**Recalculating inflated temperatures when wrong emissivites where used**

As a first approach to recalculate the Lugano dummy run for the case that the reported temperatures where inflated due to using wrong emissivity settings on the Optris thermal camera, we need to recalculate the temperatures to their real ones.

The procedure which can recalculate the temperatures is based on the following formula published by Optris which can be found on page 9 of their IR-basics document.

**U = C · [ε Tobj**^{n} +(1 – ε) · Tamb^{n} – Tpyr^{n}]

The meaning of the parameters is as follows :

---------U-----------The voltage from the thermal camera sensor

---------C-----------A constant

---------ε-----------The in band emissivity set on the Optris

---------Tobj------The temperature of the measured object in degree K

---------Tamb----The ambient temperature

---------Tpyr------The temperature of the camera sensor

---------n-----------A constant depending on the used sensor frequency band

We can use this formula for two situations, the first with the wrongly used emissivity ε1 and the accompanying measured temperature Tobj1, the second case with the correct emissivity ε2 to be used with the Optris and the correct temperature Tobj2.

Since the measured sensor voltage U is only dependent on the amount of radiation coming from the object under observation, this value is the same for both situations.

Thus we can fill in the above formula for both situation and then equate them.

This is written out below

**C · [ε1 Tobj1**^{n} + (1 – ε1) · Tamb^{n} – Tpyr^{n}] = C · [ε2 Tobj2^{n} + (1 – ε2) · Tamb^{n} – Tpyr^{n}]

Simplifying this gives :

**ε1 Tobj1**^{n} + (1 – ε1) · Tamb^{n} = ε2 Tobj2^{n} + (1 – ε2) · Tamb^{n}

Or

** ε2 Tobj2**^{n} = ε1 Tobj1^{n} + (ε2 - ε1) · Tamb^{n }

**Tobj2**^{n} = (ε1/ε2) Tobj1^{n} + (1 - ε1/ε2) · Tamb^{n}

**Tobj2 = [(ε1/ε2) Tobj1**^{n} + (1 - ε1/ε2) · Tamb^{n} ] 1/**n**

The last formula above will be used in a spreadsheet to recalculate the assumed inflated temperatures to the assumed real temperatures.

Note :

For high temperatures the term** (e1/e2)Tobj1**^{n} is much larger then the term

** (1 - ε1/ε2) · Tamb**^{n} and the last term can in that case be discarded.

(Note that the term with **Tamb** can not be discarded for lower temperatues since the errors become quite large)

This leads the to the following formula also used by the MFMP to be used for high temperatures :

**(Tobj2/Tobj1) = (ε1/ε2)1/n **

The MFMP verified this formula at higher temperatues to be working with a value of n=3.