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2.9 — How a Forecast Is Made
In 1922 an English mathematician named Lewis Fry Richardson published a book describing an attempt he had made, mostly while serving as an ambulance driver in the First World War, to calculate a weather forecast by hand. He took observations from a single day in 1910 over central Europe, divided the region into a grid, and worked the equations of fluid motion forward step by step on paper.
It took him six weeks to produce a six-hour forecast, and the answer was wildly wrong — his calculation predicted a pressure change of 145 millibars where nothing of the sort occurred. The error came from the starting data, not the method, and this was worked out decades later.
Richardson knew the method was too slow to be useful, and he did the arithmetic on what it would take: he estimated that 64,000 people working with mechanical calculators in a vast hall, each responsible for one part of the grid, with a conductor directing them by coloured light, could just about keep pace with the weather.
He was right about the method. He was only wrong about the machinery, and the machinery arrived thirty years later.
The problem: seven equations and no exact answer
The atmosphere is a fluid on a rotating sphere, and its behaviour is fully described by a set of physical laws that were known by the end of the nineteenth century:
- Conservation of momentum in three directions — Newton's second law applied to a parcel of air, including the pressure gradient force, gravity and the Coriolis effect from Chapter 2.2.
- Conservation of mass — air is not created or destroyed, so if it converges somewhere it must rise or sink.
- Conservation of energy — the first law of thermodynamics, tracking heating, cooling and the latent heat of Chapter 2.3.
- The gas law, relating pressure, density and temperature.
- Conservation of water in all three phases.
Seven equations, seven unknowns, complete in principle. The Norwegian meteorologist Vilhelm Bjerknes stated in 1904 that weather forecasting was therefore a problem of initial conditions plus computation, which was correct and completely useless at the time.
The equations cannot be solved exactly. They are non-linear — the wind carries temperature around, and the temperature pattern determines the wind — so there is no formula you can write down. They can only be solved numerically, by chopping space and time into small pieces and stepping forward.
How the numerical method actually works
Divide the atmosphere into a three-dimensional grid. Each box gets a value for pressure, temperature, humidity, and the three components of wind. A modern global model uses horizontal boxes of about 9 to 14 kilometres and 50 to 137 vertical levels, giving hundreds of millions of grid points.
Then step forward in time. For each box, compute the rate of change of each variable from the equations, using the current values in that box and its neighbours. Add that change over a short time step — typically a few minutes — and you have the state a few minutes later. Repeat, several thousand times, to get ten days.
The time step is not free to choose. If air can cross a grid box in less time than one step, the calculation becomes unstable and blows up — the same failure Richardson met. The rule limiting this is called the Courant condition, and it means halving the grid spacing forces you to halve the time step too. Halving the horizontal spacing multiplies the work by roughly sixteen: four times as many boxes horizontally, twice as many time steps, and usually more vertical levels. This is why forecast resolution has improved slowly despite computers improving enormously.
Parameterisation: the honest weak point
Many of the most important processes are smaller than a grid box, so they cannot be computed directly. A cumulus cloud is a kilometre across. A raindrop is millimetres. Turbulence near the ground is centimetres. None of it fits in a 10-kilometre box.
So these processes are parameterised — represented by formulas that estimate their average effect on the box as a whole from the variables the model does carry. If a box is warm, moist and unstable, the convection scheme estimates how much rain and how much heating that box should produce, without simulating a single cloud.
This is where forecast error mostly comes from, and it should be said plainly. Convection, cloud microphysics, boundary-layer turbulence and the effect of the land surface are all parameterised, and each scheme is a defensible approximation rather than a law of nature. Different models make different choices, which is a major reason they disagree.
Higher-resolution regional models running at 1 to 3 kilometres can resolve individual storms directly rather than parameterising them, which is why short-range thunderstorm forecasting improved sharply once such models became affordable.
The observations: where the starting numbers come from
A model is only as good as the state it starts from. Assembling that state, called data assimilation, is at least as difficult as the forecast itself.
Roughly hundreds of millions of observations arrive every day:
- Satellites provide by far the largest share — most of it not pictures but radiances, which are measurements of infrared and microwave energy at particular wavelengths from which temperature and humidity profiles are inferred. Polar-orbiting and geostationary satellites together give global coverage. GPS radio occultation is a particularly elegant trick: a satellite listens to a GPS signal passing through the atmosphere on its way from another satellite, and the bending of that signal reveals the temperature and humidity profile along the path.
- Radiosondes — balloons launched twice daily from around 800 stations worldwide, still the most accurate vertical profiles available.
- Surface stations, ships, buoys and aircraft. Commercial aircraft report automatically, which is why forecast quality measurably degraded during the 2020 aviation shutdown.
- Weather radar for precipitation structure, and Doppler radar for wind within storms.
Assimilation is not simply loading the data in. The observations are irregular, of varying quality, and cover only a fraction of the grid points. So the model's own short forecast from a few hours earlier is used as a first guess, and the new observations nudge it, weighted by how much each is trusted. The result is the best estimate of the current atmosphere given both the physics and the measurements, and modern schemes do this over a time window rather than at an instant.
Why forecasts fail: chaos, precisely stated
In 1961 Edward Lorenz was rerunning a simple weather model at MIT and, to save time, restarted it from the middle using printed values rounded to three decimal places rather than the six the machine held. The new run diverged completely from the old one within a simulated couple of months.
He had found deterministic chaos: a system that is entirely governed by fixed rules, with no randomness anywhere, in which arbitrarily small differences in starting conditions grow exponentially until the two futures have nothing in common. His 1972 talk title gave it the popular name — "Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?"
What this means for forecasting, concretely. However good the model and however dense the observations, the initial state is never exactly right. That error doubles roughly every day or two in the atmosphere. After about two weeks it has grown to the size of the weather itself, and the forecast contains no information.
This is a hard limit set by physics, not by computing power. A perfect model on an infinite computer would still hit it, because you would still need infinitely precise measurements everywhere. The practical horizon for useful day-to-day detail is around 7 to 10 days.
It does not apply to climate, and this distinction is the one most often muddled. Weather forecasting asks what the atmosphere will do on a specific day, which requires tracking a particular trajectory. Climate projection asks what the statistics will look like — the average, the distribution, the frequency of extremes — under changed conditions. You cannot say whether it will rain in Nagpur on 3 August next year, and you can say with confidence that Nagpur's August will be hotter than its January. A pot of water on a stove has chaotic, unpredictable eddies and an entirely predictable response to turning up the heat.
Ensembles: forecasting the uncertainty
Since a single forecast is one trajectory out of many possible ones, modern practice runs many.
An ensemble is 20 to 50 runs of the model, each started from a slightly different initial state within the range of observational uncertainty, and often with slightly different physics. Then you look at the spread.
If all members agree, confidence is high. If they diverge wildly by day four, the atmosphere is in a state where small differences matter and confidence is low. The ensemble forecasts its own reliability, which is the single most useful thing about it.
This is where probabilities come from, and what they mean. "70 percent chance of rain" is a statement about frequency: on all the occasions when the forecast system says 70 percent, it should rain on about 70 percent of them. That is a testable claim, and forecast centres test it continuously; a system that is right 90 percent of the time when it says 70 percent is badly calibrated and gets corrected.
A common misreading worth killing. "70 percent chance of rain" does not mean 70 percent of the area will get rain, and it does not mean rain for 70 percent of the day. Some services do publish an area-weighted version, so it is worth knowing which your source uses.
How good is it now?
Better than almost anyone realises, and the improvement is measurable and steady.
The standard benchmark is the anomaly correlation of the 500 hPa height field, which is a measure of how well the model captures the pattern of ridges and troughs steering the weather at about 5.5 kilometres altitude. A five-day forecast today is about as accurate as a one-day forecast was in 1980. Forecast skill has improved by roughly one day of lead time per decade for four decades.
And the improvement in the southern hemisphere has been dramatic, because it was starved of surface observations before satellites, and is now nearly as well forecast as the north.
Where forecasting is still weak, stated honestly: precipitation amount and timing, especially convective rain; fog; the exact track and intensity of tropical cyclones, particularly rapid intensification; and anything at the scale of an individual thunderstorm more than a few hours out.
Machine learning has arrived and it is a genuine change. Since about 2022, models trained on decades of reanalysis data — Huawei's Pangu-Weather, Google DeepMind's GraphCast and successors, and systems now being run operationally at the European Centre — have matched or beaten conventional physics-based models on many standard measures, at a tiny fraction of the computing cost. They do not solve the equations; they learn the mapping from one atmospheric state to the next. They still depend entirely on the physics-based assimilation system to tell them what today looks like, and their behaviour under conditions outside their training data is an open question. This is a fast-moving area and this book states where it stands rather than predicting where it lands.
The Indian system
The India Meteorological Department, founded in 1875 after a cyclone and a famine made the case for it, is one of the oldest weather services in the world. It runs a network of surface observatories, upper-air stations, over thirty Doppler weather radars, and its own satellites in the INSAT series. The National Centre for Medium Range Weather Forecasting runs the global model, and the Indian Institute of Tropical Meteorology in Pune does the monsoon research.
Where the challenge is specifically hard. Monsoon convection is the largest heat engine in the tropics and its representation in models is the weakest link (Chapter 2.5). Terrain from sea level to 8,000 metres within a few hundred kilometres strains any grid. And the users who need it most — small farmers — need forecasts at village scale, days ahead, for rainfall amount, which is precisely the hardest quantity.
The successes are real and worth naming. Cyclone track and landfall forecasting has become good enough to evacuate a million people accurately (Chapter 2.4). Heat-action plans triggered by forecasts, first in Ahmedabad after the 2010 heatwave and now in many cities, have measurably reduced deaths. Agricultural advisory services deliver district-level guidance by mobile phone to tens of millions of farmers.
Where this shows up in your life
Reading a forecast properly. Trust the first two or three days for detail. Treat days four to seven as trend rather than schedule. Ignore day-ten detail entirely — if a site shows you an hourly forecast for two weeks out, it is generating numbers, not information. Check whether your source gives a probability and whether it gives an ensemble spread; the spread is the honest part.
Why apps disagree. They are usually running different models — the American GFS, the European ECMWF, or a national model — with different resolutions and different parameterisations, and post-processing them differently. Disagreement between them is itself a signal that confidence should be low.
And the value of it. Weather forecasting is one of the clearest cases anywhere of pure science paying for itself many times over. Aviation, shipping, agriculture, construction, energy trading, and above all disaster warning all run on it. The 1970 Bhola cyclone killed hundreds of thousands of people. Storms of comparable strength in the same region now kill dozens. That difference is Richardson's arithmetic, done by machines, delivered by radio.
What Part 2 established
The Sun heats the tropics harder than the poles, the atmosphere and ocean redistribute that heat, and the Earth's rotation bends every flow. Out of that come the wind belts, the deserts at 30 degrees, the rainforests at the equator, the storms, the monsoon and the currents. Water carries energy from where it evaporates to where it condenses, and that transfer powers everything violent in the sky. Orbital rhythms move the whole system between glacial and interglacial states, and we live in an unusually calm interglacial that the entire human record fits inside.
What the next Part covers
That calm interglacial is where people come in. Part 3 goes back to before the calm — through the two million years in which our ancestors became human, learned to use fire, made tools, invented language and art, walked to every continent, and survived an ice age. It then covers the single most consequential change in the human story: the moment, in at least eleven separate places, when people stopped following food and started growing it — what that gained them, what it cost them, and how it produced villages, cities, priests, kings, taxes and writing within a few thousand years.