Basic cybernetics proves that we've been estimating our methane emissions wrong?

I've come across a new argument against mainstream climate science, and it makes some degree of sense to me. It's about applying the math we were taught in our Control Engineering classes to climate.
We've all seen this type of diagrams, right?

The argument goes that our methane emissions must be approximately equal to the derivative of that curve, slightly larger than that because methane in the atmosphere has a half-life of 12 years (that it would equal to the derivative if the half-life was infinite, but actually it is slightly larger than that). So it is visible from the diagram that our methane emissions reached their peak somewhen in 1980s and have been decreasing ever since (the usual explanation given for that is that grass-fed cows emit much more methane than grain-fed cows, and that there are much fewer grass-fed cows and much more grain-fed cows today than in 1980s). Makes sense, right? Well, according to this new argument I found, that's actually not the right inference, but the right inference is that our methane emissions have been approximately constant. That would be the right inference if methane in the atmosphere were a IT1-type system, but it actually isn't.
To understand why, try to derive the transfer function. Now, it might not be obvious how to derive the transfer function directly (or even how to set up the differential equations), but ask yourself: What is the impulse response (the response to the Dirac delta function)? It seems obvious that it is approximately:
g(t)=e^(ln(2)/12*t)
And since for linear time-invariant systems (which methane in the atmosphere approximately is) the transfer function is equal to the Laplace transform of the impulse response, that means the transfer function has to be:

This transfer function cannot be reduced to an IT1-type system. The transfer function of IT1 is:
G(s)=K/(Ti*s*(T*s+1))
So, what is the step-response of the methane in the atmosphere? How would a system described by its transfer function respond to constant methane emissions? Well, type this into Octave or MatLab:
pkg load control;
s=tf('s');
step(1/(s+log(2)/12))
It outputs this:

Which looks like the diagram of methane concentration in the atmosphere over time.
Does this curve converge or does it go to infinity. Well, that be determined using the final value theorem of the Laplace transform. The Laplace transform of the step response is, of course:
S(s)=1/s*G(s)
And the limit of the step response as time approaches infinity is, by the final value theorem:
lim(t->inf, s(t))=lim(s->0,s*S(s))=12/ln(2)
So, given that, according to the diagram, the amount of methane in the atmosphere rose from 1620 units to 1760 units. Therefore, our methane emissions are:
(1760-1620)/(12/ln(2))=8.0867
So it is 8.0867 units every year, and it is constant.
What do you think is wrong with that argument?
I e-mailed my Control Engineering professor [the name edited out] about it to see what he thinks, but, thus far, I received no response.

Why are you trying to guess the measurement of a gas we can factually measure?

Why are you trying to guess the measurement of a gas we can factually measure?

We can't precisely measure our methane emissions using the satellites because the spectre of methane is very similar to that of water, and water is the most abundant and most powerful greenhouse gas in the atmosphere.

CH4 has been measured directly in the environment since the 1970s.

https://en.wikipedia.org/wiki/Atmospheric_methane#Global_monitoring

They don't just estimate CH4 levels based on sources and sinks of methane (which aren't fully understood).  They measure it  - in fact it has dropped in some years.  I have no idea why you think this is "cybernetics"

Quote

CH4 has been measured directly in the environment since the 1970s.

https://en.wikipedia.org/wiki/Atmospheric_methane#Global_monitoring

They don't just estimate CH4 levels based on sources and sinks of methane (which aren't fully understood).  They measure it  - in fact it has dropped in some years.  I have no idea why you think this is "cybernetics"

They are assuming our methane emissions are approximately proportional to the first derivative of the global mathane concentrations. That would be true if it was an IT1-type system. But it's not.

They are assuming our methane emissions are approximately proportional to the first derivative of the global mathane concentrations. That would be true if it was an IT1-type system. But it's not.

They are measuring methane concentrations in the atmosphere directly. 

Measuring emissions is more difficult to get a handle on and is  done through satellite monitoring systems and estimates based on research - nothing to do with first derivatives. 

https://en.wikipedia.org/wiki/Methane_emissions#Global_methane_emissions_monitoring

Methane sinks are even more tricky to quantify. 

However we know for sure that methane concentrations keep going up and are more than double pre-industrial levels.

Measuring emissions is more difficult to get a handle on and is  done through satellite monitoring systems and estimates based on research - nothing to do with first derivatives.

Then what is this diagram you linked to showing?

It's showing the first derivative, right?

My control engineering professor responded to me, but he essentially said "I don't know.". I haven't received his permission to post his response onto an Internet forum, and, even if I did, that wouldn't really advance the conversation.

It's showing the first derivative, right?

No, it is showing how much methane concentrations in the atmosphere went up that year - directly measured.

Just go and read the wiki.

Anyway, here is what my Control Engineering professor has to say:

Kolega Samaržija,

teško da ćemo na ovaj način doći do rješenja koje tražite jer ja ne poznam ovaj proces. To je kemijski proces koji ne poznajem, niti znam na koje se sve načine metan generira, kako dolazi u atmosferu, gdje se i u kojim oblicima zadržava, s kojim drugim strukturama se veže i modificira, kojom brzinom se razgrađuje u atmosferi, … Sve te mehanizme treba poznavati kako bi se mogla modelirati količina metana u atmosferi. Teško da bi u toj procjeni mogla pomoći i mjerenja u pojedinim točkama atmosfere jer je to ogromno područje, vrlo složenog kemijskog sastava, velikih strujanja zračnih masa, … O tim stvarima pokušajte više saznati iz nekih internetskih izvora. Možda naiđete i na neke modele (matematičke opise) koji vam mogu pomoći u rješavanju ovog problema.

Pozdrav

Quote from: FlatAssembler on January 20, 2025, 10:52:21 AM

It's showing the first derivative, right?

No, it is showing how much methane concentrations in the atmosphere went up that year - directly measured.

Just go and read the wiki.

Pretending not to know what "derivative" is is not going to get you very far. It's the rate of change.

I've made a YouTube video about this problem:

Pretending not to know what "derivative" is is not going to get you very far. It's the rate of change.

It isn't a derivative you moron.  Can you not even understand a basic bar chart?  It isn't showing the rate of change, it is is showing the directly measured absolute change in methane concentrations year on year.

Quote from: FlatAssembler on January 21, 2025, 07:15:26 AM

Pretending not to know what "derivative" is is not going to get you very far. It's the rate of change.

It isn't a derivative you moron.  Can you not even understand a basic bar chart?  It isn't showing the rate of change, it is is showing the directly measured absolute change in methane concentrations year on year.

They are approximately equal. The only way they could be very different is if the function was non-continous, which it isn't.

Do you think this JavaScript program will correctly estimate our methane emissions? It uses the Gaussian method to approximate them and then uses a genetic algorithm to try to improve the approximation.
const koncentracije_metana_ocitane_s_grafa = [
  1625, 1640, 1650, 1665, 1675, 1685, 1695, 1705, 1715, 1720,
  1725, 1730, 1735, 1737, 1745, 1750, 1755, 1755, 1750, 1755
];
const krivulja =
    koncentracije_metana_ocitane_s_grafa.map((x) => { return x - 1620; });
const vrijeme_poluzivota_metana = 12; // Procjene se kreću između 9 i 12.
let jedinke = [];
const koliko_jedinki_imamo = 12;
function nasumicno_generiraj_pokusaj() {
  const niz = [];
  let trenutni = 2 * (1760 - 1600) / (vrijeme_poluzivota_metana / Math.log(2)) *
                 Math.random();
  while (niz.length < krivulja.length) {
    niz.push(trenutni);
    trenutni += Math.random() * 3 - 3 / 2;
  }
  return niz;
}
function aproksimiraj_Gaussovom_metodom() {
  const niz = [];
  for (let i = 0; i < krivulja.length; i++) {
    let razlika = krivulja[i];
    for (let j = 1; j <= i; j++)
      razlika -= niz[i - j] * Math.E **
                 (Math.log(1 / 2) / vrijeme_poluzivota_metana * j);
    razlika += Math.random() - 1 / 2;
    niz.push(razlika);
  }
  return niz;
}
jedinke.push(aproksimiraj_Gaussovom_metodom());
while (jedinke.length < koliko_jedinki_imamo ** 2)
  jedinke.push(nasumicno_generiraj_pokusaj());
function koncentracija_metana_u_atmosferi(emisije_metana_po_godinama) {
  const koncentracija_metana_u_atmosferi = [];
  for (let i = 0; i < emisije_metana_po_godinama.length; i++) {
    let zbroj = 0;
    for (let j = 0; j <= i; j++)
      zbroj += emisije_metana_po_godinama[i - j] * Math.E **
               (Math.log(1 / 2) / vrijeme_poluzivota_metana * j);
    koncentracija_metana_u_atmosferi.push(zbroj);
  }
  return koncentracija_metana_u_atmosferi;
}
function izracunaj_gresku(koncentracije_metana_u_atmosferi) {
  let zbroj = 0;
  if (krivulja.length != koncentracije_metana_u_atmosferi.length)
    throw "Neki dio programa je promijenio velicinu niza, prekidamo!";
  for (let i = 0; i < krivulja.length; i++)
    zbroj += (krivulja[i] - koncentracije_metana_u_atmosferi[i]) ** 2;
  return zbroj;
}
function krizaj_dva_pokusaja(prvi, drugi) {
  if (prvi.length != drugi.length)
    throw "Neki dio programa je promijenio velicinu niza, prekidamo!";
  const novi_niz = [];
  if (Math.random() < 1 / 2) {
    novi_niz.push((prvi[0] + drugi[0]) / 2);
    let trenutni = novi_niz[0];
    for (let i = 1; i < prvi.length; i++)
      if (Math.random() < 1 / 2) {
        trenutni += prvi[i] - prvi[i - 1];
        novi_niz.push(trenutni);
      } else {
        trenutni += drugi[i] - drugi[i - 1];
        novi_niz.push(trenutni);
      }
  } else {
    for (let i = 0; i < prvi.length; i++)
      if (Math.random() < 1 / 2)
        novi_niz.push(prvi[i]);
      else
        novi_niz.push(drugi[i]);
  }
  return novi_niz;
}
function nasumice_izmijeni(pokusaj) {
  const novi_pokusaj =
      pokusaj.map((x) => { return x + Math.random() * 3 - 3 / 2; });
  return novi_pokusaj;
}
function sortiraj_jedinke_i_zadrzi_najbolje() {
  jedinke.sort((prva, druga) => {
    return izracunaj_gresku(koncentracija_metana_u_atmosferi(prva)) <
           izracunaj_gresku(koncentracija_metana_u_atmosferi(druga));
  });
  jedinke = jedinke.slice(0, koliko_jedinki_imamo);
}
let brojac = 0;
while (brojac < 1_000) {
  sortiraj_jedinke_i_zadrzi_najbolje();
  for (let i = 0; i < koliko_jedinki_imamo; i++) {
    for (let j = 0; j < koliko_jedinki_imamo; j++) {
      if (i == j)
        continue;
      if (Math.random() < 1 / koliko_jedinki_imamo)
        jedinke.push(
            nasumice_izmijeni(krizaj_dva_pokusaja(jedinke[i], jedinke[j])));
      else
        jedinke.push(krizaj_dva_pokusaja(jedinke[i], jedinke[j]));
    }
    jedinke.push(nasumice_izmijeni(jedinke[0]));
    jedinke.push(
        nasumicno_generiraj_pokusaj()); // Biološki krajnje nerealistično
                                        // (abiogeneza se ne događa danas), ali
                                        // izgleda da program daje bolje
                                        // rezultate ovako.
  }
  for (let i = 0; i < 6; i++) // Dakle, neka najbolja jedinka ima 6+1=7 djece sa
                              // svakom drugom jedinkom.
    for (let j = 1; j < koliko_jedinki_imamo; j++)
      jedinke.push(krizaj_dva_pokusaja(jedinke[0], jedinke[j]));
  brojac++;
}
sortiraj_jedinke_i_zadrzi_najbolje();
const izlaz_simulacije = koncentracija_metana_u_atmosferi(jedinke[0]);
console.log(
    "Godine\tSimulirane emisije metana\tSimulirana koncentracija metana");
for (let i = 0; i < jedinke[0].length; i++)
  console.log(i + "\t" + jedinke[0][i] + "\t" + izlaz_simulacije[i]);
console.log(
    "Ukupno kvadratno odstupanje simulirane koncentracije metana od mjerenih podataka: " +
    izracunaj_gresku(izlaz_simulacije));

Here is what it outputs if you set it that the half-life of methane is 12 years:

Here is what it outputs if you set it that the half-life of methane is 9 years:

Either way, there doesn't seem to be a clear upward or a downward trend here. What strikes me as weird is that the 9th year is a peak if you assume the half-life is 12, but it's a valley if you assume the half-life is 9. I don't know how that's possible. Can somebody explain that to me?

Anyway, I've written a paper about the problem in Croatian language and I will try to have it reviewed by some expert. Because I really want to know whether our methane emissions are increasing or decreasing.