=Paper= {{Paper |id=Vol-1638/Paper64 |storemode=property |title=Assessing hazard probability factors related to forecasted weather conditions |pdfUrl=https://ceur-ws.org/Vol-1638/Paper64.pdf |volume=Vol-1638 |authors=Valentina G. Burmistrova,Alexandr A. Butov,Alexandr V. Zharkov,Yuliya V. Pchelkina }} ==Assessing hazard probability factors related to forecasted weather conditions == https://ceur-ws.org/Vol-1638/Paper64.pdf 
Mathematical Modeling


    ASSESSING HAZARD PROBABILITY FACTORS
       RELATED TO FORECASTED WEATHER
                 CONDITIONS

          V.G. Burmistrova1, A.A Butov1, A.V. Zharkov1, Yu.V. Pchelkina2
                      1
                        Ulyanovsk State University, Ulyanovsk, Russia
                   2
                    Samara National Research University, Samara, Russia

       Abstract. In this paper we have considered two meteorological factors as an
       object of our research: horizontal and vertical visibility. These meteorological
       factors are associated with "danger thresholds", going beyond these levels (in
       excess or decrease) is unacceptable or undesirable. The initial values of the hor-
       izontal and vertical visibility are assumed to be equal to the values presented in
       the weather forecast. The results of this development can be used to solve prob-
       lems related to aviation forecasting, for example, when making a decision re-
       garding a departure or landing of the aircraft.


       Keywords: meteorological conditions, assessment of probability, modeling,
       forecast.


       Citation: Burmistrova VG, Butov AA, Zharkov AV, Pchelkina YuV. As-
       sessing hazard probability factors related to forecasted weather conditions.
       CEUR Workshop Proceedings, 2016; 1638: 521-526. DOI: 10.18287/1613-
       0073-2016-1638-521-526


Introduction

In this paper we consider the weather report, which is transmitted in the form of re-
ports METAR (Meteorological Aerodrome Report β€” aeronautical meteorological
code for transmitting reports on the actual weather at the airport) or TAF (the weather
at a certain time in the future (normally from several hours to days).
The forecast TAF can be represented in the form of one or more sections, each of
which begins with one of the code words:
NOSIG (No significant change) – this means that there are no major weather changes
in the next 2 hours;
BECMG (Becoming) - sustained significant changes of weather conditions are ex-
pected;
TEMPO (Temporary) - temporary significant changes of weather conditions are ex-
pected.
The values of the studied meteorological factors are modeled based on the justifica-
tion of the forecast (forecast errors and probability). We took into account not only



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the bulk of the weather forecast but also a group of stable and temporary changes in
the forecast. The main objective of this work is to determine the probability of the
value of meteorological factors to go beyond the "hazard threshold".
The solution to this problem has been found under the assumption that specific im-
plementation of meteorological factors at the time of landing (or take off) is a contin-
uous random variable with values on a usual quantitative scale, and it has a normal
distribution and that the meteorological factors are independent variables.
We will consider X(t) to be some meteorological factor provided in the actual weather
report (METAR) or airport weather forecast (TAF). It is a continuous one-
dimensional value (a mark on a quantitative scale) by which we will consider parame-
ters of meteorological factors: horizontal visibility (in meters) at approaching of the
aircraft or the height of the lower threshold of the cloud (in meters) at the terminal
aerodrome [1].
The value of meteorological factor X(t) can be provided in the main part of the fore-
cast (defined as π‘‹π‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ ), and in the group of persistent changes (they are defined as
BECMG, FM), or in the group of temporal changes (defined as TEMPO), sometimes
indicating the probability of possible changes in 40% (PROB40 ) or 30% (PROB30).
The value of meteorological factors not included into the main part of the weather
forecast will be defined as π‘‹π‘‘π‘’π‘šπ‘π‘œ .
"Maximum hazard threshold" is associated with these meteorological factors - the
value of going beyond it (the excess or decrease) is unacceptable or undesirable, the
value of this level of "maximum danger level" is defined by [2]-[6].
The basic assumption of this article is the following: the definite implementation of
meteorological factor X (t) at the time of landing (or take off) is a continuous random
variable with values in the usual quantitative scale.
Our main problem is to find probability of the event that the value X(t) value will
overcome π‘‹π‘‘π‘Žπ‘›π‘”π‘’π‘Ÿ , defined as

𝑃 = 𝑃{𝑋(𝑑) > π‘‹π‘‘π‘Žπ‘›π‘”π‘’π‘Ÿ } or 𝑃 = 𝑃{𝑋(𝑑) < π‘‹π‘‘π‘Žπ‘›π‘”π‘’π‘Ÿ }.

We define 𝑃 as the probability of the weather conditions exceeding the safety thresh-
old.
We will assume that:
─ distribution of possible 𝑋(𝑑) implementations are Gaussian (normal) with parame-
  ters π‘Ž and 𝜎, [7] - [8];
─ the value π‘‹π‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ is the value of the parameter π‘Ž (i.e., the mean value, and mode
  (the highest value) of the distribution density).
─ the value 𝜎 is derived from the conditions of accuracy and statistical validity of the
  forecast implementation.


Calculating Estimated Probability of Weather Conditions

Let us define 𝐿(𝑑) as the horizontal visibility (which can be in a real situation) and
will be a random variable. The variable πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ is defined in the TAF forecast sec-


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tion. The variable of the horizontal visibility 𝐿(𝑑) ∈ Ν(πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ ; 𝜎) has a Gaussian
(normal) distribution (with mathematical expectation πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ and variance 𝜎 2 ). The
variance of 𝐿(𝑑) is determined by the forecast provision and accuracy.
According to [9], at forecasted visibility of more than 800 meters, the error is equal to
30% of the forecast and 80% of provision. In this case the following is true:

𝑃{|𝐿(𝑑) βˆ’ πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ | < 0.3 βˆ™ πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ } = 0.8                                         (1)

Hence we can derive 𝜎 = 0.23 βˆ™ πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ .
We obtain the probability of weather conditions exceeding the safety threshold in this
situation (forecast visibility of more than 800 m):

                                    πΏπ‘‘π‘Žπ‘›π‘”π‘’π‘Ÿ
𝑃{𝐿(𝑑) < πΏπ‘‘π‘Žπ‘›π‘”π‘’π‘Ÿ } = Ξ¦ (4.27 βˆ™ (                βˆ’ 1))                                   (2)
                                    πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘


where Π€(βˆ—) is the function of the standard Gaussian distribution. Geometric illustra-
tion of the formula (2) is given in Figure 1.




            Fig. 1. Determining visibility probability is below the minimum value

If the value of the forecast visibility is less than 800 meters, then the error is equal to
Β±200 π‘š and taking into account the reliability of 80% of the forecast, the following
is true:

𝑃 = 𝑃{πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ βˆ’ 200 < 𝐿(𝑑) < πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ + 200} = 0.8                                   (3)

Herewith we will obtain 𝜎 = 156 π‘š.
Then the probability of a "dangerous" weather situation is equal to:




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                                   πΏπ‘‘π‘Žπ‘›π‘”π‘’π‘Ÿ
𝑃{𝐿(𝑑) < πΏπ‘‘π‘Žπ‘›π‘”π‘’π‘Ÿ } = Ξ¦ (4.27 βˆ™ (               βˆ’ 1))                                (4)
                                   πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘


Let us define the observable value of the cloud conditions 𝐻(𝑑) , 𝐻(𝑑) ∈
𝑁(π»π‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ ; 𝜎) where π»π‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ is the value of cloud conditions provided by a fore-
cast. Forecasted clouds height means that the error is ο‚±30% of the forecast if the
forecast value lies from 300 π‘š to 3 π‘˜π‘š, and the error is ο‚±30 m if the forecast value
lies below 300 π‘š.
We will suppose the forecast is below 300 π‘š. Given the forecast accuracy of 70%,
we will claim:

𝑃 = 𝑃{π»π‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ βˆ’ 30 < 𝐻(𝑑) < π»π‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ + 30} = 0.7                                 (5)

Herewith we will obtain 𝜎 = 29 π‘š.
Then the probability of a "dangerous" weather situation is equals:
                           π»π‘‘π‘Žπ‘›π‘”π‘’π‘Ÿ βˆ’π»π‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘
𝑃{𝐻(𝑑) < π»π‘‘π‘Žπ‘›π‘”π‘’π‘Ÿ } = Ξ¦ (                        )                                   (6)
                                   𝜎

If the value π»π‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ is more than 300 π‘š, then π»π‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ is 30% for accuracy and
70% (according to [2]) and the following is true:

𝑃 = 𝑃{|𝐻(𝑑) βˆ’ π»π‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ | < 0.3 βˆ™ π»π‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ } = 0.7                                 (7)

Herewith we will obtain 𝜎 = 0.29 βˆ™ π»π‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ .
The probability of a "dangerous" weather situation equals:

                                    π»π‘‘π‘Žπ‘›π‘”π‘’π‘Ÿ
𝑃{𝐻(𝑑) < π»π‘‘π‘Žπ‘›π‘”π‘’π‘Ÿ } = Ξ¦ (3.46 βˆ™ (               βˆ’ 1))                                (8)
                                   π»π‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘


The horizontal and vertical visibilities are considered when determining the assess-
ment of weather conditions implementations beyond the safety threshold. Let the
probability of exceeding "maximum level of hazard" of the studied weather factors be
defined as 𝑃1 and 𝑃2 . Herein considered values are independent variables (as the ac-
curacy and correctness of the forecast of each value are considered) although they are
weather dependent. We will define the probability of the event as:
𝑃(𝑑) = 1 βˆ’ (1 βˆ’ 𝑃1 )(1 βˆ’ 𝑃2 )                                                       (9)

Taking into account weather conditions changes, TAF forecast may be an indicator of
possible changes (BECMG, FM or TEMPO). If changes are not forecasted, then
NOSIG indicator may emerge. In this event in TEMPO, the value πΏπ‘‘π‘’π‘šπ‘π‘œ should be
obtained from the weather forecast.
The emergence of the event from TEMPO is used with the indication within period
from 𝑇1 to 𝑇2 that is the mixture of two weather conditions at that time and in the
place: one with characteristics of the main forecast, the other from the forecast of
changes. The probability of encountering TEMPO weather at the time of aircraft
arrival equals:


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       0.5, 𝑖𝑓𝑠𝑒𝑑 π‘‘π‘œ 𝑇𝐸𝑀𝑃𝑂 π‘Žπ‘›π‘‘ π‘›π‘œπ‘‘ 𝑠𝑒𝑑 π‘‘π‘œ 𝑃𝑅𝑃𝑂𝐡,
𝑝0 = { 0.4, 𝑖𝑓𝑠𝑒𝑑 π‘‘π‘œ 𝑇𝐸𝑀𝑃𝑂 π‘Žπ‘›π‘‘ 𝑠𝑒𝑑 π‘‘π‘œ 𝑃𝑅𝑃𝑂𝐡40,                                                (10)
      0.3, 𝑖𝑓𝑠𝑒𝑑 π‘‘π‘œ 𝑇𝐸𝑀𝑃𝑂 π‘Žπ‘›π‘‘ 𝑠𝑒𝑑 π‘‘π‘œ 𝑃𝑅𝑃𝑂𝐡30.

Accordingly, the probability of finding the weather conditions forecast indicated in
the main forecast equals (1 βˆ’ 𝑝0 )
The BECMG forecast replaces the main forecast from time 𝑇2 when there is an im-
provement of weather conditions and from time 𝑇1 if there is deterioration in the
weather conditions with the forecast reliability of 80%.
For example, let the visibility distance be πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ (then the real visibility is a ran-
dom variable Gaussian distributed with parameters π‘Ž = πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ and 𝜎1 = πœ† βˆ™
πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ ), whereas the visibility distance is πΏπ‘‘π‘’π‘šπ‘π‘œ in TEMPO forecast (i.e. the new
forecasted visibility is a random value distributed with parameters π‘Ž = πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘
and𝜎1 = πœ† βˆ™ πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ ).
Then the probability of a "dangerous" weather situation is equal to:
𝑃 = 𝑃{𝐿(𝑑) < πΏπ‘‘π‘Žπ‘›π‘”π‘’π‘Ÿ } =

           1     πΏπ‘‘π‘Žπ‘›π‘”π‘’π‘Ÿ                          1       πΏπ‘‘π‘Žπ‘›π‘”π‘’π‘Ÿ
= 𝑝0 βˆ™ Ξ¦ ( βˆ™ (             βˆ’ 1)) + (1 βˆ’ p0 )Ξ¦ ( βˆ™ (                   βˆ’ 1))                   (11)
           πœ†     πΏπ‘‘π‘’π‘šπ‘π‘œ                           πœ†       πΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘


Depending on the valuesπΏπ‘“π‘œπ‘Ÿπ‘’π‘π‘Žπ‘ π‘‘ , πΏπ‘‘π‘’π‘šπ‘π‘œ we will select pre-adapted formulas (2) or
(4).
For the vertical visibility weather conditions 𝐻(𝑑) (i.e. the clouds height) formulas
are similar:

𝑃 = 𝑃{𝐻(𝑑) < π»π‘‘π‘Žπ‘›π‘”π‘’π‘Ÿ } =


            𝟏    π‘―π’…π’‚π’π’ˆπ’†π’“                              𝟏    π‘―π’…π’‚π’π’ˆπ’†π’“
= π’‘πŸŽ βˆ™ 𝚽 ( βˆ™ (             βˆ’ 𝟏)) + (𝟏 βˆ’ 𝐩𝟎 )𝚽 ( βˆ™ (                    βˆ’ 𝟏))       (12)
            𝝀    π‘―π’•π’†π’Žπ’‘π’                               𝝀    𝑯𝒇𝒐𝒓𝒆𝒄𝒂𝒔𝒕




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