Abstract

Today, meteorology is one of the most significant factors that affects   various aspects of life. The spheres of influence of weather and climate include agriculture, aviation, construction, television, radio, etc. Erroneous forecasts can lead to negative consequences. To infer, meteorology is a very relevant tool in controlling different processes in our everyday life.

The purpose of this study is to correct the results of calculations of known mathematical models that are used in weather forecasting. When evaluating the errors of the formulas, a hypothesis was formulated that the data deviates from the actual values according to a certain algorithm. Based on this hypothesis, formulas were created for more successful weather forecasting.

Initially, a purely analytical approach was used. For the research, the best models on the market were selected. The city of Atyrau was taken as an experiment. The results of the actual readings were compared with those of the models.

The second stage involved working with the Pearson coefficient. The relationship of the results was assessed.

In the final phase, improvements were proposed to the existing European weather forecast model. The weekly analysis showed significant progress.

The scientific value of this work lies in the possibility of improving existing numerical weather prediction models. The results of this project improve:

• vital activity of sensitive people;

• crop sowing planning;

• the likelihood of avoiding ice for car drivers;

• flight schedule;

• planning of major events, depending on the weather;

Introduction

Today, the importance of proper weather forecasting is undeniable. There is not a day when it is left without the attention of all people of different professions and different types of employment. Erroneous predictions cannot be ignored by the public. They lead to adverse effects ranging from declining crop yields due to land depletion and erosion to changing market demands. Proceeding from this, I came to the opinion that the weather measurement should be as close as possible to the real results. Mathematical formulas of weather forecasting should be improved in order to minimalize the negative outcomes of it.

Numerical weather forecasting requires huge computational calculations that only advanced supercomputers are capable of. With the development of atmospheric models and the progress of computer technology, it becomes more and more obvious that with an improvement in the parameterization of subgrid-scale processes and an increase in the spatial resolution, the errors of the model climate decrease [10, 11].

Nowadays, developed countries are developing their own weather forecast models. Each model has its own disadvantages and advantages. This paper will focus on the two most common models: GFS and ECMWF.

GFS - The Global Forecast System (GFS) produced by the National Centers for Environmental Prediction (NCEP).

ECMWF - The European Centre for Medium-Range Weather Forecasts (ECMWF) is an independent intergovernmental organization supported by most of the nations of Europe and is based at Shinfield Park, Reading, United Kingdom. It operates one of the largest supercomputer complexes in Europe and the world's largest archive of numerical weather prediction data.

The goals to be achieved in this work:

• analysis of two existing weather forecast models;

• collection of data on weather parameters in the city of Atyrau;

• comparison of model readings with actual data;

• calculation of the correlation coefficient between model values ​​and actual values

• improvement of the existing model

To improve the model, based on observations, a hypothesis will be proposed. There are a huge number of different parameters in weather forecasting, such as atmospheric pressure, humidity, solar radiation and others. In this paper, we will consider the 3 most important parameters in meteorology: temperature, wind speed and cloud cover.

Research part

Chapter 1. Review of the results of known research

In 1922, the British mathematician Lewis Fry Richardson first proposed the use of numerical methods for weather forecasting. He also tried to forecast the weather using these methods, but failed. Richardson's numerical weather forecast for just 6 hours was not just bad - a fantastic storm was predicted, and the real weather remained quite normal. The cause of the error was found several years later.

Richardson's unsuccessful first assault on weather forecasting was extremely productive. The following researchers took into account his mistakes, changed the model and computational scheme, and used computers. [1]

At the moment, there are a large number of models for weather forecasting. This research work will focus on two main models: GFS and ECMWF.

The Global Forecasting System (GFS) is a weather forecasting model developed by the National Centers for Environmental Forecasting (NCEP). Dozens of atmospheric and soil variables are available through this dataset, from temperature, wind and precipitation to soil moisture and atmospheric ozone concentration. The entire globe is covered by GFS with a base horizontal resolution of 18 miles (28 kilometers) between grid points, which is used by operational forecasters who predict weather up to 16 days in the future. Horizontal resolution drops to 44 miles (70 kilometers) between grid points for forecasts from one week to two weeks. It is a constantly evolving and improving weather model. Until January 2003, the GFS was known as the GFS Aviation Model (AVN) and the GFS Medium Range Forecast Model (MRF). [2]

GFS uses the spectral method for its calculations. The Global Spectral Model (GSM) uses primitive equations to express the movements of the layers of the atmosphere. It also includes sophisticated parameterization schemes for physical processes. [3]

The European Center for Medium-Range Weather Forecasts (ECMWF) is an independent intergovernmental organization supported by most of Europe and based in Shinfield Park, Reading, United Kingdom. He operates one of the largest supercomputer complexes in Europe and the world's largest archive of numerical weather forecast data. [4]

Chapter 2. Methods for solving the set goal

The purpose of this work is: correction of the results of calculations of known mathematical models that are used in weather forecasting. To achieve the goal, we will analyze the data of the GFS and ECWMF models for 7 days in the city of Atyrau. [5]; [7] Consider the three most important parameters: temperature, cloudiness, wind speed.

Let's take the time of calculation of parameters local time 14:00. Temperature is measured in degrees, wind speed is in km / h. For the probability of cloudy weather, we will use values ​​from 0 to 1, where "0" means sunny weather without clouds. Actual results were obtained from hourly readings from the Weather channel mobile app and daily measurements.

Forecast models
July 5
 July 6
 July 7
July 8
July 9
July 10
July 11
GFS model
+37
+38
+39
+38
+37
+36
+37
ECWMF model
+38
+39
+40
+39
+38
+38
+38
Actual results
+37
+39
+40
+40
+39
+38
+38
Table 1. Statistical data on temperature in Atyrau , July 2020

Forecast models
July 5
July 6
July 7
July 8
July 9
July 10
July 11
GFS model
0
0.25
0.75
0
0.25
1
0
ECWMF model
0
0.25
0.25
0.25
0.75
1
0.5
Actual results
0.5
1
0.25
0.25
0.25
0.5
0.5
Table 2 . Statistical data on cloudiness in Atyrau , July 2020

Forecast models
July 5
July 6
July 7
July 8
July 9
July 10
July 11
GFS model
16,668
12,964
11,112
11,112
14,816
20,372
11,112
ECWMF model
15
15
17
18
18
17
17
Actual results
10
18
17
20
20
21
16
Table 3 . Statistics on wind speed in Atyrau , July 2020

As we can see, there are slight errors in predictions made by forecast models.To estimate the error in the values ​​of all three parameters of each model, we will build graphs and calculate the Pearson correlation coefficient. [ 8 ]

First, consider the graphs of the parameters of the GFS model :

1.               Temperature

The correlation coefficient is denoted through  , the actual temperature through , and the temperature according to the GFS model through y, then:

-        The correlation is average, which indicates a moderate strength of connections.

2.    Cloudiness

The correlation coefficient is denoted through , the actual cloudiness through x, and the cloudiness according to the GFS model through y, and the average values through , , then:

-        Such a correlation suggests that the quantities are weakly related to each other.

3.    Wind speed

The correlation coefficient is denoted through , the actual temperature through x, and the temperature according to the GFS model through y, and the average values through , , then:

-        Correlation is very weak, values ​​are not related.

Now let's look at the ECMWF model:

1.    Temperature

The correlation coefficient is denoted through , the actual temperature through x, and the temperature according to the ECMWF model through y, then:

- The correlation is good enough. This model often shows the temperature correctly.

2.    Cloudiness

-        The correlation coefficient is denoted by , the actual Cloudy through x,and cloudiness of the model ECMWF through y, and average values over , , then:

-        There is no correlation

3.    Wind speed

The correlation coefficient is denoted by , the actual wind speed over x, and the wind speed of the model ECMWF through y, and average values over ,  , then :

Chapter 3. Results of work and their discussion

In the course of studying two models, a clear accuracy of the ECMWF model was revealed, since the correlation coefficient with the actual data was higher than that of the GFS. Since , which infares   a full correctness of the weather forecast, while it is not possible. There are variety of reasons for this phenomenon, due to the fact that weather might be affected by any process on the earth. For instance, the destruction of one construction building affects the wind speed, and this, in turn, affects the cloudiness of that region. As a consequence, clouds coating affects the probability of rain, and on change in temperature. There are other sources of miscalculation and many of them are due to the human factor (minor human influence). For this reason, the weather can not be predicted with 100% certainty . Our task is to bring the results as close as possible to reality.

Using the results, obtained through model ECMWF's, improve it by proposing a hypothesis and testing it with the actual results. To improved ECMWF model we gave a new name: meteo+.

On the model of the temperature of the ECMWF was deflected to the positive side in those days , when the difference between the cloud was positive, and vice versa. Therefore, to calculate the temperature according to the new model, we take a direct proportionality with the coefficient α = 1 h ∙ ° C / km . Such coefficients need to be selected in different ways, depending on the climate of the city.

The formula for measuring temperature according to the Meteo + model :

u 0 - wind speed

u av - average wind speed

α - coefficient

λ 0 - cloudiness

T 0 - temperature according to the ECMWF model

Calculating by the model, we get the following data

Most of the results came out roughly the same as the actual data . The only apparent deviation came out on Tuesday. Meteo + temperature deviated by 1.5 degrees.

The formula for measuring cloudiness according to the Meteo + model :

= 0, 2 h / km ∙ ° C

The new model showed improved results on Monday, Tuesday, Friday. The worst results was demonstrated on Thursday. On other days, the results are identical.

The formula for measuring wind speed according to the Meteo + model :

µ = 2.8169 km / h ∙ ° C

The results improved in about all 5 days, Saturday was unchanged and on Wednesday the result was slightly worse.

Review of the head

The chosen theme for further investigation, namely "Mathematical formulas in meteorology" is very relevant nowadays. With the improvement of the computing characteristics of computers, it is possible to calculate weather forecast with higher probability of accuracy under parameters such as temperature, cloudiness and wind speed. However, it is still impossible to predict the weather with 100% certainty.

The work done by Aigerim improves the accuracy of the calculations. It is recommended to carry out calculations a few more times, according to your author's model Meteo +. I think we should expand the number of parameters in weather forecasting, such as: probability of rain, humidity, hurricane, cyclone, and so on. The proposed hypothesis can play an important role in achieving this goal.

Head:                                                                                     Aigaraev B.K mathematics teacher

Conclusion

This study fulfilled the set goal of the work, which is the search for the best weather forecast model, its improvement and addition of mathematical formulas.

Initially, there was a brief overview of existing models for weather forecasting. As a result of lengthy comparisons, the most accurate model was chosen, namely ECMWF. This model had its miscalculations in predicting parameters such as cloudiness and wind speed. Ultimately, three formulas were proposed in order to correct the values ​​of the same model. On the basis of new and improved formulas, it is possible to compile the most competent and accurate weather forecast, which in the future can be applied in various fields and industries. The more successful model was named Meteo +.

This project can be used:

meteorologists for weather forecast

when planning events by different organizations

to improve the lives of metodependent people ;

when planning the sowing date of the crop;

vehicle drivers ;

when creating a schedule s air travel;

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