"When it is evening ye say, it will be fair weather for the sky is
red. And in the morning, it will be foul weather today for the
sky is red and lowering." (Matthew Ch. 16 v. 2)
The Biblical quotation above is an ancient example of attempting to make
meteorological predictions based on empirical observations. The idea it
contains tends to work as a forecasting scheme, and it has been expressed in
many different ways in the subsequent 2,000 years. it is one of many sayings and
rhymes of folklore that provide a qualitative description of the weather. In the 17th
century, the introduction of instruments to measure atmospheric variables meant
that meteorology became a quantitative science.
By the 19th century meteorological data was being collected all over Europe. It
was during the 1800s that many scientists began lamenting the fact that the
collection of meteorological data had far outpaced attempts to analyse and
understand this data (Lempfert 1932). Attempts had been to find patterns in tables
of meteorological, but these had usually ended in failure. One example is the efforts
to link weather to celestial motions made by the Palatine Mteorological Society of
Mannheim (the World's first such society) in the late 18th century.
The breakthrough came when scientists began plotting the growing
meteorological database on maps; the tendency of mid-latitude low-pressure systems
to advance eastward was discovered: clouds in the west provide a red sky at dawn
before bringing bad weather, clouds in the east redden the sky at sunset before
making way for clear skies. The discovery of mid-latitude eastward flow not only
explained the success of ancient folk wisdom, combined with the telegraph it also
provided a means to extend predictability beyond the horizon. The late 19th and
early 20th century saw many efforts to identify cycles in time series of weather data.
One the most prolific cycle seekers, and ultimately one of the few successful ones,
was Sir Gilbert Walker. Walker collated global weather data and spent years seeking
correlations (Walker 1923, 1924, 1928, 1937, Walker and Bliss 1932) During this
work he discovered that the pressure i,n Tahiti and in Darwin were anti-correlated.
This discovery withstood the subsequent tests and is called the "Southern
Oscillation". It is now known to be the atmospheric component of El Nine-Southern
Oscillation climate phenomenon.
Many other claims for the existence of weather cycles were made, but few
withstood the scrutiny of statistical tests of robustness. By the 1930s, mainstream
meteorology had largely given up attempting to forecast the state of the atmosphere
using statistical approaches based solely on data (Nebeker 1995). Since then,
estimates of the recurrence time of the atmosphere have implied that globally "similar" atmospheric states can only be expected to repeat on time scales far greater
than the age of the Earth (van den Dool, 1994). This result suggests that there will
never be enough data available to construct pure data-based forecast models (except
for very short lead times for which only the local state of the atmosphere is
important) .
Towards the end of the 19th century a growing number of scientists took the view
that the behaviour of the atmosphere could be modelled from first principles, that is,
using the laws of physics. The leading proponent of this view was a orwegian
physicist called Vilhelm Bjerknes. Bjerknes believed that the problem of predicting
the future evolution of the atmosphere could be formulated mathematically in terms
of seven variables: three components of air velocity, pressure, temperature, density
and humidity - each of these variables being a function of space and time.
Furthermore, using the established laws of dynamics and thermodynamics, a
differential equation could be formulated for each of these seven quantities
(Richardson 1922).
The equations describe the flows of mass, momentum, energy and water vapour.
These equations, however, form a set of non-linear partial differential equations
(PDEs) and so an analytic solution was out of the question. Meanwhile, the First
World War raged and Briton Lewis Fry Richardson was developing a scheme for
solving Bjerknes' equations of atmospheric motion. I
In 1911, Richardson developed a method to obtain approximate solutions to
PDE's. The method involved approximating infinitesimal differences as finite
differences, that is dividing space into a finite number of grid boxes and assuming the
variables are uniform within each grid box. The solution obtained is not exact, but
becomes more accurate as the number of grid boxes increases. Richardson applied
his approach to the atmospheric equations, producing a set of finite difference
equations that could be solved by straightforward arithmetic calculations.
The difficulty of Richardson's achievement cannot be understated. In creating his
scheme for numerical weather prediction (NWP), he had to ensure the problem was
formulated in terms of quantities that could be measured, sometimes developing
new methods of measurement when they were required. He also had to develop a
way of dealing with turbulence. He related vertical transfer of heat and moisture to
the vertical stability of the atmosphere, as measured by a dirnensionless quantity now
called the Richardson number. Richardson was truly years ahead of his time - and
therein lay the problem. Although his recipe was straightforward, it was also
incredibly tedious. It took him six weeks to produce a single six-hour forecast for just
two European grid points! Richardson envisaged an army of clerks doing the
calculations that would be necessary to generate an operational forecast, but this did
not happen.
The Second World War saw the invention of the digital computer. After the War,
the computer pioneer, John von Neumann, was trying to persuade the US
government of the usefulness of this new device. Though not a meteorologist, von
Neumann identified weather forecasting as an ideal application to demonstrate the
power of the computer. Thanks to Richardson, the problem had been formulated in
an algorithm that could be executed by a computer, but was impractical to do
without one. Furthermore, the potential benefits of successful weather forecasting
could be appreciated by laymen, generals and politicians. In 1950, von Neumanri's
team, led by meteorologist Jules Charney, ran the first numerical weather prediction
program on the ENIAC computer. Thus began an intimate relationship between
meteorology and the leading edge of computer science - a relationship that
continues to this day See Nebeker (1995) for a comprehensive account of the history
of weather forecasting.
Since we never know the precise state of the atmosphere, it would be foolhardy
to expect to produce a precise forecast of its future state. This has motivated
operational forecast centres to develop probability forecasts, a set of possible outcomes based on slightly different views of the current state of the atmosphere.
These probability forecasts come closer to the type of information required for
effective risk management and the pricing of weather derivatives.
Modern Numerical Forecasting
Weather forecasting can be divided, somewhat arbitrarily, into three categories: short-,
medium- and long-range. In the context of this chapter, short-range forecasting refers
to forecasting the weather over the next day or two. Medium-range forecasting covers
lead times of three days to about two weeks, while long-range, or seasonal,
forecasting aims to predict the weather at lead times of a month or more. While short
and medium-range forecasts are valuable to many users, including energy companies
in planning their operations, it is seasonal forecasts that are of most interest to the
weather derivatives markets.
Nowadays it is increasingly common for seasonal forecasts to be made using
computer models which are essentially the same as those used to produce
tomorrow's forecasts. In this section we shall outline the key features of these
models so that the reader will have some understanding of the origin of forecast
products. We also hope to familiarise the reader with some of the technical language
that meteorologists use to describe their models.
Short-range forecasts can be made using limited area models. These models use
grid boxes to cover restricted parts of the globe. In a period of two weeks, however,
weather systems can travel halfway round the globe. Therefore forecasting in the
medium range or beyond requires a global model of the atmosphere. The most
advanced global model used for operational forecasting belongs to he European
Centre for Medium Range Weather Forecasting (ECMWF)funded by 19 countries and
based in Reading, UK.2 Today; ECMWF makes daily forecasts out to 10 days. These
forecasts are distributed through the national meteorological offices of the member
countries.
The current ECMWF global model is a T511 spectral model (see Panel 1);
equivalent to a horizontal resolution of 40 km, with 60 vertical levels. The complete
model state, at a given time, is described by approximately 10 million individual
variables. The state is evolved forward in time by taking time steps of about 10 minutes. The evolution of the atmosphere is thus represented as a trajectory in the
ultra-high dimensional state space of the model.2
The approximations introduced by representing continuous fields on a finite grid
are, in a sense, well-defined or mathematical approximations, sometimes referred to
as "errors of representation". There are, however, another set of physical
approximations that all numerical weather models contain. These approximations
are called "parameterisations". Numerical models have a finite spatial resolution. As
mentioned above, the ECMWFglobal model cannot represent the details of weather,
or of topography, at scales less than 40 km. It would be nice to be able to know the
distribution of rainfall on a much smaller scale, but even having information on
weather averaged over tens of kilometres can be very useful.
Even if one is content to accept weather forecasts averaged over relatively large
regions, weather on smaller "sub-grid" scales can have a profound impact on the
weather at larger scales. For example, thunderstorms are too small for global models
to resolve, but the convection and rainfall associated with them has a major impact
on the energy balance of the atmosphere, and consequently on the weather over an
area much larger than the storm itself. Therefore, the impact of sub-grid processes
must be parameterised and included in the equations that describe the evolution of
the atmosphere at the larger scales. Essentially, a parameterisation scheme for
cumulus convection must predict the amount of convection in a grid box purely as a
function of the meteorological variables averaged over the grid box (and possibly
surrounding grid boxes), and then predict the affect that this amount of convection
will have on the time evolution of those meteorological variables. Parameterisation
schemes are usually designed based partly on a physical understanding of the
processes involved and partly on the study of empirical observations. There are many
processes that must be parameterised in numerical models of the atmosphere, eg,
surface evaporation, drag due to topography and sub-grid turbulence. Designing
better parameterisation schemes is one of the most active and important areas of
modern meteorological research.
DATA ASSIMILATION
Before a forecast can be made with a numerical model of the atmosphere, the current
state of the atmosphere, as represented within the model, must be estimated. The
process of using observations to make this estimate of the model's initial condition is called "data assimilation". The estimate of the state of the atmosphere derived from data assimilation is called the analysis.
To initialise the model, one must effectively know the value of all the relevant meteorological variables as represented on the grid points of the model. Even with the vast amounts of weather data that are collected every day (see Panel 2) there are still massive gaps in the observational data set. The simplest approach to overcoming these gaps is interpolation, as Richardson did in his early in numerical forecasting experiments (Richardson 1922). More sophisticated approaches actually combine the observations with the knowledge of atmospheric dynamics that is contained in the numerical model itself. Any state of the model can be converted into an estimate of the observations that would result if the atmosphere were in that particular state, by using the "observation function".
At ECMWF a data assimilation technique called "variational assimilation" is used. This method involves trying to find a model trajectory that leads to the closest match of the model to the actual observations that were made. The trajectory of the model is the path the model state traces out in time within the state space of the model. This space is a high dimensional space, defined by the millions of variables that describe the model state. Optimisation by variational assimilation is performed by trying to find the state of the model that leads to the best match with observations over the subsequent assimilation period. The state of the model somewhere in the middle of the assimilation period is then used as the analysis with which to initialise the forecast. A lower resolution, '1'159 model, is used for the data assimilation. Data assimilation is also used by forecasting centres to produce reanalysis
products. These products are historical reconstructions of the state of the atmosphere, projected into the model grid-point representation. They are constructed using similar techniques that are used to generate the analysis used to initialise forecasts. When producing a reanalysis, however, it is possible to use observations made after the time for which the state estimate is required, in addition to those from before, to estimate the model state. Because reanalysis products are reconstructions that are complete in time and space, they can be used to estimate weather at locations for which direct historical observations are not available. Reanalysis projects have been undertaken by ECMWFand NCEp-i
In the late 1950s the meteorologist, Edward Lorenz, was experimenting with a
numerical model of the atmosphere at the Massachusetts Institute of Technology
(MIT). During a set of experiments, he started a new run by resetting the state of the
model to the state obtained halfway through a previous run. To his surprise, the
behaviour of the atmosphere in second half of the new run was substantially different
from the second half of the initial run. Eventually Lorenz realised that, while the
computer was evaluating the model to six decimal-place accuracy, it was printing out
the model state to only three decimal-place accuracy, thus resetting the model with
the printed output had introduced a tiny discrepancy (less than one pa.rt in 1,000)
between the two runs. This error was enough to cause a big difference in their
evolutions (Lorenz 1993). While such behaviour had been known for centuries, the
existence of digital computing made such behaviour more amenable to study. In
1975 the word "chaos" was coined-to describe such sensitivity of these models to
initial conditions, a property popularly known as the "butterfly effect".
To study the model sensitivity to initial conditions, experiments in which artificial
errors, consistent with known observational uncertainty, are introduced into
computer models of the atmosphere. The model results using these alternative initial
conditions can then be compared with the original runs; the comparisons suggest
that it is unlikely that we will ever predict the precise evolution of the atmosphere
for longer than a few weeks (Lorenz 1982).
The impact of chaos on meteorology has not been entirely negative. Instead, it has led to a shift in emphasis - a shift likely to be useful to weather risk management
professionals. The atmosphere is not uniformly sensitive to initial conditions; its
sensitivity depends on its current state. On some days the atmosphere can be
approximated by a relatively predictable linear system over a short enough period of
time. On other days, the inevitable uncertainty that exists in the analysis can lead to
rapid error growth in the forecast. The key point is that the predictability of the
atmosphere depends upon the state of the atmosphere. Predicting predictability has
now become a major component of operational weather forecasting.
ENSEMBLE FORECASTING
Both the ECMWFand the US National Centre for Environmental Prediction (NCEP)
have been running daily ensemble forecasts since 1992. The idea behind ensemble
forecasting is simple; run several forecasts using slightly different initial conditions
generated around the analysis. The relative divergence of the forecasts indicates the
predictability of the atmospheric model in its current state - the greater the
divergence the lower the confidence in the forecast. Thus, ensemble forecasts should
provide a priori information on the reliability of that day's forecast. The difficulty
arises in deciding what perturbations (errors introduced into the initial condition) to
make to the analysis. Limited computing power means that only a few dozen
forecasts can be produced (even when the ensemble forecasts are produced using a
lower resolution model than the main forecasts).
The ensemble formation methods discussed above attempt to account for errors
in the initial condition. Other ensemble forecasts use models with different
combinations of parameterisation schemes for sub-grid processes, in an attempt to
estimate how the uncertainty in the choice of scheme affects the final forecast
(Stensrud 2000). Another way in which ensemble members may differ is by
introducing random terms into the dynamical equations of the model. This approach
attempts to model the uncertainty in the future evolution of the atmosphere at each
time step of the model. In such a stochastic parameterisation, the impact of sub-grid
processes on the resolved flow is not assumed to be a deterministic function of the
model state. Instead, it is a random variable. The parameters of the distribution from
which this variable is selected (such as its mean and variance) can be determined by
the model state.
Due to the non-linearity of NWP models, introduction of these stochastic terms
can actually improve the mean state of the model in addition to helping to assess the
uncertainty in its evolution (Palmer 2001). Stochastic parameterisation is a new
feature of numerical weather prediction, and is partly a manifestation of
meteorology's willingness to accept uncertainty and its attempt to quantify it.
For the purpose of pricing weather derivatives, ensemble forecasts are much
more useful than traditional single forecasts. For example, each member of an
ensemble forecast can be used to calculate the number of heating degree-days
(HDDs) accumulated in a period: this provides a rudimentary distribution of future
HDDs. At present, however, the relatively small size of the ensembles and the fact that
they represent quantities averaged over tens of kilometres rather than at individual
weather stations, means that using ensembles in this manner would be ill-advised.
We shall discuss how the predictability information contained in current ensemble
forecast might ~e extracted in the next section.
Once uncertainty in the atmospheric state has been accepted as a fact of life that
will not go away, the extension of medium-range forecasting techniques to longer range
seasonal forecasting is not a major leap. As noted above, the sensitivity of the
evolution of the atmosphere's state to its initial conditions prohibits the precise
prediction of the trajectory of this state for longer than, at best, a few weeks into the
future. This means that there is little hope of forecasting whether it will rain on a
specific day in a few months time. This does not mean, however, that useful forecasts
at lead times of several months are not possible. It is possible to predict whether a season will be wetter or colder than average at lead times of over three months and to estimate of the probability of magnitudes of change - the probability it will be atleast 1QCwarmer than average, for example.
The crucial extra ingredient required for seasonal forecasting is a computer
model of the ocean. The time scales on which the state of the ocean changes are
quite long compared to the lead time of a medium-range forecast. Because of this,
when making such a forecast, the state of the ocean can be held fixed. Beyond a
couple of weeks, however, the changing state of the ocean is an important influence
on the behaviour of the atmosphere. To make progress in seasonal forecasting, a
model of the ocean must be coupled to the atmosphere model. The atmosphere
forces the ocean through wind stress at its surface, while the ocean forces the
atmosphere by exchanging heat with it, especially through the evaporation of water
- which forms clouds - and radiation.
The sea surface temperature (SST) is an important influence on the behaviour of
the atmosphere, particularly in the tropics. It is the coupling of the ocean and
atmosphere that lies behind much inter-annual climate variation such as El Nino-
Southern Oscillation (ENSO).' ENSO is characterised by a large-scale cycle of
warming and cooling in the Eastern tropical Pacific that repeats on a time scale of 2-7
years. The warm SST of the El Nino phase of the cycle influences the atmospheric
circulation over large parts of the globe. In particular, El Nino events are associated
with heavy rainfall in Peru and Southern California, mild winters in the Eastern US
and drought in Indonesia and Northern Australia (Glantz 1996). These are all
probabilistic associations - ENSO is just one influence of the atmosphere's
behaviour, although an important one at lower latitudes (see Philander, 1990 for
further reading).
In many ways, the existence of ENSO is a blessing, it imposes some degree of
regularity on tropical climate that helps seasonal prediction. The numerical ocean
models used in seasonal forecasting are not fundamentally different from their
atmospheric cousins; they rely on the division of the oceans into finite elements,
horizontally and vertically; the equations of mass, momentum and energy
conservation are integrated numerically, and sub-grid processes are parameterised.
Seasonal forecasts at mid-latitudes are not as skilful as in the tropics. Although there
are thought to be mid-latitude climate cycles, such as the North Atlantic Oscillation
(NAO), they are not as regular and well detlned as ENSO. The existence of these
cycles has allowed the development of statistical seasonal forecasting models which
have skill at lead times of up to six months (eg, Penland and Margorian, 1993). Like
all statistical models, however, the availability of historical data is a major constraint
on their refinement. Improvements in seasonal forecasting will require better
information about tile state of the ocean. Observations of the ocean are not as dense
as atmospheric observations. Better ocean data, such as that obtained from the
TOPEX-POSEIDON satellite, and its successor jASON-1, which measure sea surface
height, should enable better estimation of the state of the ocean, and thus improved
seasonal forecasts6
Beyond seasonal time scales, forecasts that have more skill than climatology are
elusive. It is possible, however, that coupled numerical models of the ocean atmosphere
system can help to improve the climatologist distributions that are used
to assess weather risk. The instrumental records for most locations are quite short,
often only extending back a few decades. Extended runs of oceanic-atmospheric
general circulation models (GCMs) could enable better estimates of the risk of
extreme events that may only have occurred on a handful of occasions in recent
history. This is particularly true if secular changes to forcing of the ocean atmosphere
system - such as enhanced radiative forcing due to increased carbon
dioxide and other greenhouse gases (Harries et al. 2001, IPCC, 2001) - reduce the
relevance of the historical record. Before GCMs can be used for this type of risk assessment, it must be demonstrated that they can reproduce observed historical
climate variability on regional scales, not just average global temperatures.
Interpretation and Post-processing Model Output
The output generated by numerical models, should not be considered as "weather
forecasts". The output merely represents the state of the model, which contains
information about the weather but is not immediately relevant to quantities that are
actually observed. There are many ways in which extra processing of the raw forecast
products produced by forecast centres can substantially increase the value of these
forecasts. For example, even the highest resolution global models cannot resolve the
details of mountain topography or small islands, yet these physical features can have
a substantial impact on the local weather conditions. One of the ways in which
human forecasters can add value to a forecast is by using their experience of the
weather in a particular locale to predict the likely conditions there, given the larger
scale weather picture that the numerical forecast provides.
The finite resolution of numerical models, that is, the grid size, limits its
application to areas no smaller than a grid. The user of the model forecast, however,
is likely to need to know the values of forecast variables on a scale smaller than a
model grid: pricing a weather derivative may require the temperature at the London
Heathrow weather station, for example, but the ECMWF model predicts a
temperature that is averaged over a grid box of 40 km by 40 km. "Downscaling" is the
term used to describe a variety of quantitative methods that use the values of
forecasted model variables for estimating the values of specific variables on scales
smaller than the model grid.
One common method is the use of model output statistics (MOS) (see Glahn and
Lowry 1972). A small number of model variables is chosen as the set of predictors of
the desired variable. These predictors are extracted from past numerical forecasts,
and correlated with the corresponding observational record of the desired variable.
A statistical model can then be constructed that predicts the desired variable, using
the forecast variables as predictors. This statistical model should remove any
systematic biases in the numerical model. How MOS should be extended to
ensemble forecasts is not obvious.
One could use each member of the ensemble as the predictor in a traditional
statistical MOS model.
However, the forecast uncertainty represented by the ensemble must also be
downscaled. This is because the forecasts are averaged over grid boxes that are tens
of kilometres in size. The uncertainty of a forecast averaged over a grid box should
be lower than the uncertainty of a forecast at a single weather station (that averaging
reduces variability is a much exploited fact in both science and finance). So, while
traditional MOS can be used to downscale the mean of the ensemble, obtaining an
appropriate estimate of uncertainty is trickier. Methods for extracting the
predictability that exists in the ensemble forecasts are now being developed (eg,
Roulston et al., 2002).
Another method for downscaling a forecast is "nested modelling". This uses a
local area numerical model, covering a much smaller region than the global model,
but with a much higher resolution. The local model is "nested" inside the global
model. This means that it is integrated forward like the global model, but that on the
edges of the region it covers the values of its variables are obtained by using the
corresponding values from the global model (interpolated onto the local model's
higher resolution grid). An example of using a nested model in a forecast application
can be found in Kuligowksi and Barros (1999).
PANEL 1: COMPUTER MODELS: GRID POINT
VS SPECTRAL POINT
Computer models of the atmosphere come in two types: grid points and spectral
points. Grid point models represent the atmosphere as finite boxes centred on a grid
point and calculate the changes in mass, energy and momentum at each grid point
as a function of time. The size of the boxes determines the resolution of the model.
Spectral models represent the distribution of atmospheric properties as sums of
spherical harmonics. These harmonics are similar to sine and cosine functions, except
they are two dimensional andexist 01'1 the surface of a sphere. The resolution of a
spectral model is determined by the wavelength of the highest spherical harmonic
used in the model. Spectral models are denoted by labels such as 'Tfil l": this means
that the highest harmonic used has 5ll waves around each line of latitude. T5ll
model has a horizontal resolution of 1/(2 x 5ll) times the radius of the Earth (about
40 km). The state of the atmosphere, as represented in a spectral model, can be
converted to a grid-point representation using a mathematical transformation. Both
grid-point and spectral models divide the atmosphere vertically into layers.
PANEL 2: OBSERVING THE ATMOSPHERE
The World Meteorological Organization's World Weather Watch (WMO WWW)
manages the data from 10,000 land stations, 7,000 ship stations and 300 buoys fitted
with automatic weather stations. All these stations are maintained by National
Meteorological Centres.
The most important development in meteorological observation in the last 40 years
has been the advent of weather satellites. The WMO WWW incorporates data from a
constellation of nine weather satellites that provide global coverage. It is a testimony
to the importance of these satellites that weather forecasts now tend to be more skilful
in the southern hemisphere, where surface stations are quite sparse, than in the more
densely observed northern hemisphere. The fact that southern hemisphere skill is
actually slightly higher is probably because there is less land south of the equator.
Topography and other continental effects make the job of forecasting northern
hemisphere weather harder, even with the greater data coverage in the North.
PANEL 3: EVALUATION OF FORECASTS
When choosing forecast products to help manage their weather risk, users should
obviously be concerned with the skill level of the forecasts. Unfortunately, it is difficult
to assess skill when forecasting multiple variables with numerical models.
A common measure of skill used by meteorologists is the mean square error (MS
error). This is just the mean of the square of the difference between the forecasted
value and the observed value.
The mean can be an average over time for a univariate forecast (eg, temperature at
Heathrow airport), or a time-space average for a multivariate forecast (eg, the 500hPa
height field over the northern hemisphere). The MS error, however, is not a good way
to evaluate an ensemble, or probabilistic forecast. The ensemble average can be
calculated and assessed this way but such approach completely fails to take into
account the information about uncertainty inherent in the ensemble forecast,
information of great interest to risk managers.
Verifying probabilistic forecasting systems requires different measures of skill. The
class of measures most relevant to risk manages is probabilistic "scoring rules". To
use these rules the number of possible outcomes is usually a finite number of classes.
Examples of such classes might be rainlno rain or a set of temperature ranges. A
probabilistic forecast will assign a probability to each of the possible outcomes.
Scoring rules are functions of the probability that was assigned to the outcome that
actually occurred. If this probability is p then the quadratic score of the forecast is p_.
The logarithmic score is log p.7 The quadratic scoring rule forms the basis for the
"brier score" and the "ranked probability score" (Brier 1950). The logarithmic scoring
rule is connected to the information content of the forecast, and also to the returns a
gambler would expect if they bet on the forecast (Roulston and Smith, 2002).
Another method for evaluating probabilistic forecasts is the cost-loss score
(Richardson, 2000).
The cost-loss score is based on the losses that would be incurred by someone using
a probabilistic forecast to make a simple binary decision. Consider the situation where
the user must decide whether to grit the roads tonight. The cost of gritting is C. If it
does not freeze, no loss is suffered if the roads are not gritted. If it does freeze,
ungritted roads will lead to a loss, l. If P is the forecast probability of it freezing tonight,
then the expected loss if the user does not grit is pL. If p>C/l then this expected loss
exceeds the cost of gritting and a rational user would send out the gritting trucks. The
cost-loss score illustrates how important it is to have a probabilistic forecast. A user
with a C/l ratio far from 0.5 would be ill-advised to take the course of action suggested
by a best-guess forecast. Best-guess forecasts can be easily converted into
probabilistic forecasts by estimating the historical forecast error distribution (ie, the
errors of previous forecasts).
Such a conversion should always be performed before evaluating best-guess
forecasts using any skill score designed for probabilistic forecasts. Not doing so will
artificially inflate the advantage of using an ensemble system. Users that can formalise
their decision-making can directly estimate the value of forecasts by determining their
impact on decisions. Decision making processes for real users will be more complex
than the cost-loss scenario described above but, the principle of utility maximisation
should still apply. Katz and Murphy (1997) contains detailed articles on common
evaluation methods for weather forecasts.
Conclusion
Over the last 2,000 years, the prediction horizon of state-of-the-art weather forecasts
has advanced significantly from seeing one day ahead based on the colour of the sky
to almost a two-week outlook. The risk management community moved more
quickly, as in only a few years, the very concept of a weather forecast has changed
from a single best-guess of the future to a distribution of likely future weather
scenarios.
In this chapter the methods used to produce modern numerical weather forecasts
have been outlined. It was also claimed that the modelling techniques used for
making short-range and medium-range forecasts are not fundamentally different
from the approaches that must be adopted to forecast climate on seasonal scales and
beyond. The main difference is that, for longer-range forecasts, a model of ocean
dynamics must be coupled to the atmospheric model. This chapter has also stressed
the importance of ensemble forecasting, for quantifying forecast uncertainty.
Ensemble forecasting is on its way to becoming the standard approach to forecasting
the evolution of the atmosphere and the ocean. This is a welcome development for
the risk management community because ensemble forecasts contain information
about the uncertainty of forecasts, uncertainty that is synonymous with risk.
The ultimate aim of accurate probability forecasts of commercially relevant
variables is being pursued on time scales from a few hours to several months (Palmer
2002). For those who understand how to interpret it, this information will give a
competitive edge at a range of lead times, from pricing next week's electricity futures
to pricing next year's weather derivatives. Skilled and timely forecasts can be
financially rewarding.
2 All operational forecast centres frequently upgrade their forecast models. Details
of these upgrades, and of the present configuration of the ECMWF model can be
found on their website http://www.ecmwjint.
3 A description of the ECMWF and its output products can be found in Persson
(2000).
4 Details of reanalysis projects and data can be found at
http://ecmwjint/research/era/ and http://www.cdc.noaa.gov/ncePJeanalysis.
5 As noted above Gilbert Walker discovered the atmospheric Southern Oscillation
in the 1920s. The relationship between this variation and the oceanic El Nino
phenomenon was discovered in the 1960s byjacob Bjerknes, son ofVilhelm.
6 Information on these satellites is available at http://topex-wwwjpl.nasa.gov/
mission/jason-1. html.
7 Readers may be curious as to why the simplest linear skill score =p has been
omitted. It turns out that this score is "improper" in the sense that a forecaster
increases their expected score by reporting probabilities that differ from what they
believe to be true.
BIBliOGRAPHY
Brier, G. w., 1950, "Verification of Forecasts Expressed in Terms of Probabilities", Monthly Weather
Review, 78, pp. 1-3.
van den 0001, H. M., 1994, "Searching for Analogues, How Long must we Wait?" Tellus A, 46, pp.
314-24.
Glahn, H. R., and Lowry, D. A., 1972, "The Use of Model Output Statistics (MOS) in Objective Weather
Forecasting",}ournal of Applied Meteorology, 11, 1203-11.
Glantz, M. 11., 1996, Currents of Change: El Nino's Impact on Climate and Society, (Cambridge
University Press).
Harries,]. E., Brindley, H. E., Sagoo, P.]. and Bantges, R.]., 2001, "Increases in Greenhouse Forcing
Inferred from the Outgoing Longwave Radiation Spectra of the Earth in 1970 and 1997", Nature, 410,
pp. 355-7.
IPCC, Intergovernmental Panel on Climate Change, 2001, Climate Change 2001: Synthesis Report,
(cd) R. T. \Vatson, (Cambridge University Press).
Katz, R. w., and Murphy, A. H., 1997, Economic Value of Weather and Climate Forecasts, (Cambridge
University Press).
Kuligowski, R. ]. and A. P. Barros, 1999, "High-Resolution Short-Term Quantitative Precipitation
Forecasting in Mountainous Regions Using a Nested Model", Journal of Geophysical Research
(atmospheres), 104, pp. 31, 553-64.
Lempfcrt, R. G. K, 1932, "The Presentation of Meteorological Data", Quarterly Journal of the Royal
Meteorological SOCiety,58, pp. 91-102.
Lorenz, E. N., 1982, ''Atmospheric Predictability Experiments with a Large Numerical Model", Tellus, 34,
pp. 505-13.
Lorenz, E. N., 1993, The Essence of Chaos, (London: UCl Press).
Murphy, A_ H_, 1997, "Forecast Verification", in: Economic Value of Weather and Climate Forecasts,
(eds) R. \'if Katz and A. H. Murphy, (Cambridge University Press), pp. 19-70.
Nebeker, F., 1995, Calculating the Weather, (San Diego: Academic Press).
Palmer, T. N., 2000, "Predicting Uncertainty in Forecasts of Weather and Climate", Reports Oil Progress
in Physics, 63, pp. 71-116.
Palmer, T. N., 2001, "A Nonlinear Dynamical Perspective on Model Error: A Proposal for Non-Local
Stochastic-Dynamic Paramcterization in Weather and Climate Prediction Models", Quarterly journal 0/
the Royal Meteorological Society, 127, pp. 279-304.
Palmer, T. N., 2002, "The Economic Value of Ensemble Forecasts as a Tool for Risk Assessment: From
Days to Decades", Quarterly Journal 0/ the Royal Meteorological Society, 128, pp. 747-74.
Penland, c., and T. Magorian, 1993, "Prediction of Nino 3 Sea Surface Temperature Using Linear
Inverse Modeling",Journal of Climate, 6, pp. 1067-76.
Persson, A., 2000, "User Guide to ECMWF Forecast Products", European Centre for Medium Range
Weather Forecasting, Reading.
Philander, S. G., 1990, El Nino, La Nina, and the Southern Oscillation, (San Diego: Academic Press).
Richardson, L. F., 1922, Weather Prediction by Numerical Processes, (Cambridge University Press).
Richardson, D. S., 2000, "Skill and Relative Economic Value of the ECMWF Ensemble Prediction
System", Quarterly journal of the. Royal Meteorological. Society, 126, pp. 649-67.
Roulston, M. S. and L. A. Smith, 2002, "Evaluating Probabilistic Forecasts using Information Theory".
Monthly Weatber Review, 130, pp. 1653-60, 2002.
Roulston, M. S., D. T Kaplan, J. Hardenberg and L. A. Smith, 2002, "Using Medium Range Weather
Forecasts to Improve the Value of Wind Energy Production", Forthcoming, Renewable Energy
Smith, L. A., Roulston, M. S. and lIardenberg, J., 2001, "End to End Ensemble Forecasting: Towards
Evaluating the Economic Value of the Ensemble Prediction System", ECMWF Technical Memorandum
336, ECMWF, Reading.
Stensrud, D. J., J. W Bao, and T. T. Warner, 2000, "Using Initial Condition and Model Physics
Perturbations in Short-Range Ensemble Simulations of Mesoscale Convective Systems", Monthly
Weather Review, 128, pp. 2077-107.
Walker, G. T., 1923, "Correlation in Seasonal Variations of Weather VIII", Memorandum of the Indian
Meteorological Deptartment, 24, pp. 75-131.
Walker, G. T., 1924, "World Weather IX", Memorandum of the Indian Meteorological Dcptartmcnt, 24,
pp. 275-332.
Walker, G. T., 1928, "World Weather Ill", Memorandum of the Royal Meteorological Dcptartment, 17,
pp. 97-106.
Walker, G. T., 1937, "World Weather VI", Memorandum of the Royal Meteorological Society, 4, pp.
119-39.
RISK
red. And in the morning, it will be foul weather today for the
sky is red and lowering." (Matthew Ch. 16 v. 2)
The Biblical quotation above is an ancient example of attempting to make
meteorological predictions based on empirical observations. The idea it
contains tends to work as a forecasting scheme, and it has been expressed in
many different ways in the subsequent 2,000 years. it is one of many sayings and
rhymes of folklore that provide a qualitative description of the weather. In the 17th
century, the introduction of instruments to measure atmospheric variables meant
that meteorology became a quantitative science.
By the 19th century meteorological data was being collected all over Europe. It
was during the 1800s that many scientists began lamenting the fact that the
collection of meteorological data had far outpaced attempts to analyse and
understand this data (Lempfert 1932). Attempts had been to find patterns in tables
of meteorological, but these had usually ended in failure. One example is the efforts
to link weather to celestial motions made by the Palatine Mteorological Society of
Mannheim (the World's first such society) in the late 18th century.
The breakthrough came when scientists began plotting the growing
meteorological database on maps; the tendency of mid-latitude low-pressure systems
to advance eastward was discovered: clouds in the west provide a red sky at dawn
before bringing bad weather, clouds in the east redden the sky at sunset before
making way for clear skies. The discovery of mid-latitude eastward flow not only
explained the success of ancient folk wisdom, combined with the telegraph it also
provided a means to extend predictability beyond the horizon. The late 19th and
early 20th century saw many efforts to identify cycles in time series of weather data.
One the most prolific cycle seekers, and ultimately one of the few successful ones,
was Sir Gilbert Walker. Walker collated global weather data and spent years seeking
correlations (Walker 1923, 1924, 1928, 1937, Walker and Bliss 1932) During this
work he discovered that the pressure i,n Tahiti and in Darwin were anti-correlated.
This discovery withstood the subsequent tests and is called the "Southern
Oscillation". It is now known to be the atmospheric component of El Nine-Southern
Oscillation climate phenomenon.
Many other claims for the existence of weather cycles were made, but few
withstood the scrutiny of statistical tests of robustness. By the 1930s, mainstream
meteorology had largely given up attempting to forecast the state of the atmosphere
using statistical approaches based solely on data (Nebeker 1995). Since then,
estimates of the recurrence time of the atmosphere have implied that globally "similar" atmospheric states can only be expected to repeat on time scales far greater
than the age of the Earth (van den Dool, 1994). This result suggests that there will
never be enough data available to construct pure data-based forecast models (except
for very short lead times for which only the local state of the atmosphere is
important) .
Towards the end of the 19th century a growing number of scientists took the view
that the behaviour of the atmosphere could be modelled from first principles, that is,
using the laws of physics. The leading proponent of this view was a orwegian
physicist called Vilhelm Bjerknes. Bjerknes believed that the problem of predicting
the future evolution of the atmosphere could be formulated mathematically in terms
of seven variables: three components of air velocity, pressure, temperature, density
and humidity - each of these variables being a function of space and time.
Furthermore, using the established laws of dynamics and thermodynamics, a
differential equation could be formulated for each of these seven quantities
(Richardson 1922).
The equations describe the flows of mass, momentum, energy and water vapour.
These equations, however, form a set of non-linear partial differential equations
(PDEs) and so an analytic solution was out of the question. Meanwhile, the First
World War raged and Briton Lewis Fry Richardson was developing a scheme for
solving Bjerknes' equations of atmospheric motion. I
In 1911, Richardson developed a method to obtain approximate solutions to
PDE's. The method involved approximating infinitesimal differences as finite
differences, that is dividing space into a finite number of grid boxes and assuming the
variables are uniform within each grid box. The solution obtained is not exact, but
becomes more accurate as the number of grid boxes increases. Richardson applied
his approach to the atmospheric equations, producing a set of finite difference
equations that could be solved by straightforward arithmetic calculations.
The difficulty of Richardson's achievement cannot be understated. In creating his
scheme for numerical weather prediction (NWP), he had to ensure the problem was
formulated in terms of quantities that could be measured, sometimes developing
new methods of measurement when they were required. He also had to develop a
way of dealing with turbulence. He related vertical transfer of heat and moisture to
the vertical stability of the atmosphere, as measured by a dirnensionless quantity now
called the Richardson number. Richardson was truly years ahead of his time - and
therein lay the problem. Although his recipe was straightforward, it was also
incredibly tedious. It took him six weeks to produce a single six-hour forecast for just
two European grid points! Richardson envisaged an army of clerks doing the
calculations that would be necessary to generate an operational forecast, but this did
not happen.
The Second World War saw the invention of the digital computer. After the War,
the computer pioneer, John von Neumann, was trying to persuade the US
government of the usefulness of this new device. Though not a meteorologist, von
Neumann identified weather forecasting as an ideal application to demonstrate the
power of the computer. Thanks to Richardson, the problem had been formulated in
an algorithm that could be executed by a computer, but was impractical to do
without one. Furthermore, the potential benefits of successful weather forecasting
could be appreciated by laymen, generals and politicians. In 1950, von Neumanri's
team, led by meteorologist Jules Charney, ran the first numerical weather prediction
program on the ENIAC computer. Thus began an intimate relationship between
meteorology and the leading edge of computer science - a relationship that
continues to this day See Nebeker (1995) for a comprehensive account of the history
of weather forecasting.
Since we never know the precise state of the atmosphere, it would be foolhardy
to expect to produce a precise forecast of its future state. This has motivated
operational forecast centres to develop probability forecasts, a set of possible outcomes based on slightly different views of the current state of the atmosphere.
These probability forecasts come closer to the type of information required for
effective risk management and the pricing of weather derivatives.
Modern Numerical Forecasting
Weather forecasting can be divided, somewhat arbitrarily, into three categories: short-,
medium- and long-range. In the context of this chapter, short-range forecasting refers
to forecasting the weather over the next day or two. Medium-range forecasting covers
lead times of three days to about two weeks, while long-range, or seasonal,
forecasting aims to predict the weather at lead times of a month or more. While short
and medium-range forecasts are valuable to many users, including energy companies
in planning their operations, it is seasonal forecasts that are of most interest to the
weather derivatives markets.
Nowadays it is increasingly common for seasonal forecasts to be made using
computer models which are essentially the same as those used to produce
tomorrow's forecasts. In this section we shall outline the key features of these
models so that the reader will have some understanding of the origin of forecast
products. We also hope to familiarise the reader with some of the technical language
that meteorologists use to describe their models.
Short-range forecasts can be made using limited area models. These models use
grid boxes to cover restricted parts of the globe. In a period of two weeks, however,
weather systems can travel halfway round the globe. Therefore forecasting in the
medium range or beyond requires a global model of the atmosphere. The most
advanced global model used for operational forecasting belongs to he European
Centre for Medium Range Weather Forecasting (ECMWF)funded by 19 countries and
based in Reading, UK.2 Today; ECMWF makes daily forecasts out to 10 days. These
forecasts are distributed through the national meteorological offices of the member
countries.
The current ECMWF global model is a T511 spectral model (see Panel 1);
equivalent to a horizontal resolution of 40 km, with 60 vertical levels. The complete
model state, at a given time, is described by approximately 10 million individual
variables. The state is evolved forward in time by taking time steps of about 10 minutes. The evolution of the atmosphere is thus represented as a trajectory in the
ultra-high dimensional state space of the model.2
The approximations introduced by representing continuous fields on a finite grid
are, in a sense, well-defined or mathematical approximations, sometimes referred to
as "errors of representation". There are, however, another set of physical
approximations that all numerical weather models contain. These approximations
are called "parameterisations". Numerical models have a finite spatial resolution. As
mentioned above, the ECMWFglobal model cannot represent the details of weather,
or of topography, at scales less than 40 km. It would be nice to be able to know the
distribution of rainfall on a much smaller scale, but even having information on
weather averaged over tens of kilometres can be very useful.
Even if one is content to accept weather forecasts averaged over relatively large
regions, weather on smaller "sub-grid" scales can have a profound impact on the
weather at larger scales. For example, thunderstorms are too small for global models
to resolve, but the convection and rainfall associated with them has a major impact
on the energy balance of the atmosphere, and consequently on the weather over an
area much larger than the storm itself. Therefore, the impact of sub-grid processes
must be parameterised and included in the equations that describe the evolution of
the atmosphere at the larger scales. Essentially, a parameterisation scheme for
cumulus convection must predict the amount of convection in a grid box purely as a
function of the meteorological variables averaged over the grid box (and possibly
surrounding grid boxes), and then predict the affect that this amount of convection
will have on the time evolution of those meteorological variables. Parameterisation
schemes are usually designed based partly on a physical understanding of the
processes involved and partly on the study of empirical observations. There are many
processes that must be parameterised in numerical models of the atmosphere, eg,
surface evaporation, drag due to topography and sub-grid turbulence. Designing
better parameterisation schemes is one of the most active and important areas of
modern meteorological research.
DATA ASSIMILATION
Before a forecast can be made with a numerical model of the atmosphere, the current
state of the atmosphere, as represented within the model, must be estimated. The
process of using observations to make this estimate of the model's initial condition is called "data assimilation". The estimate of the state of the atmosphere derived from data assimilation is called the analysis.
To initialise the model, one must effectively know the value of all the relevant meteorological variables as represented on the grid points of the model. Even with the vast amounts of weather data that are collected every day (see Panel 2) there are still massive gaps in the observational data set. The simplest approach to overcoming these gaps is interpolation, as Richardson did in his early in numerical forecasting experiments (Richardson 1922). More sophisticated approaches actually combine the observations with the knowledge of atmospheric dynamics that is contained in the numerical model itself. Any state of the model can be converted into an estimate of the observations that would result if the atmosphere were in that particular state, by using the "observation function".
At ECMWF a data assimilation technique called "variational assimilation" is used. This method involves trying to find a model trajectory that leads to the closest match of the model to the actual observations that were made. The trajectory of the model is the path the model state traces out in time within the state space of the model. This space is a high dimensional space, defined by the millions of variables that describe the model state. Optimisation by variational assimilation is performed by trying to find the state of the model that leads to the best match with observations over the subsequent assimilation period. The state of the model somewhere in the middle of the assimilation period is then used as the analysis with which to initialise the forecast. A lower resolution, '1'159 model, is used for the data assimilation. Data assimilation is also used by forecasting centres to produce reanalysis
products. These products are historical reconstructions of the state of the atmosphere, projected into the model grid-point representation. They are constructed using similar techniques that are used to generate the analysis used to initialise forecasts. When producing a reanalysis, however, it is possible to use observations made after the time for which the state estimate is required, in addition to those from before, to estimate the model state. Because reanalysis products are reconstructions that are complete in time and space, they can be used to estimate weather at locations for which direct historical observations are not available. Reanalysis projects have been undertaken by ECMWFand NCEp-i
In the late 1950s the meteorologist, Edward Lorenz, was experimenting with a
numerical model of the atmosphere at the Massachusetts Institute of Technology
(MIT). During a set of experiments, he started a new run by resetting the state of the
model to the state obtained halfway through a previous run. To his surprise, the
behaviour of the atmosphere in second half of the new run was substantially different
from the second half of the initial run. Eventually Lorenz realised that, while the
computer was evaluating the model to six decimal-place accuracy, it was printing out
the model state to only three decimal-place accuracy, thus resetting the model with
the printed output had introduced a tiny discrepancy (less than one pa.rt in 1,000)
between the two runs. This error was enough to cause a big difference in their
evolutions (Lorenz 1993). While such behaviour had been known for centuries, the
existence of digital computing made such behaviour more amenable to study. In
1975 the word "chaos" was coined-to describe such sensitivity of these models to
initial conditions, a property popularly known as the "butterfly effect".
To study the model sensitivity to initial conditions, experiments in which artificial
errors, consistent with known observational uncertainty, are introduced into
computer models of the atmosphere. The model results using these alternative initial
conditions can then be compared with the original runs; the comparisons suggest
that it is unlikely that we will ever predict the precise evolution of the atmosphere
for longer than a few weeks (Lorenz 1982).
The impact of chaos on meteorology has not been entirely negative. Instead, it has led to a shift in emphasis - a shift likely to be useful to weather risk management
professionals. The atmosphere is not uniformly sensitive to initial conditions; its
sensitivity depends on its current state. On some days the atmosphere can be
approximated by a relatively predictable linear system over a short enough period of
time. On other days, the inevitable uncertainty that exists in the analysis can lead to
rapid error growth in the forecast. The key point is that the predictability of the
atmosphere depends upon the state of the atmosphere. Predicting predictability has
now become a major component of operational weather forecasting.
ENSEMBLE FORECASTING
Both the ECMWFand the US National Centre for Environmental Prediction (NCEP)
have been running daily ensemble forecasts since 1992. The idea behind ensemble
forecasting is simple; run several forecasts using slightly different initial conditions
generated around the analysis. The relative divergence of the forecasts indicates the
predictability of the atmospheric model in its current state - the greater the
divergence the lower the confidence in the forecast. Thus, ensemble forecasts should
provide a priori information on the reliability of that day's forecast. The difficulty
arises in deciding what perturbations (errors introduced into the initial condition) to
make to the analysis. Limited computing power means that only a few dozen
forecasts can be produced (even when the ensemble forecasts are produced using a
lower resolution model than the main forecasts).
The ensemble formation methods discussed above attempt to account for errors
in the initial condition. Other ensemble forecasts use models with different
combinations of parameterisation schemes for sub-grid processes, in an attempt to
estimate how the uncertainty in the choice of scheme affects the final forecast
(Stensrud 2000). Another way in which ensemble members may differ is by
introducing random terms into the dynamical equations of the model. This approach
attempts to model the uncertainty in the future evolution of the atmosphere at each
time step of the model. In such a stochastic parameterisation, the impact of sub-grid
processes on the resolved flow is not assumed to be a deterministic function of the
model state. Instead, it is a random variable. The parameters of the distribution from
which this variable is selected (such as its mean and variance) can be determined by
the model state.
Due to the non-linearity of NWP models, introduction of these stochastic terms
can actually improve the mean state of the model in addition to helping to assess the
uncertainty in its evolution (Palmer 2001). Stochastic parameterisation is a new
feature of numerical weather prediction, and is partly a manifestation of
meteorology's willingness to accept uncertainty and its attempt to quantify it.
For the purpose of pricing weather derivatives, ensemble forecasts are much
more useful than traditional single forecasts. For example, each member of an
ensemble forecast can be used to calculate the number of heating degree-days
(HDDs) accumulated in a period: this provides a rudimentary distribution of future
HDDs. At present, however, the relatively small size of the ensembles and the fact that
they represent quantities averaged over tens of kilometres rather than at individual
weather stations, means that using ensembles in this manner would be ill-advised.
We shall discuss how the predictability information contained in current ensemble
forecast might ~e extracted in the next section.
Once uncertainty in the atmospheric state has been accepted as a fact of life that
will not go away, the extension of medium-range forecasting techniques to longer range
seasonal forecasting is not a major leap. As noted above, the sensitivity of the
evolution of the atmosphere's state to its initial conditions prohibits the precise
prediction of the trajectory of this state for longer than, at best, a few weeks into the
future. This means that there is little hope of forecasting whether it will rain on a
specific day in a few months time. This does not mean, however, that useful forecasts
at lead times of several months are not possible. It is possible to predict whether a season will be wetter or colder than average at lead times of over three months and to estimate of the probability of magnitudes of change - the probability it will be atleast 1QCwarmer than average, for example.
The crucial extra ingredient required for seasonal forecasting is a computer
model of the ocean. The time scales on which the state of the ocean changes are
quite long compared to the lead time of a medium-range forecast. Because of this,
when making such a forecast, the state of the ocean can be held fixed. Beyond a
couple of weeks, however, the changing state of the ocean is an important influence
on the behaviour of the atmosphere. To make progress in seasonal forecasting, a
model of the ocean must be coupled to the atmosphere model. The atmosphere
forces the ocean through wind stress at its surface, while the ocean forces the
atmosphere by exchanging heat with it, especially through the evaporation of water
- which forms clouds - and radiation.
The sea surface temperature (SST) is an important influence on the behaviour of
the atmosphere, particularly in the tropics. It is the coupling of the ocean and
atmosphere that lies behind much inter-annual climate variation such as El Nino-
Southern Oscillation (ENSO).' ENSO is characterised by a large-scale cycle of
warming and cooling in the Eastern tropical Pacific that repeats on a time scale of 2-7
years. The warm SST of the El Nino phase of the cycle influences the atmospheric
circulation over large parts of the globe. In particular, El Nino events are associated
with heavy rainfall in Peru and Southern California, mild winters in the Eastern US
and drought in Indonesia and Northern Australia (Glantz 1996). These are all
probabilistic associations - ENSO is just one influence of the atmosphere's
behaviour, although an important one at lower latitudes (see Philander, 1990 for
further reading).
In many ways, the existence of ENSO is a blessing, it imposes some degree of
regularity on tropical climate that helps seasonal prediction. The numerical ocean
models used in seasonal forecasting are not fundamentally different from their
atmospheric cousins; they rely on the division of the oceans into finite elements,
horizontally and vertically; the equations of mass, momentum and energy
conservation are integrated numerically, and sub-grid processes are parameterised.
Seasonal forecasts at mid-latitudes are not as skilful as in the tropics. Although there
are thought to be mid-latitude climate cycles, such as the North Atlantic Oscillation
(NAO), they are not as regular and well detlned as ENSO. The existence of these
cycles has allowed the development of statistical seasonal forecasting models which
have skill at lead times of up to six months (eg, Penland and Margorian, 1993). Like
all statistical models, however, the availability of historical data is a major constraint
on their refinement. Improvements in seasonal forecasting will require better
information about tile state of the ocean. Observations of the ocean are not as dense
as atmospheric observations. Better ocean data, such as that obtained from the
TOPEX-POSEIDON satellite, and its successor jASON-1, which measure sea surface
height, should enable better estimation of the state of the ocean, and thus improved
seasonal forecasts6
Beyond seasonal time scales, forecasts that have more skill than climatology are
elusive. It is possible, however, that coupled numerical models of the ocean atmosphere
system can help to improve the climatologist distributions that are used
to assess weather risk. The instrumental records for most locations are quite short,
often only extending back a few decades. Extended runs of oceanic-atmospheric
general circulation models (GCMs) could enable better estimates of the risk of
extreme events that may only have occurred on a handful of occasions in recent
history. This is particularly true if secular changes to forcing of the ocean atmosphere
system - such as enhanced radiative forcing due to increased carbon
dioxide and other greenhouse gases (Harries et al. 2001, IPCC, 2001) - reduce the
relevance of the historical record. Before GCMs can be used for this type of risk assessment, it must be demonstrated that they can reproduce observed historical
climate variability on regional scales, not just average global temperatures.
Interpretation and Post-processing Model Output
The output generated by numerical models, should not be considered as "weather
forecasts". The output merely represents the state of the model, which contains
information about the weather but is not immediately relevant to quantities that are
actually observed. There are many ways in which extra processing of the raw forecast
products produced by forecast centres can substantially increase the value of these
forecasts. For example, even the highest resolution global models cannot resolve the
details of mountain topography or small islands, yet these physical features can have
a substantial impact on the local weather conditions. One of the ways in which
human forecasters can add value to a forecast is by using their experience of the
weather in a particular locale to predict the likely conditions there, given the larger
scale weather picture that the numerical forecast provides.
The finite resolution of numerical models, that is, the grid size, limits its
application to areas no smaller than a grid. The user of the model forecast, however,
is likely to need to know the values of forecast variables on a scale smaller than a
model grid: pricing a weather derivative may require the temperature at the London
Heathrow weather station, for example, but the ECMWF model predicts a
temperature that is averaged over a grid box of 40 km by 40 km. "Downscaling" is the
term used to describe a variety of quantitative methods that use the values of
forecasted model variables for estimating the values of specific variables on scales
smaller than the model grid.
One common method is the use of model output statistics (MOS) (see Glahn and
Lowry 1972). A small number of model variables is chosen as the set of predictors of
the desired variable. These predictors are extracted from past numerical forecasts,
and correlated with the corresponding observational record of the desired variable.
A statistical model can then be constructed that predicts the desired variable, using
the forecast variables as predictors. This statistical model should remove any
systematic biases in the numerical model. How MOS should be extended to
ensemble forecasts is not obvious.
One could use each member of the ensemble as the predictor in a traditional
statistical MOS model.
However, the forecast uncertainty represented by the ensemble must also be
downscaled. This is because the forecasts are averaged over grid boxes that are tens
of kilometres in size. The uncertainty of a forecast averaged over a grid box should
be lower than the uncertainty of a forecast at a single weather station (that averaging
reduces variability is a much exploited fact in both science and finance). So, while
traditional MOS can be used to downscale the mean of the ensemble, obtaining an
appropriate estimate of uncertainty is trickier. Methods for extracting the
predictability that exists in the ensemble forecasts are now being developed (eg,
Roulston et al., 2002).
Another method for downscaling a forecast is "nested modelling". This uses a
local area numerical model, covering a much smaller region than the global model,
but with a much higher resolution. The local model is "nested" inside the global
model. This means that it is integrated forward like the global model, but that on the
edges of the region it covers the values of its variables are obtained by using the
corresponding values from the global model (interpolated onto the local model's
higher resolution grid). An example of using a nested model in a forecast application
can be found in Kuligowksi and Barros (1999).
PANEL 1: COMPUTER MODELS: GRID POINT
VS SPECTRAL POINT
Computer models of the atmosphere come in two types: grid points and spectral
points. Grid point models represent the atmosphere as finite boxes centred on a grid
point and calculate the changes in mass, energy and momentum at each grid point
as a function of time. The size of the boxes determines the resolution of the model.
Spectral models represent the distribution of atmospheric properties as sums of
spherical harmonics. These harmonics are similar to sine and cosine functions, except
they are two dimensional andexist 01'1 the surface of a sphere. The resolution of a
spectral model is determined by the wavelength of the highest spherical harmonic
used in the model. Spectral models are denoted by labels such as 'Tfil l": this means
that the highest harmonic used has 5ll waves around each line of latitude. T5ll
model has a horizontal resolution of 1/(2 x 5ll) times the radius of the Earth (about
40 km). The state of the atmosphere, as represented in a spectral model, can be
converted to a grid-point representation using a mathematical transformation. Both
grid-point and spectral models divide the atmosphere vertically into layers.
PANEL 2: OBSERVING THE ATMOSPHERE
The World Meteorological Organization's World Weather Watch (WMO WWW)
manages the data from 10,000 land stations, 7,000 ship stations and 300 buoys fitted
with automatic weather stations. All these stations are maintained by National
Meteorological Centres.
The most important development in meteorological observation in the last 40 years
has been the advent of weather satellites. The WMO WWW incorporates data from a
constellation of nine weather satellites that provide global coverage. It is a testimony
to the importance of these satellites that weather forecasts now tend to be more skilful
in the southern hemisphere, where surface stations are quite sparse, than in the more
densely observed northern hemisphere. The fact that southern hemisphere skill is
actually slightly higher is probably because there is less land south of the equator.
Topography and other continental effects make the job of forecasting northern
hemisphere weather harder, even with the greater data coverage in the North.
PANEL 3: EVALUATION OF FORECASTS
When choosing forecast products to help manage their weather risk, users should
obviously be concerned with the skill level of the forecasts. Unfortunately, it is difficult
to assess skill when forecasting multiple variables with numerical models.
A common measure of skill used by meteorologists is the mean square error (MS
error). This is just the mean of the square of the difference between the forecasted
value and the observed value.
The mean can be an average over time for a univariate forecast (eg, temperature at
Heathrow airport), or a time-space average for a multivariate forecast (eg, the 500hPa
height field over the northern hemisphere). The MS error, however, is not a good way
to evaluate an ensemble, or probabilistic forecast. The ensemble average can be
calculated and assessed this way but such approach completely fails to take into
account the information about uncertainty inherent in the ensemble forecast,
information of great interest to risk managers.
Verifying probabilistic forecasting systems requires different measures of skill. The
class of measures most relevant to risk manages is probabilistic "scoring rules". To
use these rules the number of possible outcomes is usually a finite number of classes.
Examples of such classes might be rainlno rain or a set of temperature ranges. A
probabilistic forecast will assign a probability to each of the possible outcomes.
Scoring rules are functions of the probability that was assigned to the outcome that
actually occurred. If this probability is p then the quadratic score of the forecast is p_.
The logarithmic score is log p.7 The quadratic scoring rule forms the basis for the
"brier score" and the "ranked probability score" (Brier 1950). The logarithmic scoring
rule is connected to the information content of the forecast, and also to the returns a
gambler would expect if they bet on the forecast (Roulston and Smith, 2002).
Another method for evaluating probabilistic forecasts is the cost-loss score
(Richardson, 2000).
The cost-loss score is based on the losses that would be incurred by someone using
a probabilistic forecast to make a simple binary decision. Consider the situation where
the user must decide whether to grit the roads tonight. The cost of gritting is C. If it
does not freeze, no loss is suffered if the roads are not gritted. If it does freeze,
ungritted roads will lead to a loss, l. If P is the forecast probability of it freezing tonight,
then the expected loss if the user does not grit is pL. If p>C/l then this expected loss
exceeds the cost of gritting and a rational user would send out the gritting trucks. The
cost-loss score illustrates how important it is to have a probabilistic forecast. A user
with a C/l ratio far from 0.5 would be ill-advised to take the course of action suggested
by a best-guess forecast. Best-guess forecasts can be easily converted into
probabilistic forecasts by estimating the historical forecast error distribution (ie, the
errors of previous forecasts).
Such a conversion should always be performed before evaluating best-guess
forecasts using any skill score designed for probabilistic forecasts. Not doing so will
artificially inflate the advantage of using an ensemble system. Users that can formalise
their decision-making can directly estimate the value of forecasts by determining their
impact on decisions. Decision making processes for real users will be more complex
than the cost-loss scenario described above but, the principle of utility maximisation
should still apply. Katz and Murphy (1997) contains detailed articles on common
evaluation methods for weather forecasts.
Conclusion
Over the last 2,000 years, the prediction horizon of state-of-the-art weather forecasts
has advanced significantly from seeing one day ahead based on the colour of the sky
to almost a two-week outlook. The risk management community moved more
quickly, as in only a few years, the very concept of a weather forecast has changed
from a single best-guess of the future to a distribution of likely future weather
scenarios.
In this chapter the methods used to produce modern numerical weather forecasts
have been outlined. It was also claimed that the modelling techniques used for
making short-range and medium-range forecasts are not fundamentally different
from the approaches that must be adopted to forecast climate on seasonal scales and
beyond. The main difference is that, for longer-range forecasts, a model of ocean
dynamics must be coupled to the atmospheric model. This chapter has also stressed
the importance of ensemble forecasting, for quantifying forecast uncertainty.
Ensemble forecasting is on its way to becoming the standard approach to forecasting
the evolution of the atmosphere and the ocean. This is a welcome development for
the risk management community because ensemble forecasts contain information
about the uncertainty of forecasts, uncertainty that is synonymous with risk.
The ultimate aim of accurate probability forecasts of commercially relevant
variables is being pursued on time scales from a few hours to several months (Palmer
2002). For those who understand how to interpret it, this information will give a
competitive edge at a range of lead times, from pricing next week's electricity futures
to pricing next year's weather derivatives. Skilled and timely forecasts can be
financially rewarding.
2 All operational forecast centres frequently upgrade their forecast models. Details
of these upgrades, and of the present configuration of the ECMWF model can be
found on their website http://www.ecmwjint.
3 A description of the ECMWF and its output products can be found in Persson
(2000).
4 Details of reanalysis projects and data can be found at
http://ecmwjint/research/era/ and http://www.cdc.noaa.gov/ncePJeanalysis.
5 As noted above Gilbert Walker discovered the atmospheric Southern Oscillation
in the 1920s. The relationship between this variation and the oceanic El Nino
phenomenon was discovered in the 1960s byjacob Bjerknes, son ofVilhelm.
6 Information on these satellites is available at http://topex-wwwjpl.nasa.gov/
mission/jason-1. html.
7 Readers may be curious as to why the simplest linear skill score =p has been
omitted. It turns out that this score is "improper" in the sense that a forecaster
increases their expected score by reporting probabilities that differ from what they
believe to be true.
BIBliOGRAPHY
Brier, G. w., 1950, "Verification of Forecasts Expressed in Terms of Probabilities", Monthly Weather
Review, 78, pp. 1-3.
van den 0001, H. M., 1994, "Searching for Analogues, How Long must we Wait?" Tellus A, 46, pp.
314-24.
Glahn, H. R., and Lowry, D. A., 1972, "The Use of Model Output Statistics (MOS) in Objective Weather
Forecasting",}ournal of Applied Meteorology, 11, 1203-11.
Glantz, M. 11., 1996, Currents of Change: El Nino's Impact on Climate and Society, (Cambridge
University Press).
Harries,]. E., Brindley, H. E., Sagoo, P.]. and Bantges, R.]., 2001, "Increases in Greenhouse Forcing
Inferred from the Outgoing Longwave Radiation Spectra of the Earth in 1970 and 1997", Nature, 410,
pp. 355-7.
IPCC, Intergovernmental Panel on Climate Change, 2001, Climate Change 2001: Synthesis Report,
(cd) R. T. \Vatson, (Cambridge University Press).
Katz, R. w., and Murphy, A. H., 1997, Economic Value of Weather and Climate Forecasts, (Cambridge
University Press).
Kuligowski, R. ]. and A. P. Barros, 1999, "High-Resolution Short-Term Quantitative Precipitation
Forecasting in Mountainous Regions Using a Nested Model", Journal of Geophysical Research
(atmospheres), 104, pp. 31, 553-64.
Lempfcrt, R. G. K, 1932, "The Presentation of Meteorological Data", Quarterly Journal of the Royal
Meteorological SOCiety,58, pp. 91-102.
Lorenz, E. N., 1982, ''Atmospheric Predictability Experiments with a Large Numerical Model", Tellus, 34,
pp. 505-13.
Lorenz, E. N., 1993, The Essence of Chaos, (London: UCl Press).
Murphy, A_ H_, 1997, "Forecast Verification", in: Economic Value of Weather and Climate Forecasts,
(eds) R. \'if Katz and A. H. Murphy, (Cambridge University Press), pp. 19-70.
Nebeker, F., 1995, Calculating the Weather, (San Diego: Academic Press).
Palmer, T. N., 2000, "Predicting Uncertainty in Forecasts of Weather and Climate", Reports Oil Progress
in Physics, 63, pp. 71-116.
Palmer, T. N., 2001, "A Nonlinear Dynamical Perspective on Model Error: A Proposal for Non-Local
Stochastic-Dynamic Paramcterization in Weather and Climate Prediction Models", Quarterly journal 0/
the Royal Meteorological Society, 127, pp. 279-304.
Palmer, T. N., 2002, "The Economic Value of Ensemble Forecasts as a Tool for Risk Assessment: From
Days to Decades", Quarterly Journal 0/ the Royal Meteorological Society, 128, pp. 747-74.
Penland, c., and T. Magorian, 1993, "Prediction of Nino 3 Sea Surface Temperature Using Linear
Inverse Modeling",Journal of Climate, 6, pp. 1067-76.
Persson, A., 2000, "User Guide to ECMWF Forecast Products", European Centre for Medium Range
Weather Forecasting, Reading.
Philander, S. G., 1990, El Nino, La Nina, and the Southern Oscillation, (San Diego: Academic Press).
Richardson, L. F., 1922, Weather Prediction by Numerical Processes, (Cambridge University Press).
Richardson, D. S., 2000, "Skill and Relative Economic Value of the ECMWF Ensemble Prediction
System", Quarterly journal of the. Royal Meteorological. Society, 126, pp. 649-67.
Roulston, M. S. and L. A. Smith, 2002, "Evaluating Probabilistic Forecasts using Information Theory".
Monthly Weatber Review, 130, pp. 1653-60, 2002.
Roulston, M. S., D. T Kaplan, J. Hardenberg and L. A. Smith, 2002, "Using Medium Range Weather
Forecasts to Improve the Value of Wind Energy Production", Forthcoming, Renewable Energy
Smith, L. A., Roulston, M. S. and lIardenberg, J., 2001, "End to End Ensemble Forecasting: Towards
Evaluating the Economic Value of the Ensemble Prediction System", ECMWF Technical Memorandum
336, ECMWF, Reading.
Stensrud, D. J., J. W Bao, and T. T. Warner, 2000, "Using Initial Condition and Model Physics
Perturbations in Short-Range Ensemble Simulations of Mesoscale Convective Systems", Monthly
Weather Review, 128, pp. 2077-107.
Walker, G. T., 1923, "Correlation in Seasonal Variations of Weather VIII", Memorandum of the Indian
Meteorological Deptartment, 24, pp. 75-131.
Walker, G. T., 1924, "World Weather IX", Memorandum of the Indian Meteorological Dcptartmcnt, 24,
pp. 275-332.
Walker, G. T., 1928, "World Weather Ill", Memorandum of the Royal Meteorological Dcptartment, 17,
pp. 97-106.
Walker, G. T., 1937, "World Weather VI", Memorandum of the Royal Meteorological Society, 4, pp.
119-39.
RISK
Feel free to write