34 Weather Forecasting

Somenath Dutta

1. Learning Outcomes
2. Introduction
3. Weather Forecasting- an Initial and Boundary Value Problem
4. Ranges of Forecasting
5. Forecasting Methods
6. Forecast Verification
6.1. Why Forecast Verifications?
6.2. What makes a Forecast “good?”
6.3. Types of Forecast and Verifications
6.3.1. Verification Skill Scores for Probability of Precipitation (Pop)
6.3.2. Verification Skill Scores for Yes/No Rainfall Forecast
6.3.3. Verification Skill Scores for Quantitative Precipitation Forecast
6.3.4. Verification Skill Scores for Temperature Forecasts
7. Importance of weather forecasting in different aspects of life
8. Summary

1. Learning outcomes

After studying this module, you shall be able to:

• Define weather at a place
• Understand the basic concept of initial and boundary value problems and its use in weather forecasting
• Know the different ranges of weather forecasts vis-a-vis its applicability Understand the simple to complex types of weather forecasting methods Understand forecast verification techniques
• Appreciate the importance of weather forecasting in different aspects of life

2. Introduction

Weather at a place is defined as the state of the atmosphere, prevailed at that place, over a short period of time. It is generally expressed as like “it’s too hot now”, “it’s very windy today”, “it’s very dry now”, “today is a rainy day” etc. In these examples, weather at a place has been expressed in terms of the state of certain atmospheric variables which are here air temperature, wind, humidity and rainfall, respectively.

The weather of any given region has a considerable impact on the water, sunlight and temperature of an ecosystem. These factors play very important roles by influencing the types of plant and animal wildlife that can survive in the area. Certain weather patterns can also cause dangerous storms and natural disasters. Hence, weather information helps farmers to plan when to sow or harvest their crops, helps pilots to know when to take off or land, helps sailors at sea to timetable their journeys, helps people to plan what dress to put on for the day (e.g., they will know whether or not to put on a sweater or a jacket and whether or not to carry an umbrella) and helps the government to prepare for disasters like floods, drought, very strong winds among others.

The main goal of a Meteorological office is to issue weather forecasts in different time scales to various user agencies like aviation, marine, agriculture, water management, builders, tourism industry, planners and to general public. The forecast requirement varies from detailed weather forecasts in time scales of a few hours to days and to more general indication of the broad weather pattern of succeeding months, seasons or even beyond. For example, aviation industry needs weather information in time scales of a few hours or a day whereas agriculture sector demands weather  forecasts in time scales of a week. Weather forecasts of a month or a season in advance are required mostly by planners.

3. Weather Forecasting – an Initial and Boundary Value Problem

To address the time evolution of some field, we approach through initial value problems, which are special type of differential equations as given below:

Let us consider an arbitrary field variable , which varies spatially as well as temporarily. So, it is a function of spatial variable, say ‘x’ and of time variable ‘t’ also. We can write it mathematically as . Now, suppose, present value of this field at a spatial location ‘x’ is known and we wish to find out its value at some future time at this location. For that we should have the rule telling how this field variable varies with time. This rule is given in terms of following differential equation. , given, ; here x is the spatial variable, t is time variable,  is a known function and  is the unknown function, to be determined at any arbitrary time t > 0. General solution of this equation will involve an arbitrary constant of integration, which can be uniquely found out using the information [] about the present value of the field and hence we shall get the particular solution of the field corresponding to the given information about its present value. Physically this problem may be interpreted as to attempt to tell about the future values of the function   at any spatial points ‘x’, given it’s present value at all points.

In mathematical literature, a boundary value problem is a differential equation with a set of given conditions/constraints,   for   example,   ,    with . Here, x,y both are spatial variables, may be distance measured along West-East and south-north direction;  is a known function and  is an unknown function which is to be determined at any internal point. Physically this problem may be interpreted as to attempt to tell about the values of the function  at any internal point of a domain knowing it’s value at all points on the boundary of the domain. Thus any physical problem, which involves finding value of a variable at future time, given complete knowledge of its present value may be framed as an initial value problem.Weather forecasting is to estimate the future state of the atmosphere from the complete knowledge about its present state. At the same time, forecasting weather at over a domain requires knowledge about the value of weather parameters at the boundary of the domain. It is known that weather ‘WX’ at a place, with position  , at time, is a function of weather parameters at that place at that time. If the value of basic meteorological parameters such as pressure, wind, temperature, humidity, precipitation, sunshine and cloud cover at a place () at a time ′t′ are known, then by establishing a suitable functional relation, weather at the place () at time ′t′ can be determined. Thus the problem of forecasting weather has been translated to the problem of forecasting above meteorological parameters. Now question is whether it is possible to predict these parameters at a place for future. The answer to this question lies on the fact, whether these variables satisfy following type equation or not: to solve , given, . Accordingly, weather forecasting can be recognized as an initial value as well as a boundary value problem.

4. Ranges of Forecasting

Following WMO, depending on the validity period of a forecast, meteorological forecasting ranges are given below:

5. Forecasting Methods

Weather forecasting basically consists of two steps. The first step is to have an accurate assessment of the present/initial state of the atmosphere. This helps in identifying the different weather systems and their horizontal and vertical state. As we know, there are a variety of phenomena  occurring in the atmosphere having different space and time scales. The characteristic sizes of these motions vary from a fraction to centimeter to several thousands of kilometers, with time scales of a fraction of a section to several weeks. Each of the various scales of motions has a varying degree of influence upon all the others and it is important to properly observe, analyze and account them in atmospheric studies and weather forecasting. As weather has no political boundary, one needs weather data from a fairly large region; the area from which data required increases with the duration of forecast made, since weather systems from one part of a region may travel and affect the weather condition over a far off region in course of time. The second step in weather forecasting is to utilise a suitable technique to predict the future state of the atmosphere.

Primarily, there are three methods, viz., (i) synoptic method, (ii) statistical method and (iii) numerical weather prediction (NWP) or dynamical method. These methods are discussed below:

In the synoptic method, a forecaster attempts to predict the future changes in the state of atmosphere from its initial state using his theoretical knowledge and experience. Here, various weather charts, commonly known as synoptic charts are analyzed at a fixed time to understand the three dimensions of the atmospheric state. This method can generate forecast for a broad period over a broad region but cannot generate time and location specific objective forecast. That way this method is subjective and the success of the forecast depends heavily on the knowledge, experience and skill of the forecaster. The synoptic method is widely used in short range weather forecasting, especially in tropics.

The statistical technique is based on correlation and regression analysis. By analyzing the past weather records for a long period, useful relationship can be established relating the occurrence of one weather event with another or a number of other weather elements. For example, ‘All-India Rainfall’ behaviour of southwest monsoon is related with El-Nino, Southern Oscillation, Winter Snow Cover over Himalayas, and surface Pressure over Northwest Australia etc. This technique is mostly used in long range forecasting.

The third method of weather forecasting which is more promising and has become more popular in recent decades is the NWP. In this method, predictions are made by solving the hydro-dynamical equations which govern the atmospheric motions using powerful computers. The method being objective, has gained tremendous boost with the progress in technology – observational, telecommunication and computer, and understanding of various atmospheric processes. While the synoptic method has inherent difficulties to provide realistic weather forecasts beyond 2-3 days, the NWP technique can be expected to provide objective forecasts for much longer time periods.

6. Forecast Verification

Verification is the process of comparing forecasts to relevant observations. Verification is one aspect of measuring forecast goodness. It measures the quality of forecasts (as opposed to their value).

6.1. Why Forecast Verification?

A forecast is like an experiment – given a set of initial conditions, a hypothesis is made that certain weather shall prevail after a certain time at a given place. One wouldn’t consider an experiment to be complete until its outcome is known. In the same way, one can’t consider a forecast experiment to be complete until it is found out whether the forecast was successful or not. The three most important reasons to verify forecasts are:

To monitor forecast quality – how accurate are the forecasts and are they improving over time?

To improve forecast quality -the first step toward getting better is discovering what wrong you are doing.

To compare the quality of different forecast systems – to what extent does one forecast system give better forecasts than another and in what ways is that system better?

6.2. What makes a Forecast “good”?

Allan Murphy, a pioneer in the field of forecast verification, wrote an essay on what makes a forecast “good” (Murphy, 1993). He distinguished three types of “goodness”:

Consistency – the degree to which the forecast corresponds to the forecaster’s best judgement about the situation, based upon his/her knowledge base

Quality – the degree to which the forecast corresponds to what actually happened

Value – the degree to which the forecast helps a decision maker to realize some incremental economic and/or other benefit

Since we are interested in forecast verification, let’s look a bit closer at the forecast quality. Murphy described nine aspects (called “attributes”) that contribute to the quality of a forecast. These are:

Bias – It is the correspondence between the mean forecast and mean observation.

Association – It is the strength of linear relationship between the forecasts and observations (for example, the correlation coefficient measures this linear relationship).

Accuracy – It is the level of agreement between the forecast and the truth (as represented by observations). The difference between the forecast and the observation is the error. The lower the error, the greater is the accuracy.

Skill – It indicates the relative accuracy of the forecast over some reference forecast. The reference forecast is generally an unskilled forecast such as random chance, persistence (defined as the most recent set of observations, “persistence” implies no change in condition) or climatology. Skill refers to the increase in accuracy due purely to the “smarts” of the forecast system. Weather forecasts may be more accurate simply because the weather is easier to forecast – skill takes this into account.

Reliability – It is the average agreement between forecast values and observed values. If all forecasts are considered together then the overall reliability is the same as the bias. If the forecasts are stratified into different ranges or categories, then the reliability is the same as the conditional bias, i.e., it has a different value for each category.

Resolution – It is the ability of the forecast to sort or resolve the set of events into subsets with different frequency distributions. This means that the distribution of outcomes when “A” was forecast is different from the distribution of outcomes when “B” is forecast. Even if the forecasts are wrong, the forecast system has resolution if it can successfully separate one type of outcome from another.

Sharpness – It is the tendency of the forecast to predict extreme values. To use a counter-example, a forecast of “climatology” has no sharpness. Sharpness is a property of the forecast only, and like resolution, a forecast can have this attribute even if it’s wrong (in this case it would have poor reliability).

Discrimination – It is the ability of the forecast to discriminate among observations, that is, to have a higher prediction frequency for an outcome whenever that outcome occurs.

Uncertainty – It is related to the variability of the observations. The greater the uncertainty, the more difficult the forecast will tend to be.

Traditionally, forecast verification has emphasized accuracy and skill. It’s important to note that other attributes of forecast performance also have a strong influence on the value of the forecast.

6.3.   Types of forecast and verification

6.3.1. Verification Skill Scores for Probability of Precipitation (Pop)

Brier  Skill  score:  To  understand  Brier  Skill  Score,  let  us  consider  the  forecast  and  observed probabilities  of  precipitation  for  ‘n’  events.  If    and    are  respectively  forecast  and  observed probabilities of precipitation for ith precipitation event, then, Brier Score is defined as, . It ranges from 0 to 1. Brier Skill score is defined as,

6.3.2. Verification Skill Scores for Yes/No Rainfall Forecast

This type of observation or forecast of an event in Yes/No type is known as Dichotomous. Verification of forecast of the occurrence of such events uses 2X2 contingency table as given below:

The possible cases include following:

A – event forecast and event occurred. This is known as hits.

B – event forecast to occur, but didn’t occur, known as false alarm.

C – event couldn’t been forecast to occur, but occurred. This is known as misses.

D – event didn’t occur and also didn’t forecast to occur. This is known as correct negative.

The total number of cases (N) is given by; N = A + B + C + D. Then different skill scores for yes/no forecast are defined as;

(a) Ratio Score (RS) = [(Hits + Correct Negatives)/Total] x 100 = [(A+D) / (A + B + C + D)] X 100

(b) Probability of Detection (POD) = Hits / (Hits + Misses) = A / (A + C)

(c)  False Alarm Rate (FAR) = False Alarms / (Hits + False Alarms) = B / (A + B)

(d) BIAS = (Hits + False Alarms) / (Hits + Misses) = (A + B) / (A + C)

It indicates whether the forecast system has a tendency to under forecast (BIAS < 1) or over forecast (BIAS    > 1)

(e) True skill score (TSS) is defined as;

= [A/(A + C)] + [D/(B + D)] – 1

= [A/(A + C)] – [B/(B + D)]

 (f) Critical Success Index (CSI) or Threat Index is defined as; CSI = Hits / (Hits + False Alarms + Misses) = A / (A + B + C)

The perfect score is 1 and 0 means no skill.

(g) Hanssen and Kuipers Score (HKS) is defined as; HKS = [{Correct forecast – (Correct forecast)random} /

{N – (Correct forecast)random unbiased}] = (AD – BC) / (A + C)( B + D) HKS varies from -1 to +1, -1 means no skill and +1 means that the forecast is prefect.

(h) Heidke skill score (T) is defined as;

T = [{Correct forecasts – Correct forecasts by chance}/{Total –Correct forecasts by chance}] = 2(AD – BC) / (A + C)(C + D) + (A + B)(B + D) Heidke skill score varies from -1 to 1. -1 means no skill and +1 indicates the prefect forecast.

6.3.3. Verification Skill Scores for Quantitative Precipitation Forecast

In India, rainfall events may quantitatively be categorized into six types, viz., light, moderate, rather heavy, heavy, very heavy and extremely heavy, following IMD’s standard criteria. Observed and forecast quantitative description of rainfall events may be given in following matrix form:

Then Threat score (TS) is defined as, Hence, threat score for extremely heavy rain = , for moderate rain=  etc. The value of threat score decreases if one tries to over-forecast a particular category of rainfall in order to catch the occurrences of that category.

6.3.4. Verification Skill Scores for Temperature Forecasts

Skill score for temperature forecast is defined in terms of accuracy of the forecast of interest and that of climatological forecast. Again, these accuracies are defined in terms of mean square error or root mean square error. Hence, those will be discussed first.

Mean error (ME) in forecasting air temperature is equal to (, where,  is the mean of forecast value of air temperature and  is the mean of observed value of the same. If ME = 0, then the forecasts are unconditionally unbiased. A positive bias (ME > 0) indicates that the predicted temperature is on higher side on average, where as negative bias (ME < 0) indicates that it is cooler.

Similarly, mean absolute error (MAE) in temperature forecast is given by, .   Mean square error (MSE) in temperature forecast is given by, . MSE is a measure of accuracy of the forecast. The root mean square error (RMSE) of the forecast is defined as square root of the MSE. For completely accurate forecasts, both MSE and RMSE become zero.

Now we are in a position to define skill score for temperature forecast. Skill score (SS) for the temperature forecast is defined as, .where,   and  are the accuracy of the forecast of interest and of the climatological forecast, respectively. Here, measures of accuracy such as MAE, MSE andRMSE. If can be replaced by any of the is used as reference, then , is computed as, where  is the long-term climatological value of the observed temperatures.

7. Importance of weather forecasting in different aspects of life

It is obvious that knowing the weather in advance can be important for individuals and organizations. Accurate weather forecasts can tell a farmer the best time to plant, an airport control tower what information to send to planes that are landing and taking off, and residents of a coastal region when a cyclonic storm might strike.

Aviation sector is especially sensitive to the weather. Fog or exceptionally low clouds can prevent many aircraft from landing and taking off. There are many aviation weather hazards, viz., turbulence, icing and thunderstorm. Severe turbulence due to updrafts and downdraft associated with thunderstorm is a significant weather hazards. Icing can cause severe damage to an aircraft in flight. Volcanic ash is also a significant problem for aviation, as aircraft can lose engine power within ash clouds. On a day-to-day basis airliners are routed to take advantage of the jet stream tailwind to improve fuel efficiency. Air crews are briefed prior to takeoff on the weather conditions to expect en route and at their destination. Additionally, airports often change the runway to be used to take advantage of a headwind. This reduces the distance required for takeoff and eliminates potential crosswinds. Hence, accurate forecast of above mentioned weather is very essential for aviation industry.

Commercial and recreational use of waterways can be limited significantly by wind, wave periodicity and heights, tides and precipitation. Each of these factors can influence the safety of marine transit. Consequently, a variety of codes have been established to efficiently transmit detailed marine weather forecasts to vessel pilots via radio.

Weather forecasting of wind, precipitations and humidity is essential for preventing and controlling wildfires. Different indices have been developed to predict the areas having higher risk to experience fire from natural or human causes. Conditions for the development of harmful insects can be predicted by forecasting the evolution of weather too.

Electricity and gas companies rely on weather forecasts to anticipate demand which can be strongly affected by the weather. In winter, severe cold weather condition can cause a surge in demand of heater and similarly, in summer a surge in demand of air conditioning systems. By anticipating a surge in demand, utility companies can purchase additional supplies of power or natural gas before the price increases, or in some circumstances, supplies are restricted through the use of brownouts and blackouts. For such anticipation they require the information about air temperature which enables them to determine whether there will be strong use of heater or air conditioner/cooler. Such information is based on a daily average temperature of 65 °F (18 °C). Cooler temperatures force heating degree days (one per degree Fahrenheit) while warmer temperatures force cooling degree days.

Besides, forecast weather information is very important during military/air force/naval operation.

8. Summary

• Weather at a place at any instant of time is the state of the atmosphere, prevailing at that time over that place, demonstrated by the feelings of hot or cold, rainy or non-rainy, humid or dry etc.
• Since forecasting weather at a place involves requirement of knowledge about present state of the atmosphere at that place and its surroundings and attempt of foretelling about future state of the atmosphere, hence forecasting weather can be thought of as an initial value problem as well as a boundary value problem.
• Operationally there are six ranges of weather forecasting, based on the forecast validity period. These are: now casting, very short range forecasting, short range forecasting, medium range forecasting, extended range and long range forecasting.
• There are many techniques for verification of weather forecast issued. Among them some important techniques are: Verification Skill Scores for Probability of Precipitation, Verification Skill Scores for Yes/No Rainfall Forecast, Verification Skill Scores for Quantitative Precipitation Forecast and Verification Skill Scores for Temperature Forecasts.
• Information about weather forecasts of different ranges are of immense use for different important and essential sectors of the society.