<p>Validation is done by <a href="https://en.wikipedia.org/wiki/Backtesting" rel="noreferrer">hindcasting</a>, not by reversing the time axis, which seems to be what you're asking about.  In hindcasting, we take the state at some time in the past, apply our weather models to that state (and the state before it), run the forecast model, and compare that to the reference state¹ ahead of that point in time.</p>

<p>There is a concept called <em>backtracking</em>, which is (for example) used to calculate where particulates have been first emitted (so we measure some plume, and calculate this originated 18 hours ago at a particular source).  But this assumes knowledge of present <em>and past</em> winds, and is therefore different from what you ask.</p>

<p>Reversing the time axis would involve water falling up to the clouds and cyclones moving equatorward and turning into potential vorticity.  It seems involved and I'm not sure what benefit would be.  I don't think this can be  done with existing models, and it would be a lot of effort to make it work.</p>

<hr>

<p>¹ It's not as simple as that, because the full "actual state" also involves modelling to "fill in the gaps" in time and space, between all the times and places where we have measurements.  This is known as <a href="https://en.wikipedia.org/wiki/Meteorological_reanalysis" rel="noreferrer">re-analysis</a>.</p>

CC BY-SA 4.0
2018-07-09T13:32:01.977Z
Maybe "will be done" instead of "can be done". It seems to me that, given enough research effort, it could be done. But the benefit is questionable, so it likely won't be done.
Ian MacDonald
CC BY-SA 4.0
2018-07-09T13:36:23.967Z
@IanMacDonald Right.  I just meant that it's not supported by current models, but of course, it can theoretically be developed given enough time and expertise.  Edited for clarity.
gerrit
CC BY-SA 4.0
2018-07-09T12:52:16.827Z
2018-07-09T13:35:36.047Z
2018-07-09T13:35:36.047Z
gerrit
<p>A completely different approach to the one leftroundabout pointed out is to use recurrent neural networks. These were made to predict the future development of a time series by first learning the hidden model itself and then using it to guesstimate the future values. The advantage of this method is, that not even the slightest knowledge about meteorology is needed, all the modelling of the weather system is done by the algorithm training the neural network.</p>

<p>In fact there was a Kaggle competition with the task to predict rainfall from past data: <a href="http://simaaron.github.io/Estimating-rainfall-from-weather-radar-readings-using-recurrent-neural-networks/" rel="nofollow noreferrer">http://simaaron.github.io/Estimating-rainfall-from-weather-radar-readings-using-recurrent-neural-networks/</a> .
The winner used recurrent neural networks.</p>

<p>In the case of predicting past values from current values the same architecture can be used, as it can learn the backward model as directly as a forward model.</p>

CC BY-SA 4.0
2018-07-12T07:07:16.300Z
And can those be used to reverse the time axis?  I imagine that a purely statistical approach could, but you're not actually addressing the core question in this answer.
gerrit
CC BY-SA 4.0
2018-07-10T17:24:05.290Z
2018-07-14T09:34:22.803Z
2018-07-14T09:34:22.803Z
Metlam
<p>In a nutshell, no.</p>

<p>The reason why is because of the <a href="https://en.wikipedia.org/wiki/Butterfly_effect" rel="nofollow noreferrer">Butterfly Effect</a>.  In a system where the current state depends on the previous state in an iterative way, you can get chaotic effects.  Chaotic effects can magnify extremely tiny inputs to gigantic changes over time.</p>

<p>This was first noted by the excellent mathematician and meteorologist Edward Lorenz.  <a href="http://www.stsci.edu/~lbradley/seminar/butterfly.html" rel="nofollow noreferrer">This is a decent explanation</a> of how he came to notice that the equations predicting the weather are extremely sensitive to current conditions.  You simply can't build a computer with enough sensitivity to do a good job. </p>

<p>Since tiny fluctuations can cause huge effects over time, you have to ask yourself - how much information can your simulation encompass?  Lorenz showed that tiny things can change the entire landscape over time.  To be accurate, a simulation would have to take into account every source of small changes - sunspot activity, the wobble of the moon, the gravitational tug from Pluto...the list is endless.  </p>

<p>So unfortunately for a chaotic system like weather, you can't predict with any accuracy previous or future states.  </p>

CC BY-SA 4.0
2018-07-10T00:54:58.973Z
While Chaos Theory is indeed valid (leftaroundabout goes into great summarization of the mathematics of how it really works)... we can still make very good predictions at a certain length.  Your answer makes it sound like all forecasting is hopeless.
JeopardyTempest
CC BY-SA 4.0
2018-07-10T00:56:10.083Z
In particular, *"you can't predict with any accuracy previous or future states"*... for example, NHC 48 hour tropical cyclone track forecasts are down to about 75 miles average error (whereas persistence+climatology is around 225 miles)... and continues to show improvement by on the order of 25% per decade.  In the grand scheme, the window of predictability is pretty limited... but you certainly **CAN** with some accuracy predict some future states.
JeopardyTempest
CC BY-SA 4.0
2018-07-10T14:25:22.653Z
@JeopardyTempest, you are correct.  Some prediction is of course possible, I did somewhat overstate the case when I said that any accuracy for all states is impossible.  Mathematically that is true, but in practice it is not.  You can predict some future states with the caveat that you understand that it is an approximation and not an exact representation, and with weather being a chaotic system it will diverge from your expectations fairly quickly.
BoredBsee
CC BY-SA 4.0
2018-07-09T20:49:39.397Z
2018-07-09T20:49:39.397Z
BoredBsee
<p>The underlying equations for fluid-dynamic models are <a href="https://en.wikipedia.org/wiki/Hyperbolic_partial_differential_equation" rel="noreferrer">hyperbolic partial differential equations</a>. They can generally be written in the form
$$
  \frac{\partial}{\partial t} u(t) = D(u(t))
$$
where $D$ in some way evaluates the current state of the system and its spatial derivatives.</p>

<p>A numerical simulation then <em>integrates</em> this differential equation, to extrapolate from a start state $u(t_0) = u_0$ the time-dependent $u(t)|_{t&gt;t_0}$.</p>

<p>Well, if we can do that, then surely we could also solve in the inverse time direction, by considering the equation
$$
  \frac{\partial}{\partial t} u(-t) = -D(u(t))
$$
and running the integrator with $\tilde t = -t$, $\tilde D = -D$?</p>

<p>Actually, you quickly run into problems when you try that. The operator $D$ can be characterised by its <a href="https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant" rel="noreferrer">Jacobian</a>, which basically tells you how pertubations in the state influence the derivative. Specifically, the complex <em>eigenvalues</em> of the Jacobian can tell you whether a small deviation will <strong>a)</strong> amplify over time (positive real part), or <strong>b)</strong> decay (negative real part), or <strong>c)</strong> just oscillate (purely imaginary).</p>

<p>For physical systems the eigenvalues tend to be mostly <strong>c)</strong> or <strong>b)</strong>: you get a lot of wave-like solutions which propagate / oscillate over the system, and tend to <em>decay</em> over time. <strong>a)</strong> however is more tricky: if you start with a small deviation from the start state, the system will over time deviate ever more and and more. Now, this kind of thing is by no means unheard of especially in meteorology; it's the essence of a <a href="https://en.wikipedia.org/wiki/Chaos_theory" rel="noreferrer">chaotic system</a>. Storms can emerge and grow stronger over time, but only by scooping up energy that's already stored in the system. At some point they'll stop.</p>

<p>OTOH, you always have a lot of <em>consistently negative</em> real-part eigenvalues. These correspond to <em>dissipative effects</em>: small-scale pertubations generally are smoothed out to zero by the physical effects, e.g. winds have friction, mixing of air of different temperature averages out the differences, etc.. If you now run the simulation backwards, you turn those negative real parts into positive real parts, and that means the system is suddenly <em>massively chaotic</em> on all length scales. Small pertubation arise out of numerical uncertainties, and grow over all bounds. You would not only end up with states different from the actual weather a week ago, but with states that are completely unlike anything the weather has ever been like – huge, erratic temperature fluctuations and small vortices with crazy wind speeds.</p>

CC BY-SA 4.0
2018-07-09T15:52:56.397Z
Thanks a lot for filling in the mathematical details for me.
BmyGuest
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2018-07-09T15:55:50.440Z
If I want to put it into less-mathematical more intuitive phrasing, can one simply say: "Many (slightly) different situations would evolve to the same/similar weather situation (and the maths "smooths" over this) - so reverting doesn't get you to one of them (and the maths exaggerates the error)" ?
BmyGuest
CC BY-SA 4.0
2018-07-09T16:06:53.213Z
Not really. As I said, weather _is_ in fact chaotic, which is why normal weather models also can't give high-precision predictions for the long-term future. Just the deviations never amplify _out of bounds_; calculating forwards always gives you a state that is at least physically _plausible_. In the reverse direction that is not given however. To put it another way: going forward, “many (even **completely**) different situations would evolve to a very similar weather situation”.
leftaroundabout
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2018-07-09T16:09:05.017Z
That's what I essentially meant. Thanks, I think I've understood.
BmyGuest
CC BY-SA 4.0
2018-07-09T17:03:24.173Z
@BmyGuest You should change the accepted answer from mine (which is conceptual has some context) to this one (which is a lot more definitive).
gerrit
CC BY-SA 4.0
2018-07-09T20:06:44.590Z
What sort of timescales are involved for "reverse forecasting" to become worthless?  Can we predict what the weather was five minutes ago?  An hour ago?  Yesterday?
Mark
CC BY-SA 4.0
2018-07-09T20:32:47.003Z
@Mark it'll depend on the level of detail you're trying to resolve. When you just run the dynamics backwards naïvely, I'd estimate you can do about 20-100 simulation time-steps before the state is dominated by unbounded oscillations, depending on the solver. Via the [CFL condition](https://en.wikipedia.org/wiki/Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition), you can calculate what time span that is; it scales linearly with the size of the smallest feature you're resolving.
leftaroundabout
CC BY-SA 4.0
2018-07-10T04:58:12.560Z
So in other words, **yes**, @BmyGuest, that's exactly what you were saying. We can't project backwards because our present modeling shows many different original states can converge at what we see now, so we quickly lose the ability to project backwards to a single value with accuracy. No, left, the bit about the impossible scales created doesn't really affect anything he's saying or interested in.
lly
CC BY-SA 4.0
2018-07-10T04:59:54.347Z
Imho, @gerrit should have more faith in himself. His answer actually covered what you were getting at: the way we can use known values to validate prediction methodologies.
lly
CC BY-SA 4.0
2018-07-10T09:28:30.220Z
@lly but also many different states can arise from what we see now (with only pertubations within the measurement uncertainty) when calculating _forwards_, because of chaos. So that's not really a qualitative difference between the time directions. The difference is that when looking backwards, even many states that are _almost everywhere completely different_ (and possibly completely unphysical) will converge to the same current state.
leftaroundabout
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2018-07-10T11:50:27.453Z
TLDR: No, because entropy.
workoverflow
CC BY-SA 4.0
2018-07-10T18:21:39.467Z
@gansub yes. For most meteorological phenomena, Navier-Stokes behave mostly like the [Euler equations](https://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)), which are one of the classical examples of hyperbolic PDE. Only in the high-viscosity limit do Navier-Stokes turn into the parabolic diffusion equation, but that's AFAIK not relevant for weather prediction.
leftaroundabout
CC BY-SA 4.0
2018-07-11T10:34:39.910Z
@gansub _pretty sure_ the primitive equations are hyperbolic! Generally speaking, if you can write something in conservative/advective form with finite velocities, then it's hyperbolic. Sure enough you use _hyperbolic solvers_ of the FD or FV class, don't you? (At least for DNS, I don't not much about LES etc..)
leftaroundabout
CC BY-SA 4.0
2018-07-11T13:53:03.010Z
Actually the $\frac{\partial}{\partial t} u(t) = D(u(t))$ form is so general that it includes parabolic equations as well, so the point is somewhat moot. But I persist: it _is_ hyperbolic. The Rossby number has nothing to do with that. The Reynolds number _has_, but this is in meteorology always so high that we're in hyperbolic territory.
leftaroundabout
CC BY-SA 4.0
2018-07-09T15:43:57.057Z
2018-07-09T15:53:11.380Z
2018-07-09T15:53:11.380Z
leftaroundabout
<p>If one applies today's state-of-the-art weather forecast computations "backwards", i.e. computing how the systems was X days <em>before</em> the current situation based on knowledge of today's situation, at which point do "predictions" deviate significantly from the (known) sitautions ?</p>

<p>In other words, are those computations reversible and even possible?
And if so, is the error level comparabliy (i.e. time-symmetric) or not?</p>

<p>Edit:
Imagine one would get the data from an Earth-like planet through an observation (satelite etc.) at a given moment, would current weather-models allow to compute how the weather <em>was</em> before that time?</p>

CC BY-SA 4.0
2018-07-09T09:32:40.917Z
I've never heard that anyone would do this. Any sources implying that someone does? I would guess that most models can't do this.
Communisty
CC BY-SA 4.0
2018-07-09T09:35:11.947Z
@Communisty No, but from a programmers/mathematicians perspective that would be an interesting idea to validate forecast methods, I think. And I was also interested *if* these methods are invertible or not. So, I *assumed* somebody else might have had this idea as well (for sure) and at least tried it. Unless, of course, it is not doable. Which would be the other answer...
BmyGuest
CC BY-SA 4.0
2018-07-09T10:25:37.610Z
The problem I'd think is A definitely causes B, but the inverse, B can often have many causes of A, C, D, etc.  I'd think that's a problem in most mathematical modeling of time?
JeopardyTempest
CC BY-SA 4.0
2018-07-09T10:27:34.547Z
@JeopardyTempest Sure, but some mathematical algorithms are reversible, some within limits and some simply aren’t. I don’t know into which category the ones used for weather predictions fall - hence me asking.
BmyGuest
CC BY-SA 4.0
2018-07-09T11:58:30.773Z
Validation is done by [hindcasting](https://en.wikipedia.org/wiki/Backtesting), not by reversing the time axis, which seems to be what you're asking about.  Reversing the time axis would involve water falling up to the clouds and cyclones moving equatorward and turning into potential vorticity.  It seems involved and I'm not sure what benefit would be.
gerrit
CC BY-SA 4.0
2018-07-09T12:18:36.410Z
@gerrit 99% the answer I want (-I agree to the practical point, still would be curious if there is *mathematical* forking point in the alogirthm which would prevents it from working or not. Or rather: Can one - given the current system - compute the previouse system (f.e. if some planet atmosphere on a remote planet would be 'snap-shot' visited, could one compute "backwards"?.  but can you make your comment into an answer so that I can accept and close?
BmyGuest
CC BY-SA 4.0
2018-07-10T06:37:17.047Z
Not enough info for an additional answer, but worth noting is the reason why we can still somewhat trust weather forecasts is because rather than performing one state-of-the-art-technology simulation, several are performed with slightly differing initial conditions (accounting for the fact that the current state of the atmosphere can never be faithfully measured). This technique is called [ensemble forecasting](https://en.wikipedia.org/wiki/Ensemble_forecasting) and may further explain why we can't simply use the algorithms backwards to find out what the weather was like in the past.
Toivo Säwén
CC BY-SA 4.0
2018-07-10T06:48:56.473Z
@ToivoSäwén Ah, makes sense. Didn’t know it is done like this though, so thanks for adding this info.
BmyGuest
<meteorology>
Can the previous weather be computed from the current situation?
BmyGuest
# Can the previous weather be computed from the current situation?

(Asked by: username_0 on 2018-07-09)

If one applies today's state-of-the-art weather forecast computations "backwards", i.e. computing how the systems was X days *before* the current situation based on knowledge of today's situation, at which point do "predictions" deviate significantly from the (known) sitautions ?

In other words, are those computations reversible and even possible?
And if so, is the error level comparabliy (i.e. time-symmetric) or not?

Edit:
Imagine one would get the data from an Earth-like planet through an observation (satelite etc.) at a given moment, would current weather-models allow to compute how the weather *was* before that time?

**Tags:** meteorology

> **Comments:**
> **username_1**: I've never heard that anyone would do this. Any sources implying that someone does? I would guess that most models can't do this.  
> **username_0**: @username_1 No, but from a programmers/mathematicians perspective that would be an interesting idea to validate forecast methods, I think. And I was also interested *if* these methods are invertible or not. So, I *assumed* somebody else might have had this idea as well (for sure) and at least tried it. Unless, of course, it is not doable. Which would be the other answer...  
> **username_2**: The problem I'd think is A definitely causes B, but the inverse, B can often have many causes of A, C, D, etc.  I'd think that's a problem in most mathematical modeling of time?  
> **username_0**: @username_2 Sure, but some mathematical algorithms are reversible, some within limits and some simply aren’t. I don’t know into which category the ones used for weather predictions fall - hence me asking.  
> **username_3**: Validation is done by [hindcasting](https://en.wikipedia.org/wiki/Backtesting), not by reversing the time axis, which seems to be what you're asking about.  Reversing the time axis would involve water falling up to the clouds and cyclones moving equatorward and turning into potential vorticity.  It seems involved and I'm not sure what benefit would be.  
> **username_0**: @username_3 99% the answer I want (-I agree to the practical point, still would be curious if there is *mathematical* forking point in the alogirthm which would prevents it from working or not. Or rather: Can one - given the current system - compute the previouse system (f.e. if some planet atmosphere on a remote planet would be 'snap-shot' visited, could one compute "backwards"?.  but can you make your comment into an answer so that I can accept and close?  
> **username_4**: Not enough info for an additional answer, but worth noting is the reason why we can still somewhat trust weather forecasts is because rather than performing one state-of-the-art-technology simulation, several are performed with slightly differing initial conditions (accounting for the fact that the current state of the atmosphere can never be faithfully measured). This technique is called [ensemble forecasting](https://en.wikipedia.org/wiki/Ensemble_forecasting) and may further explain why we can't simply use the algorithms backwards to find out what the weather was like in the past.  
> **username_0**: @username_4 Ah, makes sense. Didn’t know it is done like this though, so thanks for adding this info.  

---

## 4 answers

### Answer by username_8 on 2018-07-09. Score: 19 | Accepted answer

The underlying equations for fluid-dynamic models are [hyperbolic partial differential equations](https://en.wikipedia.org/wiki/Hyperbolic_partial_differential_equation). They can generally be written in the form
$$
\frac{\partial}{\partial t} u(t) = D(u(t))
$$
where $D$ in some way evaluates the current state of the system and its spatial derivatives.

A numerical simulation then *integrates* this differential equation, to extrapolate from a start state $u(t\_0) = u\_0$ the time-dependent $u(t)|\_{t>t\_0}$.

Well, if we can do that, then surely we could also solve in the inverse time direction, by considering the equation
$$
\frac{\partial}{\partial t} u(-t) = -D(u(t))
$$
and running the integrator with $\tilde t = -t$, $\tilde D = -D$?

Actually, you quickly run into problems when you try that. The operator $D$ can be characterised by its [Jacobian](https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant), which basically tells you how pertubations in the state influence the derivative. Specifically, the complex *eigenvalues* of the Jacobian can tell you whether a small deviation will **a)** amplify over time (positive real part), or **b)** decay (negative real part), or **c)** just oscillate (purely imaginary).

For physical systems the eigenvalues tend to be mostly **c)** or **b)**: you get a lot of wave-like solutions which propagate / oscillate over the system, and tend to *decay* over time. **a)** however is more tricky: if you start with a small deviation from the start state, the system will over time deviate ever more and and more. Now, this kind of thing is by no means unheard of especially in meteorology; it's the essence of a [chaotic system](https://en.wikipedia.org/wiki/Chaos_theory). Storms can emerge and grow stronger over time, but only by scooping up energy that's already stored in the system. At some point they'll stop.

OTOH, you always have a lot of *consistently negative* real-part eigenvalues. These correspond to *dissipative effects*: small-scale pertubations generally are smoothed out to zero by the physical effects, e.g. winds have friction, mixing of air of different temperature averages out the differences, etc.. If you now run the simulation backwards, you turn those negative real parts into positive real parts, and that means the system is suddenly *massively chaotic* on all length scales. Small pertubation arise out of numerical uncertainties, and grow over all bounds. You would not only end up with states different from the actual weather a week ago, but with states that are completely unlike anything the weather has ever been like – huge, erratic temperature fluctuations and small vortices with crazy wind speeds.

> **Comments:**
> **username_0**: Thanks a lot for filling in the mathematical details for me.  
> **username_0**: If I want to put it into less-mathematical more intuitive phrasing, can one simply say: "Many (slightly) different situations would evolve to the same/similar weather situation (and the maths "smooths" over this) - so reverting doesn't get you to one of them (and the maths exaggerates the error)" ?  
> **username_8**: Not really. As I said, weather _is_ in fact chaotic, which is why normal weather models also can't give high-precision predictions for the long-term future. Just the deviations never amplify _out of bounds_; calculating forwards always gives you a state that is at least physically _plausible_. In the reverse direction that is not given however. To put it another way: going forward, “many (even **completely**) different situations would evolve to a very similar weather situation”.  
> **username_0**: That's what I essentially meant. Thanks, I think I've understood.  
> **username_3**: @username_0 You should change the accepted answer from mine (which is conceptual has some context) to this one (which is a lot more definitive).  
> **username_9**: What sort of timescales are involved for "reverse forecasting" to become worthless?  Can we predict what the weather was five minutes ago?  An hour ago?  Yesterday?  
> **username_8**: @username_9 it'll depend on the level of detail you're trying to resolve. When you just run the dynamics backwards naïvely, I'd estimate you can do about 20-100 simulation time-steps before the state is dominated by unbounded oscillations, depending on the solver. Via the [CFL condition](https://en.wikipedia.org/wiki/Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition), you can calculate what time span that is; it scales linearly with the size of the smallest feature you're resolving.  
> **username_10**: So in other words, **yes**, @username_0, that's exactly what you were saying. We can't project backwards because our present modeling shows many different original states can converge at what we see now, so we quickly lose the ability to project backwards to a single value with accuracy. No, left, the bit about the impossible scales created doesn't really affect anything he's saying or interested in.  
> **username_10**: Imho, @username_3 should have more faith in himself. His answer actually covered what you were getting at: the way we can use known values to validate prediction methodologies.  
> **username_8**: @username_10 but also many different states can arise from what we see now (with only pertubations within the measurement uncertainty) when calculating _forwards_, because of chaos. So that's not really a qualitative difference between the time directions. The difference is that when looking backwards, even many states that are _almost everywhere completely different_ (and possibly completely unphysical) will converge to the same current state.  
> **username_11**: TLDR: No, because entropy.  
> **username_8**: @gansub yes. For most meteorological phenomena, Navier-Stokes behave mostly like the [Euler equations](https://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)), which are one of the classical examples of hyperbolic PDE. Only in the high-viscosity limit do Navier-Stokes turn into the parabolic diffusion equation, but that's AFAIK not relevant for weather prediction.  
> **username_8**: @gansub _pretty sure_ the primitive equations are hyperbolic! Generally speaking, if you can write something in conservative/advective form with finite velocities, then it's hyperbolic. Sure enough you use _hyperbolic solvers_ of the FD or FV class, don't you? (At least for DNS, I don't not much about LES etc..)  
> **username_8**: Actually the $\frac{\partial}{\partial t} u(t) = D(u(t))$ form is so general that it includes parabolic equations as well, so the point is somewhat moot. But I persist: it _is_ hyperbolic. The Rossby number has nothing to do with that. The Reynolds number _has_, but this is in meteorology always so high that we're in hyperbolic territory.  

---

### Answer by username_3 on 2018-07-09. Score: 13

Validation is done by [hindcasting](https://en.wikipedia.org/wiki/Backtesting), not by reversing the time axis, which seems to be what you're asking about. In hindcasting, we take the state at some time in the past, apply our weather models to that state (and the state before it), run the forecast model, and compare that to the reference state¹ ahead of that point in time.

There is a concept called *backtracking*, which is (for example) used to calculate where particulates have been first emitted (so we measure some plume, and calculate this originated 18 hours ago at a particular source). But this assumes knowledge of present *and past* winds, and is therefore different from what you ask.

Reversing the time axis would involve water falling up to the clouds and cyclones moving equatorward and turning into potential vorticity. It seems involved and I'm not sure what benefit would be. I don't think this can be done with existing models, and it would be a lot of effort to make it work.

---

¹ It's not as simple as that, because the full "actual state" also involves modelling to "fill in the gaps" in time and space, between all the times and places where we have measurements. This is known as [re-analysis](https://en.wikipedia.org/wiki/Meteorological_reanalysis).

> **Comments:**
> **username_5**: Maybe "will be done" instead of "can be done". It seems to me that, given enough research effort, it could be done. But the benefit is questionable, so it likely won't be done.  
> **username_3**: @username_5 Right.  I just meant that it's not supported by current models, but of course, it can theoretically be developed given enough time and expertise.  Edited for clarity.  

---

### Answer by username_6 on 2018-07-10. Score: 3

A completely different approach to the one leftroundabout pointed out is to use recurrent neural networks. These were made to predict the future development of a time series by first learning the hidden model itself and then using it to guesstimate the future values. The advantage of this method is, that not even the slightest knowledge about meteorology is needed, all the modelling of the weather system is done by the algorithm training the neural network.

In fact there was a Kaggle competition with the task to predict rainfall from past data: <http://simaaron.github.io/Estimating-rainfall-from-weather-radar-readings-using-recurrent-neural-networks/> .
The winner used recurrent neural networks.

In the case of predicting past values from current values the same architecture can be used, as it can learn the backward model as directly as a forward model.

> **Comments:**
> **username_3**: And can those be used to reverse the time axis?  I imagine that a purely statistical approach could, but you're not actually addressing the core question in this answer.  

---

### Answer by username_7 on 2018-07-09. Score: 1

In a nutshell, no.

The reason why is because of the [Butterfly Effect](https://en.wikipedia.org/wiki/Butterfly_effect). In a system where the current state depends on the previous state in an iterative way, you can get chaotic effects. Chaotic effects can magnify extremely tiny inputs to gigantic changes over time.

This was first noted by the excellent mathematician and meteorologist Edward Lorenz. [This is a decent explanation](http://www.stsci.edu/~lbradley/seminar/butterfly.html) of how he came to notice that the equations predicting the weather are extremely sensitive to current conditions. You simply can't build a computer with enough sensitivity to do a good job.

Since tiny fluctuations can cause huge effects over time, you have to ask yourself - how much information can your simulation encompass? Lorenz showed that tiny things can change the entire landscape over time. To be accurate, a simulation would have to take into account every source of small changes - sunspot activity, the wobble of the moon, the gravitational tug from Pluto...the list is endless.

So unfortunately for a chaotic system like weather, you can't predict with any accuracy previous or future states.

> **Comments:**
> **username_2**: While Chaos Theory is indeed valid (username_8 goes into great summarization of the mathematics of how it really works)... we can still make very good predictions at a certain length.  Your answer makes it sound like all forecasting is hopeless.  
> **username_2**: In particular, *"you can't predict with any accuracy previous or future states"*... for example, NHC 48 hour tropical cyclone track forecasts are down to about 75 miles average error (whereas persistence+climatology is around 225 miles)... and continues to show improvement by on the order of 25% per decade.  In the grand scheme, the window of predictability is pretty limited... but you certainly **CAN** with some accuracy predict some future states.  
> **username_7**: @username_2, you are correct.  Some prediction is of course possible, I did somewhat overstate the case when I said that any accuracy for all states is impossible.  Mathematically that is true, but in practice it is not.  You can predict some future states with the caveat that you understand that it is an approximation and not an exact representation, and with weather being a chaotic system it will diverge from your expectations fairly quickly.  

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