Design choice: mcps

[martin-rdz]

Switch from a mixture of photon counts and mcps to mcps consistently in the code.

ToDo @HolgerPollyNet : clarify if the factor should be 50 or 150

[martin-rdz]

It's c/2 = 150 m/μs (travel time of light back and forth).
Divided by 7.5m range resolution, thats 20 μs^{-1}, which is equivalent to 50ns sampling duration per height bin.

So the equation should be

$$ \text{PCR} = \frac{\text{signal}}{\text{mshots} \, \cdot \, \text{sample\_duration\_per\_height\_bin }} = \frac{\text{signal}}{\text{mshots}} \, \frac{c}{2} \, \frac{1}{\text{hRes}} $$

Implicitly the 1e-6 in μs seems to be omitted to directly give the megacounts.

[ulysses78]

$$ \text{sample\_duration\_per\_height\_bin } = \frac{2 \cdot \text{h}_\text{res}}{\text{c}} = \frac{2 \cdot \text{7.5m}}{\text{300000000 m/s}} = \text{50} ns $$

[ulysses78]

Nice to have it in such clear and understandable formula :-)

[HavardStridBuholdt]

Similarly to the MCPS/PCR and photon count mixture, we also should be more consistent on the RCS usage in the future. In the current version we calculate the RCS from the PCR signal in the prepossessing and save it as an output. However this is never used later in the code and the RCS is instead recalculated from the photon count (BG corrected) signal whenever it is needed.

Additionally, at the moment we also aggregate the signals differently. The RCS is averaged while everything else is summed. We should find a more consistent way of doing this. Either sum everything or averaged everything.

[martin-rdz]

Funnily enough the 150 is also used while loading the data

PicassoPy/ppcpy/io/readPollyRawData.py

Line 104 in e7afdf6

data_dict['measurement_height_resolution']['var_data'] = data_dict['measurement_height_resolution']['var_data']*0.15

An upcoming change will document this more clearly.

About the other questions. When using the PCR or RCS, only averaging makes sense.

[HolgerPollyNet]

but the 150 are in km? summing up makes indeed sense if you need to calcualte the statistical error but if you transfer the alser shots you can always calculate back...

[martin-rdz]

After thorough analysis, we realized, that the SNR calculation of the temporally aggregated signal cannot be done on the pcr / mcps signal.

[HavardStridBuholdt]

Correction:

It is in theory possible to calculate the SNR for PCR signal if the shift in domain of the noise from photon count to PCR is correctly accounted for. In photon count domain the variance of the signal is defined as $Var[P_i] = P_i + 2 BG_i$ where $P_i$ is the background corrected signal at time $i$ and $BG_i$ is the background at time $i$.

Assuming that we know the variance of $P_i$ we can then calculate the variance of $PCR_i$. As mentioned above $PCR = k \frac{P_i}{m_i}$ where $m_i$ is the measurement shots at time $i$ and $k=\frac{c}{2 \times hRes}$. Since $m_i$ is not stochastic, the variance of $PCR_i$ can then be described as:

$$Var[PCR_i] = Var[k \frac{P_i}{m_i}] = \frac{k^2}{m^{2}_{i}} Var[P_i] = \frac{k^2}{m^{2}_{i}} (P_i + 2 BG_i)$$

Which gives us the following formula for the SNR for PCR-signals:

$$SNR(PCR_i) = \frac{PCR_i}{\sqrt{Var[PCR_i]}} = \frac{k \frac{P_i}{m_i}}{\sqrt{\frac{k^2}{m^{2}_{i}} (P_i + 2 BG_i)}} = \frac{P_i}{\sqrt{P_i + 2 BG_i}} = SNR(P_i) $$

As one can see, this boils down to the same formula for as for calculating the SNR from photon count signal, which makes sens as the transition between photon count and PCR is only a linear transformation. However, to calculate the SNR for our PCR-signal through the formulas above, we would need the variance of the PCR-signal which we currently only have and analytical definition of through the photon count signal $P_i$. Thus, we would need to convert back to photon count in order to calculate the SNR from PCR-perspective. This conversion back and forward can be avoided by simply plugging the PCR-signal into the photon count SNR formula and multiplying with a correction factor of $\sqrt{\frac{m_i}{k}}$ afterwards.

The bigger issue lay in temporally aggregating PCR-signals. One can not simply aggregate PCR-signals the same way as photon count signal through summing or taking the mean temporally. Instead one have to sum / take the mean of $P_i$ and $m_i$ individually before recalculating the PCR for the whole aggregated temporal period. For this it is not possible to avoid converting back and forward between PCR and Photon count.

Because of the above mentioned issues, it was decided to stick with the current logic with using photon count signals and only converting to PCR when needed to avoid unnecessary conversions.