Arquivo da tag: Probabilidades

What Is Entropy, Really? (Wired)

wired.com

Never mind the “heat death of the universe.” Entropy is just a bridge between two levels of reality.

Original article

Rhett Allain

Jul 24, 2026 7:00 AM


Entropy is one of the most maligned and misunderstood concepts in science. Maybe you’ve heard it defined as the “amount of disorder” in a system. And the second law of thermodynamics says the entropy of a closed system always increases over time. So you might think, why should you clean up your office if it will only get messier?

That might be true, but you can’t blame it on entropy. The messy-room metaphor is often used to introduce the idea (it’s usually a teenager’s bedroom—can you relate?), but it’s misleading. See, disorder doesn’t mean messiness or chaos; it refers to the number of ways the parts of a system can be arranged without changing the overall state of the system.

For example, say your “system” is just a box full of air. Inside, at the microscopic level, the gas molecules are bouncing around like bumper cars. Now, imagine you could map the location and velocity of each particle at a given instant. That would be one possible arrangement, or microstate, but there are an infinitude of others, and they’re changing trillions of times a second.

Of course, you can’t really see this stuff. Instead, what you observe are overall, macro-level properties like air pressure; if you sealed the box at sea level, that would be 14.7 pounds per square inch. And unless you add energy to the system, say by heating it, that doesn’t change. So all those microstates correspond to the macrostate of 14.7 psi. Get it?

In other words, entropy is all about the link between the invisible atomic realm and the visible, measurable realm of objects, the world we inhabit. You could say it’s a conceptual and mathematical bridge between two levels of reality. I mean, c’mon, that’s pretty cool.

Now, out of all possible outcomes, which ones actually occur? That’s basically random, so it’s a matter of probability. In fact, probability is fundamental to the idea of entropy, and this is what the messy-room image fails to capture. So I’m going to use a different analogy: rolling dice. Einstein once said “God doesn’t play dice with the universe.” Let’s just see about that, shall we?

Rolling the Bones

Imagine you roll a six-sided game die. You get a number from 1 to 6, right? There are six possible outcomes, or states. If you roll the 20-sided die in Dungeons & Dragons, there are 20 possible states. If you want to wow your D&D pals, you could casually remark that this die has a higher entropy—because it has more possible outcomes.

Now say you’re determining a character’s abilities in D&D, and you roll three six-sided dice. The three values can add up to anything between 3 and 18, but the various sums are not equally likely. For maximum dexterity, say, you need an 18. Well, there’s only one way to achieve that: Each die must come up a 6.

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But if moderate dexterity is enough for you, you might be fine with a level 10. That’s easier to get, because there are more combinations that add up to 10—six unique sets of numbers to be exact:

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And if we roll the dice one at a time and take sequence into account, there are even more permutations. Take 6-3-1 on the left. You could get the same three values in five other ways: 1-6-3, 3-1-6, 3-6-1, 6-1-3, 6-3-1. (Yes, these are different outcomes because time’s arrow moves in one direction.) All in all, there’s 27 different ways to roll a 10.

If we look at all possible results for three dice, there are 216 distinct microstates. But what matters for the game is the sum of the three values—that’s our macrostate. So the odds of rolling an 18 are 0.4 percent (1 out of 216), while the odds of rolling a 10 are 12.5 percent (27 out of 216).

We can say the 10 state has a higher entropy because there are more ways it can occur. And because there are more ways it can occur, it’s more likely to occur. See? There’s no mysterious force increasing the entropy of a system. It’s just that states with higher entropy have a higher probability. It’s actually kinda simple, really.

Opposite World

Now let’s think about this in terms of energy. Say you take a glass of cold water with a temperature of 50 degrees Fahrenheit, and you drop a hot, 120-degree ball of copper into it. What happens? Well, from experience, you’d expect the water to get warmer and the ball to get cooler, until they equalize at some temperature between 50 and 120 degres. That’s called thermal equilibrium.

But what do we mean when we say something gets warmer? We mean that its atoms and molecules increase in kinetic energy—they get more jiggly. Say the water gains 50 joules of thermal energy. Then, since energy is always conserved, we know that the copper ball cools off, losing the same 50 joules of energy.

But wait. What if, on a particular Tuesday, you dropped the hot ball in the cold water and the ball got hotter, increasing in thermal energy by 10 joules, while the water lost 10 joules and got colder? Did you just break physics? Nope. Energy is still conserved. You might find this disturbing, but it could happen. Why? Entropy. It’s one possible distribution of energy—just an extremely unlikely one. Basically, you won the Lotto.

An Object Lesson

Now imagine you have a tiny little solid. It’s so tiny, it has only three atoms. (Remember the three-dice analogy?) Quantum mechanics tells us that atoms can only have certain energy levels—just like a die can roll a 2 or a 3 but not a 2.5. The point is that if the total energy of these three atoms is 10 units, then as we saw, there are 27 ways those 10 units of energy can be distributed.

Let’s take this just one step further: Say we have two tiny objects, A and B, with different amounts of thermal energy. Object A consists of two atoms (dice) and has a total energy of 3 units. B has three atoms (dice) and 7 units of energy. This puts the total energy in the system (A and B) at 10 units. Here, a picture will help:

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Now suppose we put objects A and B in contact so that thermal energy can transfer between them. This means there could be many different arrangements of energy in the system, as long as the total remains 10 (for conservation of energy). Here are four of the many possible combos:

Image may contain Game Electronics Mobile Phone Phone and Domino

What if object A has just 2 units of energy? There’s only one way this will work—each die must be a 1 (since 1 + 1 = 2). That means object B must have a total of 8, and there are 21 ways that could happen. What if A has 4 units and B has 6 units? In this case there are 30 possible combinations. You could say this state is “less ordered” or “more disordered,” and it’s more likely.

The Law of Large Numbers

This leads us to one of the definitions of entropy as a measure of the number of ways you can arrange energy in a system. We can write entropy (S) as the natural log of the number of microstates (Ω) multiplied by Boltzmann’s constant (kb):

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Of course, our model with objects A and B is silly because most things aren’t made of two or three atoms. For a reality check, a single drop of water contains about 1.7 sextillion (1.7 x 1021) H2O molecules. So what happens when we scale up the numbers by many orders of magnitude?

The same ideas still hold, but now we have a vastly larger number of microstates, and the probability distribution gets way more concentrated around the one with the highest entropy. So while the hot ball could get hotter in cold water, it’s so unlikely that it has never happened.

And the state of thermal equilibrium, where they end up at the same temperature, is so entirely likely that, in practice, we treat it as a certainty. In fact, the second law of thermodynamics says heat always flows from a hotter object to a cooler object. But it isn’t a “law”; it’s just the odds. Turns out God really does play dice with the universe.

Como é calculada chance de chuva que serviços de meteorologia divulgam (Folha de SP/BBC)

www1.folha.uol.com.br

Quanto mais específicos e precisos forem os dados atmosféricos coletados na área observada, mais precisa será a probabilidade.

Rafael Abuchaibe

29 de abril de 2023


Se você é daqueles que não sai de casa sem antes conferir a previsão do tempo, certamente já se perguntou por que a porcentagem de chuva oferecida pela maioria dos serviços de meteorologia nem sempre corresponde ao que você vê pela janela.

“Porque representa as chances de chover na sua cidade“, alguém já deve ter respondido, quase surpreso com o quão básica parecia ser a resposta à sua pergunta:

“E as estatísticas nunca são 100% precisas.”

Outros, tendo indagado um pouco mais sobre o assunto, podem ter dito que o que o percentual representa é a área do território em que vai chover durante um determinado período de tempo (por exemplo, “das 9h às 12h”).

E para colocar mais lenha na fogueira, você deve ter visto alguns vídeos do TikTok explicando que o que a porcentagem reflete é a certeza dos meteorologistas de que vai chover em uma determinada área, com base em medições de fatores como temperatura, pressão atmosférica e velocidade do vento.

Diante de explicações tão variadas e distintas para algo que parece ser tão simples, a BBC News Mundo, serviço de notícias em espanhol da BBC, resolveu buscar uma explicação mais exata para o que aquele número representa —e percebeu que, de certa forma, todo mundo tem razão.

Probabilidade de precipitação

Para poder estabelecer o que essa porcentagem realmente significa, vamos começar revisando a definição dada pelo Serviço Meteorológico dos EUA:

“A probabilidade de precipitação representa simplesmente a probabilidade estatística de que haja 0,01 polegadas [0,25 mm] ou mais de precipitação [seja chuva, neve ou granizo] em uma determinada área dentro do período de tempo especificado.”

A porcentagem leva em consideração diferentes fatores para expressar em um valor estatístico a probabilidade de ocorrer precipitação em um determinado ponto.

“Vejamos um exemplo do que essa probabilidade significa”, diz o serviço meteorológico em sua definição.

“Se a previsão para um determinado distrito diz que há 40% de probabilidade de chuva para esta tarde, isso significa que há 40% de chance de chover em algum lugar do distrito entre meio-dia e 18h”, acrescenta.

Com base nessa definição, quanto mais específicos e precisos forem os dados atmosféricos coletados na área observada, mais preciso será o percentual de probabilidade.

Isso explica por que os dados fornecidos por diferentes serviços meteorológicos variam (embora não muito).

Duas medições, mesmo resultado

Para poder fazer uma previsão, um analista meteorológico multiplica dois fatores: a certeza que tem de que um sistema de precipitação vai se formar ou se aproximar, calculado por meio de medições atmosféricas, pela extensão —área física— que se espera que tal precipitação tenha no território analisado.

A esse resultado, basta mover duas casas decimais, e a probabilidade de precipitação é obtida.

Isso indica que é possível chegar à mesma porcentagem de precipitação tendo valores diferentes para cada fator.

Para ver essa ideia na prática, vamos voltar ao nosso exemplo do distrito com 40% de probabilidade de precipitação: se um analista tivesse 80% de certeza de que vai chover naquele distrito (medindo a velocidade do vento, a temperatura do ar, a umidade etc.), mas só espera que o sistema de precipitação cubra 50% da área, ele dirá que há uma “probabilidade de 40% de chuva” durante esse período de tempo.

Por outro lado, se outro analista estimasse que a precipitação iria cobrir 100% da área analisada, mas só tivesse 40% de certeza de que essa precipitação iria atingir o distrito, ele obteria o mesmo resultado: “40% de probabilidade de chuva em qualquer ponto do distrito durante esse período de tempo.”

Pequenas variações entre os sistemas

Cada meteorologista terá seus próprios modelos de medição e coleta de dados para calcular a probabilidade de precipitação nos locais que analisa —e alguns serão mais precisos que outros.

O importante é identificar o quão precisos são os métodos de coleta de dados atmosféricos que cada serviço possui na área específica em que você se encontra, algo que pode ser feito comparando-os e analisando qual deles se adequa melhor à realidade que você observa pela janela.

E, claro, não se pode esquecer que, por se basear em modelos probabilísticos, a meteorologia está longe de ser infalível.

Se você confiar apenas na previsão do tempo, é inevitável que um dia, por melhor que seja o sistema que usa, você saia de casa sem guarda-chuva com base na previsão do aplicativo —e seja pego na rua por um temporal.

Este texto foi originalmente publicado aqui.

How can I become a fossil? (BBC Future)

How to be fossilized (Credit: Getty) Less than one-10th of 1% of all species that have ever lived became fossils. But from skipping a coffin to avoiding Iran, there are ways to up your chances of lasting forever.

By John Pickrell

15 February 2018

Every fossil is a small miracle. As author Bill Bryson notes in his book A Short History of Nearly Everything, only an estimated one bone in a billion gets fossilised. By that calculation the entire fossil legacy of the 320-odd million people alive in the US today will equate to approximately 60 bones – or a little over a quarter of a human skeleton.

But that’s just the chance of getting fossilised in the first place. Assuming this handful of bones could be buried anywhere in the US’s 9.8 million sq km (3.8 million square miles), then the chances of anyone finding these bones in the future are almost non-existent.

Fossilisation is so unlikely that scientists estimate that less one-tenth of 1% of all the animal species that have ever lived have become fossils. Far fewer of them have been found.

As humans, we have a couple of things going for us: we have hard skeletons and we’re relatively large. So we’re much more likely to make it than a jellyfish or a worm. There are things, however, you can do to increase your chances of success.

Taphonomy is the study of burial, decay and preservation – the entire process of what happens after an organism dies and eventually becomes a fossil. To answer the question of how to become a fossil, BBC Future spoke with some of the world’s top taphonomists.

1. Get buried, and quickly

“It’s really a question of maintaining a good condition of the body after death – long enough to be buried under sediment and then altered physically and chemically deep underground to become a fossil,” says Sue Beardmore, a taphonomist and collections assistant at the Oxford University Museum of Natural History.

“To be preserved for millions of years, you must also survive the first hours, days, seasons, decades, centuries, and thousands of years,” adds Susan Kidwell, a professor at the University of Chicago. “That is, you must survive the initial transition from the ‘taphonomically active zone’… to a zone of permanent burial, where your remains are unlikely to be exhumed.”

There are almost endless ways that fossilisation can fail. Many of these happen at, or down to 20-50cm below, the soil or seafloor surface. You don’t want your remains to be eaten and scattered by scavengers, for example, or exposed to the elements for too long. And you don’t want them to be bored into or shifted around by burrowing animals.

The sand and mud deposits of Canada’s Badlands quickly buried bones The sand and mud deposits of Canada’s Badlands quickly buried bones, making the area one of the world’s richest hunting grounds for dinosaur fossils (Credit: Getty)

When it comes to rapid burial, sometimes natural disasters can help – such as floods that dump huge amounts of sediment or volcanic eruptions that smother things in mud and ash. “One theory for the occurrence of dinosaur bone beds is firstly drought conditions, that killed the dinosaurs, followed by floods that moved the sediments to bury them,” Beardmore says.

Of course, the fact that human bodies are typically buried six feet under (unless cremated) gives you another leg up here. But that isn’t enough on its own.

2. Find some water

Obviously the first step is dying, but you can’t die just anywhere. Picking the perfect environment is key. Water is one important thing to consider. If you die in a dry environment, once you’ve been picked over by scavengers, your bones will probably weather away at the surface. Instead, most experts agree you need to get swiftly smothered in sand, mud and sediments – and the best places for that are lakes, floodplains and rivers, or the bottom of the sea.

“The palaeoenvironments that we often see the best fossils come out of are lake and river systems,” says Caitlin Syme, a taphonomist at the University of Queensland in Brisbane, Australia. The important thing is the rate at which fresh sediments are burying things. She recommends rivers flowing from mountains which cause erosion and therefore carry a lot of sediment. Another option is a coastal delta or floodplain, where river sediment is rapidly dumped as the water heads out to sea.

Ideally, you also want an ‘anoxic’ environment: one very low in oxygen, where animals and microorganisms that would digest and disturb your remains can’t survive.

Kidwell recommends avoiding about 50cm below the seafloor, “the maximum burrowing depth of shrimp, crabs and worms that might irrigate the sediments with oxygenated water”, which would promote decomposition and stir up the body.

“You want to end up quickly after death in a spot that is relatively low elevation, so that it is a sink for sediment, and preferably with standing water – a pond, lake, estuary or ocean – so that anoxic conditions might develop,” she says.

A 150 million year old archaeopteryx (Credit: Getty)

Choose the right conditions and you, too, could be preserved for as long as this 150 million year old archaeopteryx (Credit: Getty)

In rare cases, fossils created in these kind of still, anoxic conditions preserve their soft tissues like skin, feathers and internal organs. Examples include the many exquisite feathered dinosaurs from China or the Bavarian quarries that produced the fossils of the earliest bird, archaeopteryx.

Once your fossil gets below the biologically active surface layer, then it’s stable and will continue to be buried more deeply as further sediments accumulate, Kidwell says. “The risk for destruction then shifts to a completely different geological timescale, namely that of tectonism.”

The question, then, is how long before the sediments encasing the corpse are turned to more permanent stone… and are lifted by geological activity to a height where erosion can expose the remains.

3. Skip the coffin

Now we come to the thorny technicality of what a fossil actually is – and what kind of fossil you want your body to be.

Very generally, anything up to around 50,000 years old is what’s known as a ‘subfossil’. These are largely still made up of the original tissues of the organism. Extinct Pleistocene megafauna found in caves – such as giant ground sloths in South America, cave bears in Europe, and marsupial lions in Australia – are good examples.

However, if you want your remains to become a fossil that lasts for millions of years, then you really want minerals to seep through your bones and replace them with harder substances. This process, known as ‘permineralisation’, is what typically creates a fully-fledged fossil. It can take millions of years.

As a result, you might skip the coffin. Bones permineralise most rapidly when mineral-rich water can flow through them, imbuing them with things like iron and calcium. A coffin might keep the skeleton nicely together, but it would interfere with this process.

There is a way a coffin might work, though. Mike Archer, a palaeontologist at the University of New South Wales, suggests burial in a concrete coffin filled with sand and with hundreds of 5mm holes drilled into the sides. This then needs to be buried deep enough that groundwater can pass through.

“If you want to be a classic bony fossil, a bit like something from Dinosaur Provincial Park in Canada, then something like a [coarse] river sand would be pretty good,” says Syme. “All the soft tissues would be destroyed and you’d be left with this beautifully articulated skeleton.”

In terms of the minerals, calcium ions which can precipitate into calcite, a form of calcium carbonate, are especially good. “These can start to cement or cover the body which will protect it in the long run, because given time it will most likely be buried at a greater depth,” Syme says.

Deliberately seeding your corpse with the appropriate minerals, such as calcite or gypsum, might be a way to accelerate this. Encouraging the growth of tough iron-rich minerals would also be sensible as they withstand weathering well in the long run.

If you want to personalise your fossil further, add colour with some copper

If you want to personalise your fossil further, add colour with some copper (Credit: Alamy)

Silicates, from the sand, are also a nice durable mineral to have incorporated. Archer even suggests getting buried with copper strips and nickel pellets if you fancy fossilised bones and teeth with a nice blue-green colour to them.

4. Avoid the edges of tectonic plates

If you made it through the first few hundred thousand years and minerals begin to replace your bones, congratulations! You’ve successfully become a fossil. As sediments build up on top and you get pushed deeper into Earth’s crust, the heat and pressure will aid the process further.

But it’s not a done deal yet. Your fossil might still shift to such depths that it could be melted by the Earth’s heat and pressure.

Don’t want that to happen? Steer clear of the edges of tectonic plates, where the crust is going to eventually get sucked under the surface. One such subduction zone is Iran, where the Eurasian Plate is rising over the Iranian Plate.

5. Get discovered

Now you need to think about the potential for rediscovery.

If you want somebody to chance upon your carefully preserved fossil one day, you need to plan for burial in a spot that currently is low enough to accumulate the necessary sediments for deep burial – but that will eventually be pushed up again. In other words, you need a place with uplift where weathering and erosion will eventually scour off the surface layers, exposing you.

The Dead Sea may be a good place to preserve your fossil

Good for more than floating, the Dead Sea may be a good place to preserve your fossil (Credit: Getty)

One good spot might be the Mediterranean Sea, Syme says; it’s getting shallower as Africa is pushed towards Europe. Other small, inland seas that will fill with sediment are good bets, too.

“Perhaps the Dead Sea,” she says. “The high salt would preserve and pickle you.”

6. Or go rogue

We’ve covered the standard method for hard, durable fossils with bone largely replaced by rock. But there are some oddball methods to consider, too.

Top of the list is amber. There are astounding fossils perfectly preserved in this gemstone made of tree resin – such as recent finds of birds, lizards and even a feathered dinosaur tail in Myanmar. “If you can find a large enough amount of tree sap and get covered in amber, that’s going to be the best way to preserve your soft tissues as well as your bones,” Syme says. “But it’s obviously pretty difficult for such a large animal.”

Can’t find enough amber? The next option is tar pits of the kind that have preserved sabre-toothed cats and mammoths at La Brea in Los Angeles. Although here you would mostly likely end up disarticulated, your bones jumbled in with other animals. There’s also freezing on a mountain or in a glacier, like Ötzi the iceman, found in the European Alps in 1991.

Where Ötzi the iceman met his fate

Where Ötzi the iceman met his fate may not seem very comfortable, but it proved key for preserving his remains (Credit: Alamy)

Another route might be natural mummification, with your body left to dry in a cave system. “There are a lot of cave system remains that get covered with calcium from groundwater, which also forms stalactites and stalagmites,” Syme says. “People like caving and so if the cave systems still exist in the future, they might happen upon you.”

One final method to preserve your corpse almost indefinitely, though not in the form of a fossil, would be launching you into space – or leaving you on the surface of a geologically inert celestial body with no atmosphere, such as the Moon.

“The vacuum of space would be very good if you want your body to remain perpetually non-decaying,” Syme says. She adds that you could attach a radio beacon if you want to get found again in the distant future.

7. Leave a little something extra

Assuming you are found millions of years hence, what else might be preserved alongside you?

Plastics (fidget spinners, anyone?), other oil-derived products that don’t biodegrade and inert metals, like alloys, gold and rare metals of the kind found in mobile phones, all might last as long.

Will mobile phones be one of the artefacts we leave for future generations?

Will mobile phones be one of the artefacts we leave for generations far in the future? (Credit: Getty)

Glass is durable too, and can withstand high temperatures and pressures. You can imagine finding the “outlines or shape of smartphones,” Syme says. Archer notes that the durability of glass means you could chisel ‘ENJOY!’ on a small sheet of glass in a concrete coffin with your body and it would be there to find with your fossil.

“To be 100% sure I would use diamond,” Syme adds – it’s immensely stable. Using a laser, you could etch a letter explaining the lengths you went to to get fossilised.

If you also want to pre-plan your archaeological context, Syme believes bitumen highways and the foundations of skyscrapers are contenders. “We’ve dug down deep into the ground to build these things. You’ll be able to see… the layouts of cities still there,” she says.

Remember, the words you write will fade and your deeds will be forgotten. But a fossil? That, perhaps, could last forever.

Understanding the Historical Probability of Drought (Science Daily)

Jan. 30, 2013 — Droughts can severely limit crop growth, causing yearly losses of around $8 billion in the United States. But it may be possible to minimize those losses if farmers can synchronize the growth of crops with periods of time when drought is less likely to occur. Researchers from Oklahoma State University are working to create a reliable “calendar” of seasonal drought patterns that could help farmers optimize crop production by avoiding days prone to drought.

Historical probabilities of drought, which can point to days on which crop water stress is likely, are often calculated using atmospheric data such as rainfall and temperatures. However, those measurements do not consider the soil properties of individual fields or sites.

“Atmospheric variables do not take into account soil moisture,” explains Tyson Ochsner, lead author of the study. “And soil moisture can provide an important buffer against short-term precipitation deficits.”

In an attempt to more accurately assess drought probabilities, Ochsner and co-authors, Guilherme Torres and Romulo Lollato, used 15 years of soil moisture measurements from eight locations across Oklahoma to calculate soil water deficits and determine the days on which dry conditions would be likely. Results of the study, which began as a student-led class research project, were published online Jan. 29 inAgronomy Journal. The researchers found that soil water deficits more successfully identified periods during which plants were likely to be water stressed than did traditional atmospheric measurements when used as proposed by previous research.

Soil water deficit is defined in the study as the difference between the capacity of the soil to hold water and the actual water content calculated from long-term soil moisture measurements. Researchers then compared that soil water deficit to a threshold at which plants would experience water stress and, therefore, drought conditions. The threshold was determined for each study site since available water, a factor used to calculate threshold, is affected by specific soil characteristics.

“The soil water contents differ across sites and depths depending on the sand, silt, and clay contents,” says Ochsner. “Readily available water is a site- and depth-specific parameter.”

Upon calculating soil water deficits and stress thresholds for the study sites, the research team compared their assessment of drought probability to assessments made using atmospheric data. They found that a previously developed method using atmospheric data often underestimated drought conditions, while soil water deficits measurements more accurately and consistently assessed drought probabilities. Therefore, the researchers suggest that soil water data be used whenever it is available to create a picture of the days on which drought conditions are likely.

If soil measurements are not available, however, the researchers recommend that the calculations used for atmospheric assessments be reconfigured to be more accurate. The authors made two such changes in their study. First, they decreased the threshold at which plants were deemed stressed, thus allowing a smaller deficit to be considered a drought condition. They also increased the number of days over which atmospheric deficits were summed. Those two changes provided estimates that better agreed with soil water deficit probabilities.

Further research is needed, says Ochsner, to optimize atmospheric calculations and provide accurate estimations for those without soil water data. “We are in a time of rapid increase in the availability of soil moisture data, but many users will still have to rely on the atmospheric water deficit method for locations where soil moisture data are insufficient.”

Regardless of the method used, Ochsner and his team hope that their research will help farmers better plan the cultivation of their crops and avoid costly losses to drought conditions.

Journal Reference:

  1. Guilherme M. Torres, Romulo P. Lollato, Tyson E. Ochsner.Comparison of Drought Probability Assessments Based on Atmospheric Water Deficit and Soil Water Deficit.Agronomy Journal, 2013; DOI: 10.2134/agronj2012.0295