperature at both equations and assuming an emissivity ratio, which is often close to 1.

The effective wavelength Λ increases rapidly as the two measurement wavelengths and become closer together. While selecting wavelengths that are close to each other can help ensure a more stable and predictable emissivity ratio between them, this also results in a significantly higher value of the effective wavelength. A higher effective wavelength, however, makes the calculated temperature much more sensitive to uncertainties in the emissivity ratio. This creates a fundamental trade-off in two-color pyrometry: using closely spaced wavelengths improves the reliability of emissivity assumptions but amplifies the impact of any error in those assumptions on the resulting temperature.

From Detector Signals to Temperature: Mathematical Basis of the Ratio Method

Another form to express the ratio dependency is as follows. The ratio signal between the channels can also be expressed by the following. It is assumed that the two wavelengths are very narrow, which are measured, and the equation simplifies after considering Planck’s law, considering that the measured bandwidth is monochromatic [math]Δλ ≪λ[/math] and the wavelength [math]λ_B≈ λ_T[/math] and the emissivity values are close to each other ,[math]ε_1 (λ_T )≈ε_2 (λ_B )[/math]. As the detectors measure the received infrared flux proportionally, the expression for the ratio method is obtained from the definition of Planck’s law in the following equation for two wavelengths [math]λ_T[/math] and [math]λ_B[/math].

[math]U_T/U_B ~(ϕ_(λ_T ) ( λ_1,T_obj ))/(ϕ_(λ_B ) ( λ_2,T_obj ) )=(ε_1 (λ_T ))/(ε_2 (λ_B ) ) (M^∘ (λ_T,T_obj ) )/(M^∘ (λ_B,T_obj ) )=⋯= ε_slope (λ_B^5)/(λ_T^5 ) (e^(c_2/(λ_B ∙T))-1)/(e^(c_2/(λ_T ∙T))-1) ≈ε_slope (λ_B/λ_T )^5 e^(c_2/(λ_T ∙T)- c_2/(λ_B ∙T))=ε_slope (λ_B/λ_T )^5 e^( c_2/T (λ_B- λ_T)/(λ_B λ_T ))= ε_slope (λ_B/λ_T )^5 e^(- c_2/TΛ )[/math]

Figure 1 illustrates the detector signals for a typical ratio pyrometer (R1M), shown as a function of target temperature. The two solid lines represent the spectral signal responses of the individual detectors in channels 1 and 2, which operate at different infrared wavelengths. As governed by Planck’s law, each single-color channel receives exponentially increasing radiation intensity with rising temperature, though the magnitude and slope differ due to the wavelength-dependent sensitivity.

The dotted line shows the logarithmic ratio of these two channels. Unlike the raw signals, the ratio response exhibits a near-linear behavior over a wide temperature range, enabling temperature measurement without requiring exact knowledge of the emissivity—assuming the emissivity ratio between both channels remains constant.

Based on the above formula, the temperature of an object can be rearranged to the following equation. Here again for further simplification, Wien approximation has to be applied.

[math]T= c_2/(Λ [ln⁡〖(ε_slope )+ ln⁡((λ_B^5)/(λ_T^5 ))-ln⁡(U_T/U_B ) 〗 ] )[/math]

How Emissivity Slope Errors Distort Two-Color Temperature Readings

Unfortunately, there is a price to pay for the ability of the ratio pyrometer to be able to cancel out the emissivity term. The temperature error depends strongly on the uncertainty in the emissivity ratio. If the two wavelengths are too close together, Λ becomes very large, making the result more sensitive to errors. However, over small wavelength ranges, the slope might be more stable or better known, which can improve accuracy overall. A slight change in the emissivity ratio affects the temperature estimate. The sensitivity can be shown by solving the ratio equation for temperature, differentiating the temperature by the slope, and considering the Wien approximation to remove some exponential and logarithmic forms.

[math]ΔT/T≈(-ΛT)/c_2 Δε_slope[/math]

Even a few percent uncertainty in the emissivity ratio can produce tens of degrees of error at high temperatures. In practical terms, a seemingly minor emissivity ratio change can shift the calculated temperature by a large amount.

Many commercial systems advertised as measuring “true” temperature rely on two detectors operating at separate wavelengths and use a “ratio” mode to compensate for emissivity differences. However, accurate temperature results depend on the emissivity ratio between the two wavelengths being known or remaining stable. This approach comes at a cost: the method is highly sensitive to deviations in the emissivity ratio. In some cases, a 1 °C difference between the two detector signals can result in a 10 °C shift in the calculated two-color temperature. In general, a ratio pyrometer is truly emissivity-independent only if the emissivity ratio remains constant during measurement.

Figure 2 illustrates the impact of emissivity slope variation on temperature measurement accuracy in two-color pyrometers. Although ratio pyrometers compensate for constant emissivity, deviations still arise when the emissivity changes differently at each wavelength; therefore, the slope changes.

Why Ratio Pyrometers Can Show Larger Errors Than Single-Color Devices

In addition to the slope changes, the system accuracy of each channel must be considered. As the ratio temperature is based on the individual’s temperature. To do so, the ratio temperature equation is calculated based on the assumption of a deviated temperature reading of a single channel and evaluated. Due to the fact that the single-color uncertainties are passed through the amlomst linear ratio pyrometer’s emissivity ratio sensitivity equation, the ratio temperature can be have a higher deviation than same device in single-color mode.

Figure 3 compares the system accuracy on a perfect blackbody calibration source and compares a conventional single-color pyrometer with an assumed accuracy of 0.3% of the reading with an assumed calibration accuracy of 0.1% per channel for the ratio pyrometer. If a true target temperature of 1000 °C, Channel 1 reads 999 °C and Channel 2 reads 1000.0 °C—both within typical single-channel calibration uncertainty, the ratio temperature might read a deviation of up to 4°C.

Two-colour pyrometers can amplify small channel mismatches and small errors in the assumed emissivity ratio. This follows directly from the effective wavelength. Choosing closely spaced wavelengths often stabilises the emissivity ratio close to the assumption that they are equal to one, but it also increases the effective wavelength and therefore the temperature’s sensitivity to slope and inter-channel differences.
As a rule of thumb: The two-colour temperature can shift markedly: for approximately every 1 °C difference between channels, the ratio result can move by up to ~10 °C.

Figure 4 combines the effect of system accuracy and slope changes. Interestingly, a short-wavelength single-color pyrometer can keep up with a dual-color pyrometer.

Figure 5 illustrates the measurement deviation for a ratio pyrometer, showing how deviations in the assumed spectral emissivity ratio and system accuracy impact temperature readings. The x-axis represents object temperature, the y-axis shows the relative change in slope of the emissivity ratio (±40%), and the z-axis indicates the resulting error in the measured temperature. Even moderate slope mismatches can lead to substantial deviations, particularly at low temperatures. This emphasizes the sensitivity of ratio pyrometry to emissivity slope errors and underscores the need for careful spectral emissivity characterization, especially when operating at lower temperatures.

When Two-Colour Pyrometry Works—and When It Doesn’t

Overall, ratio pyrometers are not necessarily more accurate than single-color pyrometers unless the emissivity ratio is stable and well-founded, and important. It correctly debunks the misconception that two-colour devices are automatically emissivity-independent. Errors in the assumed emissivity ratio—especially due to changes in surface condition, wavelength-dependent emissivity behavior, or angle of observation—can lead to significant measurement deviations. While they are still sensitive to spectral emissivity variations, ratio pyrometers offer important advantages in harsh or variable environments: they can compensate for uniform signal losses caused by dirty optics, viewing windows, or partially obscured targets. They are also less affected by target size variations or alignment shifts, making them well-suited for moving or partially covered objects. When the emissivity ratio is known or reasonably stable, ratio pyrometry enables robust, non-contact temperature measurement in applications where conventional single-color methods would struggle.