# Description

The $$M68$$ index was developed by Käse (1969) in East Germany based on the Nesterov index in order to predict fire danger in pure pine stands.

The $$M68$$ is used by the German meteorological service (DWD) for fire danger forecast throughout Germany.

The $$M68$$ requires daily air temperature and vapour pressure deficit at 13:00 as input data. This index is cumulative, the index values being daily added from the 15th of February until the 30th of September. Two correction factors are included in spring, late summer and fall: one taking into account vegetation condition and another taking into account rain and snow (Käse 1969).

# Formula

The $$M68$$ index is based on following principle (Käse 1969):

$\displaystyle \sum_{15 Febr}^{30 Sept}(T_{13}+10)\cdot{\Delta}e_{13}$

where $$T_{13}$$ is air temperature [°C], and $$\Delta{e}_{13}$$ vapor pressure deficit [mm of mercury] at 13:00. The index calculation should start on the 15th of Februrar, or after snowmelt, and finish on the 30th of September, or at the end of the fire season. If $$T$$ ≤ -9.9 °C, then the $$M68$$ equals zero.

The value is daily corrected for precipitation, snow cover and finally for vegetation condition. Only values corrected for precipitation and snow cover are used for the calculation of the next day's index.

The precipitation corrected index $$pM68$$ index is calculated as follows (Käse 1969, DWD):

$pM68_t= k_1\cdot pM68_{t-1}+max\Big[0,\ k_2(T_{13_t}+10)\cdot\Delta{e_{13_t}}\Big]$

for $$t$$ between 15 Febr. and 30 Sept., and $$pM68=0$$ for $$t$$ = 14 Febr.

and the final index $$M68$$ (corrected also for vegetation condition) :

$M68_t = k_3 \cdot pM68_t$

with

$k_1= \begin{cases} 1, & \mbox{if }P_t<{1} \\ 0.5, & \mbox{if }1\leqslant{P_t}<{5} \ \ \mbox{ or } {Snowcover}\geqslant{1\ cm} \mbox{ since }t\\ 0.25, & \mbox{if }5\leqslant{P_t}<{10} \ \ \mbox{ or } {Snowcover}\geqslant{1\ cm} \mbox{ since }t-1\\ 0, & \mbox{if }{P_t}\geqslant{10} \ \ \mbox{ or } {Snowcover}\geqslant{1\ cm} \mbox{ since }t-2\\ \end{cases}$

$k_2= \begin{cases} 0, & \mbox{if }P_t\geqslant{20} \ \ \mbox{ or } {Snowcover}\geqslant{1\ cm} \mbox{ since }t-2\\ 0.5, & \mbox{if }P_t<20 \mbox{ and }\big( P_{t-1},\ P_{t-2}\mbox{ or }{P_{t-3}}\big)\geqslant{20} \\ 1, & \mbox{if }P_t,\ P_{t-1},\ P_{t-2}\mbox{ and }{P_{t-3}}<{20} \\ \end{cases}$

$\begin{array}{ccc} k_3= \begin{cases} 3, & \mbox{for } t < t_1 \\ 2, & \mbox{for }t_1 \leqslant t \leqslant t_2 \\ 1, & \mbox{for }t_2 < t < t_3 \\ 0.5, & \mbox{for } t \geqslant t_3 \\ \end{cases} & with & \begin{array}{llll} t_1, \mbox{phenological phase "Birch, first leaves"} \\ t_2, \mbox{first occurrence of rainfall } \geqslant \mbox{ 5mm after the start of the phenological phase "Robinia, first blossom"} \\ t_3, \mbox{first occurrence of rainfall } \geqslant \mbox { 5mm after 14th of August, at the latest 1 September} \end{array} \end{array}$

The $$M68$$ is supposed to be calculated on a daily basis. The meteorological data used for its calculation have to be recorded at 1 pm.

The index calculation has to be started on the 15 of February, or once the snow has melt. No particular starting value is stipulated.

# Index interpretation

The $$M68$$ is interpreted as follows (Käse 1969):

Index values Fire danger class Interpretation
$$M68$$ ≤ 500 0 Probablity for fire occurrence < 3%
500 < $$M68$$ ≤ 2000 1 Probablity for fire occurrence < 20%
2000 < $$M68$$ ≤ 4000 2 Probablity for fire occurrence ≥ 20% and < 40%
4000 < $$M68$$ ≤ 7000 3 Probablity for fire occurrence ≥ 40% and < 60%
$$M68$$ > 7000 4 Probablity for fire occurrence ≥ 60%

# Modifications

The DWD uses a modified formula and the values of vapour pressure deficit in hektoPascals [hPa] instead of mm Hg.

$pM68_t= k_1\cdot pM68_{t-1}+max\Big[0,\ k_2\cdot\dfrac{T_{13_t} + k_4}{10}\cdot\Delta{e_{13_t}}\Big]$

with

$k_4= \begin{cases} 10, & \mbox{if }H_{13_t}\leqslant{26} \\ 20, & \mbox{if }26 < H_{13_t}\leqslant{66} \\ 30, & \mbox{if }H_{13_t}>{66} \\ \end{cases}$

It therefore uses other values for the fire danger classes, also considering wind speed and spatial differences in fire susceptibility (A, B and C):

Index values Wind speed
<=8 m/s >8 m/s
Fire danger class
A B C A B C
≤ 50 - - - - - -
51 - 100 - - - 1 1 -
101 - 150 1 - - 1 1 1
151 - 200 1 1 1 2 1 1
201 - 300 1 1 1 2 2 1
301 - 500 2 2 1 3 3 2
501 - 700 3 2 2 3 3 3
> 700 4 3 3 4 4 4

# References

Original publication:
Käse (1969)

Other publication:
Allen et al. (1998)

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### Symbols

 Variable Description Unit $$T$$ air temperature °C $$T_{dew}$$ dew point temperature °C $$H$$ air humidity % $$P$$ rainfall mm $$U$$ windspeed m/s $$w$$ days since last rain (or rain above threshold) d $$rr$$ days with consecutive rain d $$\Delta t$$ time increment d $$\Delta{e}$$ vapor pressure deficit kPa $$e_s$$ saturation vapor pressure kPa $$e_a$$ actual vapor pressure kPa $$p_{atm}$$ atmospheric pressure kPa $$PET$$ potential evapotranspiration mm/d $$r$$ soil water reserve mm $$r_s$$ surface water reserve mm $$EMC$$ equilibrium moisture content % $$DF$$ drought factor - $$N$$ daylight hours hr $$D$$ weighted 24-hr average moisture condition hr $$\omega$$ sunset hour angle rad $$\delta$$ solar declination rad $$\varphi$$ latitude rad $$Cc$$ cloud cover Okta $$J$$ day of the year (1..365/366) - $$I$$ heat index - $$R_n$$ net radiation MJ⋅m-2⋅d-1 $$R_a$$ daily extraterrestrial radiation MJ⋅m-2⋅d-1 $$R_s$$ solar radiation MJ⋅m-2⋅d-1 $$R_{so}$$ clear-sky solar radiation MJ⋅m-2⋅d-1 $$R_{ns}$$ net shortwave radiation MJ⋅m-2⋅d-1 $$R_{nl}$$ net longwave radiation MJ⋅m-2⋅d-1 $$\lambda$$ latent heat of vaporization MJ/kg $$z$$ elevation m a.s.l. $$d_r$$ inverse relative distance Earth-Sun - $$\alpha$$ albedo or canopy reflection coefficient - $$\Delta$$ slope of the saturation vapor pressure curve kPa/°C $$Cc$$ cloud cover eights $$ROS$$ rate of spread m/h $$RSF$$ rate of spread factor - $$WF$$ wind factor - $$WRF$$ water reserve factor - $$FH$$ false relative humidity - $$FAF$$ fuel availability factor - $$PC$$ phenological coefficient -

 Suffix Description $$-$$ mean / daily value $$_{max}$$ maximum value $$_{min}$$ minimum value $$_{12}$$ value at 12:00 $$_{13}$$ value at 13:00 $$_{15}$$ value at 15:00 $$_{m}$$ montly value $$_{y}$$ yearly value $$_{f/a}$$ value at fuel-atmosphere interface $$_{dur}$$ duration $$_{soil}$$ value at soil level

 Constant Description $$e$$ Euler's number $$\gamma$$ psychrometric constant $$G_{SC}$$ solar constant $$\sigma$$ Stefan-Bolzmann constant