Theoretical background on pwsFWI

== Introduction ==
 
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NOTE: This Fire Weather Index is an original development and bears no relation to any other existing fire weather index and the resulting (absolute) value can not be compared to any of those. It is the resulting warning level which can be compared and which should be used.
NOTE: This wiki article differs in some important aspects from the original blog An effort for a simpler fire weather index on which it is based.
The Fire Weather Index for a personal weather station, in short pwsFWI, is probably one of the most complex modules of CumulusUtils. Not so much for the calculations, which once you know what to do are not that complex, but more for the interpretation and understanding of what is shown. This Wiki article will show the science background to pwsFWI.
For external reference on the theory behind the pwsFWI you can go the The Hot-Dry-Windy Index: A New FireWeather Index site of Michigan State University who adopted it as a tool for climatological analysis. The 2018 article on which I based the software is The Hot-Dry-Windy Index: A New Fire Weather Index. This is an open access article under the Creative Commons Attribution License. 
On my (HansR) blog you may read several posts on the pwsFWI but also on the other fire weather indices in the world notably the Canadian FWI. This complex FWI is used in many places but it is so complex that it is not summarized in equations, but referenced by the articles. The complexity of this FWI was one of the main reasons to create pwsFWI. Other fire indexes such as the Ångström index and the Chandler Burning Index, though often seen on amateur weather sites, have been looked at but these are not discriminating enough.
After describing different indices for fire weather, in reverse order: the Canadian FWI (Contains a longer literature list), At the end of the heatwave, The Ångström index and the FMI index and The Chandler Burning Index, the following was concluded:
#Wet wood does not burn easily (if at all);
#Moisture content of the fuel (wood) is of great importance;
#Wind does not spark fire but assists drying and is dangerous for propagation;
#Rain contradicts drying but not immediately;
#Drying timber in a forest is not a single day event.
== Some remarks on Humidity ==
One addition must be made: there are woods which burn more easily than other woods (eucalyptus, pinus etc…), but they also burn only when they are dry. Goodrick et.al.Alan F. Srock, Joseph J. Charney, Brian E. Potter and Scott L. Goodrick, The Hot-Dry-Windy Index: A New FireWeather Index. Atmosphere 2018, 9, 279 state very clearly that:
 […] we want to isolate the effects of weather on a wildland fire, so we define a fire weather index (FWI) as an index that includes only weather inputs and thus does not include explicit or implicit information about the state of wildland fuels or topography.
 […] However, when a fire index is formulated using an explicit or implicit fuel relationship, the effects of the fuel-based parameters add variability due to the embedded non-linear computations. So, if a “true” FWI fails to identify a dangerous fire event, the index failed either because the weather was not adequately represented in the index or the weather did not have a significant impact on the fire. If a non-weather-only fire index fails to identify a fire event, it could be because of the aforementioned reasons but also could be due to the fuels and/or topography information in the calculation, which makes determining attribution, efficacy, and failure modes for the index more difficult.
This means, they take only meteorology into account.
I assume the same attitude here.
Hamadeh et.al.Nizar Hamadeh, Ali Karouni, Bassam Daya, Pierre Chauvet. Using Correlative Data Analysis to Develop Weather Index That Estimates the Risk of Forest Fires In Lebanon: Assessment versus Prevalent Meteorological Indices. International Journal of Physical Science Research. Vol.1, No.2, pp.14- 38, August 2017. made an analysis of the correlation between meteorological measurements and fire occurrences. In their paper, the team produced graphs, indicating the probability of fire occurring for a meteorological parameter. They more or less contradict the observations above concerning humidity and wind.
But it is not these parameters alone, but in interaction they play their role. Relative humidity and wind together dry out the fuels and it does occur over days or weeks, therefore there is no direct relation between those parameters and fire occurrence.
Generally, people will go out – for a picnic or whatever – with nice weather light winds, no rain etc… And on a picnic, a thrown cigarette can light a fire if the fuel is dry enough. Therefore, nice picnic weather characteristics will have higher probability with fire occurrence. Yes, I assume implicitly most wildfires have human origin (maybe as a result of stupidity, definitely not always malicious).
== Humidity – the science ==
Relative Humidity is defined as the ratio of the vapor pressure and the saturation vapor pressure:
 {RH}\ {{=}}\ \frac{P}{P_{Sat}} (1)
Relative humidity is measured directly by the weather station and as such is easy to use. However, the point is that the relative humidity does not cover the whole story. The same RH will require another pressure to become saturated at different temperatures. In other words: at lower temperatures, the same RH requires less evaporation to saturate than at higher temperatures. Or, if looked at a single temperature, the drive to evaporate (to dry the fuel), is given by the difference between the current vapor pressure and the saturated vapor pressure. We call this the vapor pressure deficit (VPD). In formula:
 {VPD}\ {{=}}\ {P_{Sat}}\ -\ {P} (2)
Reworking 1 and 2 gives:
 {VPD}\ {{=}}\ {P_{Sat}}\ -\ {P_{Sat}} \times {RH} 
which can be written as:
 {VPD}\ {{=}}\ {P_{Sat}} \times (1\ -\ {RH}) 
The vapor pressure(s) can be calculated (approximately) by formulas well known in meteorologySee Vapor Pressure , Relative Humidity, Saturation Vapor Density, Vapor Pressure of Water and Clausius-Clapeyron Relation.. 
The Antoine equation:
 {P_{Sat}}\ {{=}}\ 10^{A\ +\ \frac{B}{({C}\ +\ {T})}}
 Where A = 8.07131, B = 1730.63 en C = 233.426 are the coefficients of the Antoine equation as found here. 
 Psat is in mmHg and T in °C, so we need a correction to convert to hPa which is 1,3332239.
Or by the August-Roche-Magnus equation (which is used in the pwsFWI):
 P_{Sat}\ {{=}}\ {A} \times {e}^{\frac{{B}\times{T}} {({C}\ +\ {T})} }
 Where A = 6.112, B= 17.62 and C= 243.12 are the coefficients of the August-Roche-Magnus equation as found in the Guide to Meteorological Instruments and Methods Guide to Meteorological Instruments and Methods of Observation (WMO-No. 8, CIMO Guide), In Chapter 4. Measurement Of Humidity, Annex 4.B. Formulae For The Computation Of Measures Of Humidity.. Psat in hPa and T in °C. The coefficients are slightly different from the ones found in the Wikipedia.
With this estimator for PSat we can work straight on to an FWI for personal weather stations (which from here on I will call pwsFWI or rough pwsFWI). 
 {pwsFWI}\ {{=}}\ {VPD} \times {Windspeed}
The VPD must be seen as the largest driving factor in evaporation and as such in the drying of the fuel(s). For the Meteorological variables in the calculation per day, the daily high or total (T, Wind, Rain) or daily low (RH) values are taken.
== Some reasoning ==
The whole process of evaporation and drying is implicit. I will not try to quantify the waterbalance exactly which would result in a quantification of the Evapotranspiration. Evaporation is an extremely complex process, too much for this wiki page, see some pages on the wikipedia Evaporation, Hydrology (agriculture) and the FAO Irrigation and drainage paper 56., which is far beyond the scope of a fire weather index. But even for professionals it is a complex phenomenon, as it is influenced by vegetation, crop, soil etc… In short, the whole environment. That would be a bit too much although I think in principle it can be done. 
High VPD implies also high evaporation.
High temperatures are a measure of the energy input into the system and has no influence directly on ignition of fire. The VPD reflects this, as temperature is input to the equation: increasing temperature increases the saturation pressure of water and thus reflects increased drying. The energy required for this drying process comes from the sun and thermodynamic calculations could be made. Again, this is beyond the scope of a fire weather index for a personal weather station (though a nice hobby in itself).
Wind, although not a direct factor in igniting fire, contributes very much to the drying process and after ignition, it contributes to the propagation. Strong winds will quickly create dangerous fire weather. The longer the period of dry (and strong) wind and high temperatures, the drier the fuel will be and the easier it will ignite.
So, relative humidity (fraction), wind (in km/h) and temperature (in °C) are the measurements used for the daily calculation as shown above. Furthermore the effect of the duration of dry and (possibly windy) weather needs to be taken into account and that is done through a process of testing and adjusting (a kind of calibration) until the output delivers consistent warning levels comparable with experience and existing fire weather indices.
== Quenching and smoothing the fire weather index ==
The process of calibration resulted in two factors to be taken into account to dampen the heavy fluctuations which occur in practice with the pwsFWI as calculated as above through the rough equations.
#Duration
#Rain amount
When the index suddenly becomes high after a period with low index, it means that the fuel will be wet and needs some time to dry. That needs to be taken into account such that with a constant temperature and wind, the index will reach a stable level (in practice of course it will always vary). 
It must be understood, that processes in nature take time. Drying is not instantaneous and neither is the absorption of water by wood. When rain occurs, the drying process will be stopped and possibly reversed. This also will not be immediate and a single shower will have little effect. Semi-arid locations may have evaporation of 9 mm/day or higher. This means that a shower there needs to be more than 9 mm, to actually quench the drying process (for a day) and a lot more for reversal. So a heuristic quenching process has been implemented as followsThis quantification has been heavily played with/modified during calibration but is stable now:
#Rain less than 1 mm is completely ignored
#Rain between 1 and 5 mm results in a 50% reduction of pwsFWI as calculated (the rough pwsFWI)
#Rain more than 5 mm results in a 67% reduction of the pwsFWI as calculated (the rough pwsFWI)
Then for a period of five days (the so called smoothing days) for each day:
#The average of the last five days of the calculated (the rough) pwsFWI is calculated and if no rain has fallen (or less than 1 mm) that average is the new pwsFWI (or smoothed pwsFWI) as presented to the user
#If on one day an amount of rain between 1 and 5 mm has fallen, that day is skipped so that its value of pwsFWI does not contribute to the average of the five days
#if on one day an amount of more than 5 mm has fallen, that day and the next day are skipped so that their values of pwsFWI do not contribute to the average of the the five days
And finally a minor contribution, a fine tuning, is made to the pwsFWI by adding the length of the dry period in days to pwsFWI because long dry periods take all moisture from deep within thick wood. Thick wood however does not easily ignite by a spark, not even when it is dry.
== Word of thanks ==
H/T for the original testers and contributors of information who worked with me to achieve the final result. 
You can find the listing in the blog : Sites which carry the Fire Weather Index pwsFWI. Most of them still carry the Fire Weather Index page but because updating the software is not so very important for them, they fell of the map which replaced the list and became part of CumulusUtils. Once they upgrade they will automatically be shown on the map again.
== References ==
Category:CumulusUtils