Modeling the Troposphere

By Andy May and Philip Mulholland

IGRA2 radiosonde 1991–2025 means (10 hPa bins, 10° latitude slices) show systematic differences between observed mid-tropospheric temperatures, and the CMIP 5 & 6 (Coupled Model Intercomparison Project, see IPCC, 2021) ensemble means as shown by McKitrick & Christy (2020) and Po-Chedley et al. (2022). The difference between the CMIP models and the observations is most noticeable in the tropics as discussed here. While multiple reasons, such as too much weight on the warming effect of additional CO₂, anomalous cloud feedback, or bad weather balloon data, have been proposed (IPCC, 2021, p. 443), no one really knows why they are so poor at reproducing tropospheric temperature profiles.

In previous posts (see here and here) we focused on modeling the troposphere using models based on moist adiabatic theory, a methodology similar to the one used in CMIP6. Our method had three free variables, surface temperature, surface dew point, and overall lapse rate. These models did a good job of reproducing observed tropical air temperatures and dewpoint up to around 250 hPa (~10.6 km). The models did not require any other input, such as greenhouse gas levels or measurements of radiation in and out at the top of the atmosphere. But above ~250 hPa they performed poorly. This opened the possibility that the atmospheric temperature profile, in the tropics below 250 hPa was only dependent upon surface conditions; but above this altitude, a different model, with different forcings was required.

Encouraged by these results, we worked on extending the model to other latitudes and developing a new model to use higher in the troposphere. This post presents a composite model for the global troposphere, it is similar in concept, but not in detail to the composite moist adiabatic/RAE (RAE: radiative advective equilibrium) model of (Cronin & Jansen, 2016). The anchor for our model is the molar density intersection (or simply the “intersection”) discovered by Michael and Ronan Connolly (Connolly & Connolly, 2014a) & (Connolly M. , 2025). It turns out to be the easiest and most general way to divide the troposphere into a lower part that can be modeled using moist adiabatic theory and an upper part that is much more complicated. We call the upper troposphere model the frost-point model or “FPM.”

Tropospheric structure

The intersection (May, 2025) occurs below the traditional tropopause as defined by the WMO (WMO & Ashford, 1957, p. 137). The distance below the tropopause varies from 50 to over 200 hPa (hectopascals of pressure = one millibar) due to the occasional complexity of the lapse rate change from cooling with altitude in the troposphere to warming with altitude in the stratosphere. The tropopause is generally thought to be the altitude where the air temperature stops declining and begins to increase, that is, a lapse rate of zero. While the intersection is easily reproduced and precise, the WMO definition (WMO & Ashford, 1957) of the tropopause is vague and complicated (two full paragraphs) and can result in multiple tropopauses in the mid-latitudes as shown in figure 1. In the tropics the sharper “cold-point” form of the tropopause is seen (see figure 3 in May 2025 or see figures 1 & 2 here).

Figure 1. The 1991-2025 mean IGRA2 radiosonde February intersection and tropopause between 40°N and 50°N (Lat_slice = 4). Upper left, an illustration showing how the intersection is determined. The dots are 10 hPa mean pressures for the latitude band and month. Upper right shows the tropopause WMO region and how it compares to the intersection. Lower left shows the February intersection by latitude in meters. The lower right plot is a map of the intersection in meters. The mean February ITCZ is shown with a black line. The circles on the map identify the locations of the weather stations mapped. Click on the figure to see in a higher resolution. Data source: (May, 2025).

The upper left plot shows how the intersection is determined. It is the intersection of two best fit lines, an upper one above 125 hPa and a lower from 300 to 750 hPa. The fact that molar density forms a line is not surprising and follows from the ideal gas law, since molar density equals pressure divided by temperature times the gas constant, but the change in slope at the intersection is surprising. This change implies a change in state or gas composition, or both. For more details see here or Connolly & Connolly 2014a.

The Molar Density Intersection

The molar density intersection changes altitude continuously by latitude (see the lower left plot in figure 1), from month to month, and the peak altitude in the tropics moves with the Intertropical Convergence Zone (ITCZ) (May, 2025). The 1991-2025 mean IGRA2 radiosonde characteristics of the air at the intersection in February between 40 and 50°N are listed in figure 1. The abbreviations are: “rh” means relative humidity and “q” means specific humidity. Notably the air is quite dry, and the mean temperature is -51°C.

As noted above, the lower portion of the troposphere, below the intersection, can be accurately modeled using only three free variables, the surface temperature, overall lapse rate, and dew-point depression (temperature – dewpoint). We call this model the DPAH model (Dewpoint Anchor Hypothesis), and it is based on moist-adiabatic theory (May, 2025). Below the intersection, the troposphere behaves like a moist‑adiabatic system because water vapor is abundant enough to control buoyancy, lapse rate, and infrared opacity. Above the intersection, temperatures fall so low that any water vapor left is deposited on ice crystals suspended in the air, breaking the assumptions of moist‑adiabatic theory.

Even though liquid droplets can survive to −38 °C in the absence of ice nuclei, the intersection zone is far colder, around −56 °C on average. At these temperatures, the saturation mixing ratio over ice is extremely small (often below 0.05 g/kg). Any remaining water vapor is rapidly deposited onto existing ice crystals. The air is simply too cold to sustain meaningful water vapor concentrations. The mean specific humidity at the intersection is 0.03 (±0.02) g/kg and the saturation mixing ratio with respect to ice lies between 0.03 and 0.06 g/kg (Stull 2026, saturation-vapor-pressure formulae). The radiosonde mean specific humidity is 0.03±0.02g kg^-1, which coincides with the lower bound of Stull’s range, thus the actual humidity is at or below the minimum possible saturation value.

Key Measurements at the Molar Density Intersection (1991-2025 means)
Latitude Slice (- is south)	Mean Temperature °C	Mean height (m)	Mean Relative Humidity (%)	Mean Specific Humidity (g/kg)	Mean Pressure (hPa)	Mean Molar Density (mol/m³)
-70	-59.9	8,624.3	32.0	0.02	288.2	16.5
-60	-57.0	8,731.7	32.5	0.03	290.5	16.5
-50	-51.7	9,404.6	31.9	0.03	278.5	15.5
-40	-52.2	10,359.2	23.9	0.02	251.9	14.1
-30	-53.9	11,650.2	14.4	0.01	213.7	12.4
-20	-59.6	13,219.3	15.7	0.01	173.4	10.5
-10	-66.2	14,113.0	28.8	0.01	153.5	9.5
0	-67.3	14,281.1	36.1	0.01	150.1	9.4
10	-64.4	13,892.7	23.6	0.01	157.7	9.7
20	-61.3	13,453.6	23.6	0.01	168.2	10.2
30	-55.5	12,036.4	23.6	0.02	204.1	11.9
40	-51.7	10,702.0	25.5	0.03	241.4	13.6
50	-51.4	9,517.0	26.6	0.03	275.7	15.2
60	-48.9	8,753.1	37.2	0.06	306.9	16.7
70	-50.8	8,384.7	28.9	0.03	317.2	17.3
80	-51.2	8,139.2	38.3	0.05	324.3	17.7
Mean	-56.4	10,953.9	27.7	0.03	237.2	13.5
Standard Deviation	5.8	2,198.6	6.8	0.02	60.8	2.9

Table 1. Key radiosonde 1991-2025 mean measurements at the molar density intersection by latitude slice.

Table 1 suggests that at the intersection, ice deposition has removed nearly all the water vapor and the air is not just cold but is water vapor exhausted. It appears that the intersection marks the altitude where supersaturation collapses because ice deposition has caught up.

Since the mean molar density in table 1 is below 15 mol/m³, it is reasonable to assume that radiative cooling dominates in the region. The collapse of water vapor concentration coincides with a shift from moist‑adiabatic to radiatively dominated behavior as discussed by (Cronin & Jansen, 2016). Here, outgoing longwave radiation to space from non-condensing greenhouse gases dominates.

At the intersection, the moist‑adiabat framework loses validity because its key ingredient—water vapor—is no longer present in meaningful quantities. We hypothesize that this transition from water vapor to ice has something to do with the molar density intersection. At the intersection and above it, we need a different model, the FPM model.

The Connolly’s hypothesized that the formation of oxygen and nitrogen multimers (Connolly & Connolly, 2014b) began to occur at the molar density intersection and that they accounted for the change in the molar density versus pressure slope by altering the molar density. For the Connolly multimerization mechanism to explain the molar density-pressure phase change, the implied multimer concentrations would necessarily produce detectable, altitude‑localized spectroscopic and radiative anomalies. These are not observed, despite extensive modern measurements of the radiation emissions of multimers and dimers. In contrast, the conventional explanation—transition from moist convective to radiatively controlled, dry stratified conditions—accounts for the change in molar density-pressure slope without requiring an unobserved layer of multimers.

Dimer (O₂-O₂and O₂-N₂) collision complexes (or CIAs, collision induced absorption complexes) occur at all pressures and not at a specific altitude. This phenomenon is well-studied (Adkins et al., 2023) & (Finkenzeller & Volkamer, 2022) because the CIAs affect radiation transfer and spectroscopy within the atmosphere. The dimers do occur in the upper atmosphere, but they have not been observed as a condensed phase or a heavy bulk gas that would change the macroscopic molar density slope.

The Connolly multimer hypothesis is certainly not disproven, just not observed in nature. Our idea (again unproven) is that the loss of free water vapor content is the cause of the intersection and acknowledge that more work is needed to determine the root cause. In any case the observed, and easily calculated, molar density intersection provides a convenient upper limit to the applicability of the moist adiabatic model or DPAH.

The modeling results

The DPAH moist-adiabatic model works very well almost everywhere up to the intersection. Above that point, our FPM model produces mixed results depending upon the latitude slice chosen. The tropics model well, as shown in figure 2.

Figure 2. Temperature, dewpoint, and lapse rates for the equator to 10°N latitude slice (“0 slice”). Model results are shown as red dashed lines and radiosonde observations are shown in blue.

The middle latitudes also model reasonably well, although the frequent double tropopauses in this region can cause problems for the FPM portion.

Figure 3. Temperature, dewpoint, and lapse rates for the 50 to 60°N latitude slice (“50 slice”). Model results are in red and radiosonde observations are shown in blue.

The only real blow up is in the North Polar Region where modeling is very difficult.

Figure 4. Temperature, dewpoint, and lapse rates for the 80 to 90°N latitude slice. Model results are in red and radiosonde observations are shown in blue. This is the worst match between the models and observations.

The critical variables are temperature, dewpoint, and lapse rate. We splice the FPM and DPAH model output at the molar density intersection and then compute R² for the combined curves. The R² values are excellent for temperature and the dewpoint in nearly every slice but mixed for the lapse rate as shown in table 2.

R² for the total column (1010 to 10 hPa)
Latitude slice	R2_T	R2_Td	R2_lapse
-70	98.9%	99.0%	90.5%
-60	99.2%	40.5%	65.7%
-50	98.4%	99.7%	55.6%
-40	98.4%	99.8%	68.1%
-30	99.7%	99.6%	78.4%
-20	99.4%	99.8%	79.1%
-10	96.9%	99.0%	66.0%
0	99.1%	99.8%	53.5%
10	97.5%	99.1%	72.6%
20	98.9%	96.9%	77.4%
30	98.3%	95.8%	89.3%
40	96.7%	94.0%	86.4%
50	98.5%	99.8%	82.1%
60	98.7%	99.2%	83.8%
70	98.3%	98.9%	84.3%
80	96.8%	98.6%	71.8%

Table 2. R² values for the full column (the models are spliced at the molar density intersection) for temperature, dewpoint, and lapse rate.

Methods

To do this work, the IGRA2 radiosonde data were quality controlled and then binned into 10 hPa bins for each 10° latitude slice. Table 3 lists each model parameter, for both models, and provides a short physical description, and ranks the parameters from most to least important based on the variance of the optimized parameter values across all runs. The table also includes the correlations (R²) between each parameter and the column-integrated diagnostics (T, Td, lapse rate), but these R² values do not affect the ranking.

The variance shown in table 3 is not the traditional variance, it is the variance in the optimized parameter values. Table 3 ranks the ten optimized parameters across the 16 latitude slices. The largest variance is 1.09×10⁸ (strato_amp); the smallest is 0.0 (latent_heat_rel). Figure 5 displays the corresponding Z-scores.¹ Z-scores greater than 2 occur at isolated latitudes (e.g., strato_amp at 20°N, td_coeff at 30°N, radiation_out at 50°N). These statistics describe the latitude dependence of the fitted coefficients; they do not establish which processes are “fundamental” or “compensatory.” The DPAH segment uses only three free parameters (T0, dpd, ELR); the FPM segment uses seven. The higher parameter count of FPM is from empirical “curve fitting;” it is not proof that these seven independent physical mechanisms operate or are the only mechanisms at work.

Table 3. The model variables, ranked by optimized parameter variance across latitude bands. The highest variance is “strato_amp” or the strength of stratospheric warming versus latitude. The weakest, or the most consistent across latitudes, is latent heat release.

As shown in table 3, the FPM and DPAH models use different variables and have no variables in common. This suggests that atmospheric cooling above and below the intersection work via completely different processes. Below, water vapor does the work; and above radiation and wind transport do most of the work. Indeed, our FPM model suggests that above the intersection heat is mostly carried away by radiation (radiation_out) and wind (env_loss).

Somewhat counterintuitively, the variables in table 3 that have low scores are important everywhere, those with high scores are only important in some regions (high variance). The special high variance variables required to get a reasonable fit are a bit of a mystery and there are too many of them. The “strato_amp” variable is there mostly to help get the shape of the tropopause; it may have little physical meaning. The “td_coeff” variable controls the shape of the dewpoint as it relates to relative humidity. The “env_loss” variable is always important and is a factor meant to compensate for the very high wind speeds in some portions of the upper atmosphere (May, 2025, fig. 12,13). While the DPAH model is grounded in only three physically meaningful variables, the FPM model is more of a curve fitting exercise. However, it is still very helpful, since it helps us visualize how the upper troposphere might work, and emphasizes how different it is from the lower troposphere. Figure 5 compares the variables listed in table 3 by their Z-score.^[1]

Figure 5. Dimensionless Z-scores for all variables listed in Table 3. Scores above zero mean the variable exceeds its global mean and scores below are smaller than the global mean.

The striking thing about figure 5 is the variation in extremes by latitude. The important variable at 20°S is latent heat release and at 20°N it is the stratospheric warming trend. At 30°N, the main desert-band latitude is the dewpoint coefficient. At 50°N it is lower than normal precipitation and extreme radiation out. Thus, in the upper troposphere FPM model region, the factors controlling heat loss vary tremendously by latitude.

In the desert belt (~30°N), “td_coeff” has a large negative Z-score, meaning the optimizer reduces dewpoint sensitivity to RH relative to the global mean. Here, the lower troposphere is already extremely dry, and the dewpoint depression is large, so the model does not need to amplify dewpoint–RH coupling. The dryness is “baked in” to the column, and “td_coeff” is correspondingly suppressed.

At 50°N, “radiation_out” has a very large positive Z‑score, indicating unusually strong radiative cooling in the upper troposphere at this latitude. This aligns with the high land/ocean ratio and relatively low humidity there. The “env_loss” variable has negative Z‑scores at 40–60°N, meaning the optimizer relies less on the wind/mixing term in these mid‑latitudes. The “precip” variable is also low at 50°N suggesting lower than normal precipitation in that latitude band. Next, we examine the models individually.

The Models

To understand how the DPAH and FPM model parameters behave across latitudes, we examined two complementary diagnostics. First, we compute the variance of each optimized parameter across latitudes, which indicates how strongly the model relies on that parameter to adjust to regional conditions. Parameters with high variance are structurally important locally because the optimization varies them widely.

Second, we compute correlations between each parameter and the R² metrics (temperature, dewpoint, lapse rate) to assess how strongly each parameter influences model skill. Separately, we compute a normalized importance measure for each parameter at each latitude, defined as the absolute deviation from its mean divided by its standard deviation. This normalized importance highlights where each parameter becomes unusually influential across latitude. Figures 6 and 7 plot this normalized importance for each model.

The importance statistics and Z scores reveal which parameters are globally stable and physically fundamental, and which ones vary strongly with latitude because they are responding to real latitude-specific atmospheric differences. The variance, as in table 3, tells us how much a specific variable changes globally, the Z-score tells where it changes and by how much, and the importance shows how much each variable influences the solution at a specific latitude slice.

Figure 6. The importance of the DPAH model variables. Red is the dewpoint depression, green is the lapse rate, and T0 is the surface temperature. The normalized importance of each is plotted.

In figure 6 we see that the optimized variables in the DPAH model travel together once normalized. All are maximal in the tropics and at the poles. They are less significant in the middle latitudes where zonal winds and storms are very important. This is the sort of pattern expected when the troposphere below the molar density intersection is very constrained by intersection height, the observed lapse rate (surface to intersection), the dewpoint depression, and moist adiabatic theory. The DPAH model is anchored by real physical constraints, so the importance factor reflects real atmospheric structure.

Surface temperature (T0) is extremely important at the poles and in the tropics. The lapse rate (ELR) is important over most latitudes, but extremely important at the poles where the intersection is very low. The dewpoint depression (dpd) peaks (positively or negatively) in regions where the humidity structure and the intersection height change quickly. It is significant that it has a peak at 20-30°S and a valley at 20-30°N. The latitudes from 20°S-30°S are dominated by oceans and those from 20-30°N by deserts.

Figure 7. The FPM model variables. It took seven variables to model the tropopsphere above the intersection, way too many to be physically meaningful.

Figure 7 tells us that the troposphere above the intersection is very complex. “strato_amp” or the rapidity of the temperature increase at the base of the stratosphere is very important at 20°N, which is where the height of the intersection drops most rapidly (see figure 1). At 20°S “latent_heat_release” is the dominant variable since this is about the latitude with maximal ocean surface. The “latent_heat_release” variable is also important at 20°N where the intersection height drops most quickly.

The “td_coeff” variable, or the dewpoint to relative humidity factor, is maximal at 30°N, near the aforementioned desert belt where the Sahara, SW USA/Northern Mexico, Gobi, etc. exist. In desert regions entrainment (or the mixing of air parcels with the surrounding air) dilutes rising plumes of moisture, ice formation aloft is reduced, and the dewpoint depression increases throughout the column, as seen in figure 7. The spike in “td_coeff” is physically significant and important.

The “p-trans” variable is the distance of the troposphere to the stratospheric transition above the intersection, it is maximal in the Antarctic and the high northern latitudes where the lower troposphere is very thin. The “env_loss” variable is also high in the polar regions; it estimates the loss of heat to the environment. This can be due to strong winds or melting ice. The “radiation_out” variable also increases slightly.

The noticeable “strato_amp” peak at 20°N, when it is a weak variable at other latitudes is a bit surprising. However, the cold point at 20°S is lower than at 20°N, and to compensate for this the optimizer may have boosted this variable.

Conclusions

DPAH is a reasonable model, based on moist adiabatic theory, that adequately describes heat transfer (or cooling) in the troposphere below the intersection. It reproduces the observed mean temperature and dew-point profiles from the surface to the molar-density intersection with the values listed in Table 2. No explicit CO₂ term appears in the DPAH equations. That absence is a property of the chosen functional form and of the 1991–2025 radiosonde means; it does not constitute a measurement of the radiative contribution of CO₂, if any.

Above the intersection the seven-parameter FPM fit is required to reach a comparable correlation. The increase in the number of free parameters is an empirical result of the optimization; it indicates that a low-parameter description of the observed upper-tropospheric means was not obtained with the functional forms tested. The radiosonde data themselves supply no direct measurement of the relative magnitudes of radiative, advective, or latent-heat fluxes at those altitudes. The result suggests that the cooling mechanism above the intersection is more complex than below it.

This is consistent with the failure of the CMIP climate models in the 300 to 100 hPa region, the region that contains the intersection. It has been proposed that this failure is due to an overemphasis on the greenhouse effect (McKitrick & Christy, 2020), (Po-Chedley et al., 2022), and (IPCC, 2021, p. 443).

Above the intersection there is almost no water vapor, so the driving force below the intersection is gone. Above the intersection CO₂ and other well-mixed greenhouse gases probably play a significant role, below the intersection these models suggest they do not.

The FPM model that we use has too many variables. This suggests that the upper troposphere may be inherently multi-process and has no single overriding physical root across all latitudes.

With so many processes at play in the upper troposphere, the FPM model became more an exercise in curve fitting than a physical model. In the early days of this project, we experimented with a more general, and less empirical Markov model for the upper troposphere. It failed, supporting the idea that a low-parameter physical model is not possible.

The polar regions are dominated by freezing, thawing, and radiation emissions to space. The Northern and Southern Hemispheres act differently because ocean surfaces dominate in the Southern Hemisphere and land surfaces in the Northern. We found it very interesting that the desert regions around 30°N were so visible in the model results and “radiation_out” was so prominent at 50°N where the land/ocean ratio is maximal.

As expected, “p_trans,” or the distance from the molar density intersection to the beginning of the stratosphere, is lowest in the tropics and highest in the middle latitudes where double WMO tropopauses are common. At the scale considered in this post, the DPAH model is fairly solid and shows that in the lower troposphere heat loss is dominated by water vapor processes. Above the molar density intersection, where water vapor is essentially zero, the FPM model has too many degrees of freedom and too many variables to be a proper model, but it is illuminating. It matches observations reasonably well by changing the variable emphasized from latitude to latitude in a reasonable way. Thus, while not a physically meaningful model on its own, it is a good learning tool.

The cause of the sharp molar density intersection is still a mystery. It is unlikely to be multimer formation since multimers occur at all altitudes and are very short lived (Welsh, 1962) & (Hartmann et al., 2011). The dehydration of the air by freezing is also a gradual process, but it could be the cause of the intersection or part of the cause.

This post benefited from conversations with Dr. Ronan Connolly; we are grateful for his help. He disagrees with our assessment of the multimerization theory and argues that our critiques have already been addressed in Connolly & Connolly (2014) We disagree, but interested readers are welcome to read his paper and form their own opinion.

The bibliography can be downloaded here.

Parameter value minus global parameter mean, divided by the global standard deviation. The scaled Z-scores in figure 5 can be used to compare the importance of each parameter across latitudes. A Z‑score measures how many standard deviations a parameter at a given latitude lies above or below its global mean. Thus, Z = +2 indicates the parameter is two standard deviations higher than average at that latitude, while Z = −1 indicates it is one standard deviation lower. Because Z‑scores are dimensionless, they allow direct comparison across parameters with very different physical units and magnitudes. Large positive or negative Z‑scores identify latitudes where a parameter plays an unusually strong or unusually weak role in the model solution. ↑

The Tropospheric Model R code can be downloaded here:
https://andymaypetrophysicist.com/wp-content/uploads/2026/08/Tropospheric_model_R_code.zip

You will also need the R code from May 2025 to prepare the IGRA2 data for the model. All the links you need are here:
https://andymaypetrophysicist.com/2025/12/19/the-story-behind-my-paper-on-the-itcz-and-the-hadley-circulation/

This post was just to present the results.

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59 Comments

Philip Mulholland

August 4, 2026 6:06 pm

Thanks Anthony for the opportunity to present this work here.

Sparta Nova 4

Reply to Philip Mulholland

August 5, 2026 5:43 am

Excellent work.

Izaak Walton

August 4, 2026 6:40 pm

So what is the model? Table 3 shows that there are 10 parameters used but what is the function of those 10 variables that gives the temperature? If you want people to judge the accuracy of the model then you have to actually present it rather than a selected set of results.

-15

Andy May

Author

Reply to Izaak Walton

August 5, 2026 3:29 am

Izaak, Here is the model:
https://andymaypetrophysicist.com/wp-content/uploads/2026/08/Tropospheric_model_R_code.zip

I’ll also put a link in the post to download it. You will also need the R code from my paper to prepare the IGRA2 data for the model. All the links you need are here:
https://andymaypetrophysicist.com/2025/12/19/the-story-behind-my-paper-on-the-itcz-and-the-hadley-circulation/

This post was just to present the results. The whole modeling project has taken over 9 months, and the results and modeling techniques are spread out over a number of posts. Links to many of them are in this post. I’ve put up links to my R code before, but very few downloads occur, so this time I thought I’d wait until someone ask.

gyan1

August 4, 2026 6:46 pm

This is exactly the kind of research needed to understand better what is going on in Earth’s atmosphere. Climate establishment myopia about CO2 explicitly ignores contrary data.

jonesingforozone

August 4, 2026 8:26 pm

The complaint “…model has too many degrees of freedom…” is not found in any literature except this post. Models generally suffer from zero degrees of freedom due to their interpolative nature, meaning the models can’t predict outside the sample data. See Wiki: Degrees of freedom and What Are Degrees of Freedom?

-13

Philip Mulholland

Reply to jonesingforozone

August 4, 2026 8:49 pm

@jonesingforozone

The complaint “…model has too many degrees of freedom…” is not found in any literature except this post.

Huh? This is a standard user-beware health warning that has been around for decades.

The worry that a model has “too many degrees of freedom” (i.e., too many free parameters relative to the data) is textbook statistics and machine-learning folklore. It is exactly the concern John von Neumann captured with his famous line: “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.” Fermi later used that quip to dismiss an over-parameterized theory of Dyson’s, and the same idea has been repeated ever since whenever people warn about overfitting, residual degrees of freedom collapsing toward zero, or models that can interpolate anything.

So our comment is not unique to this post; it is one of the oldest and most common cautionary notes in the literature. Put simply all we are saying is “let the reader beware”.

The second half of your comment is also only half-right.

Models generally suffer from zero degrees of freedom due to their interpolative nature, meaning they can’t predict outside the sample data.

An interpolating model does drive residual degrees of freedom to zero by construction: it fits the training points exactly. Classical residual-df theory therefore supplies no residual variance estimate and cannot guarantee good out-of-sample performance. That is why the classical statistical advice was “don’t interpolate noisy data.”
Modern theory and practice, however, show that the classical implication does not always hold. Over-parameterized interpolators can still generalize (the “benign overfitting” / double-descent regime). The effective complexity of the fitted function is often far smaller than the nominal parameter count because of implicit regularization induced by the optimization algorithm, the architecture, or explicit penalties. In short:

Yes, interpolators have residual df = 0 (or negative).No, that fact alone does not imply they “can’t predict outside the sample data.”

See Wiki: Degrees of freedom and What Are Degrees of Freedom?

Pointing to the general definition does not make the complaint rare or idiosyncratic; it is precisely the definition that generates the warning in the first place!

jonesingforozone

Reply to Philip Mulholland

August 4, 2026 9:21 pm

In statistics and science, degrees of freedom (often abbreviated as df) refer to the number of independent pieces of information or values in a calculation that are free to vary without breaking a set rule or constraint. If the actual data is outside the rule or constraint, all bets are off unless you can also prove continuity.

Now, that explanation is simpler than the one posted above, isn’t it?

-12

Philip Mulholland

Reply to jonesingforozone

August 4, 2026 9:33 pm

Now, that explanation is simpler than the one posted above, isn’t it?

Yes, that is the basic definition, and it is the same one that has been used for decades.
It still leads straight to the original warning: when a model has a large number of free parameters relative to the number of observations, residual degrees of freedom drop toward zero. That is precisely when the model becomes flexible enough to fit almost any pattern in the training data (the “elephant” case), and why the complaint about too many degrees of freedom is routine rather than novel.
Your extra claim about data falling “outside the rule or constraint” and needing to “prove continuity” is not part of the standard statistical account. Continuity is a separate modelling assumption; it is not required to discuss residual degrees of freedom or the overfitting risk that follows from low residual df.

jonesingforozone

Reply to Philip Mulholland

August 4, 2026 11:10 pm

A twice-differentiable region is always continuous. A proof would be required if any discontinuities arise due to the constraint boundary or boundaries within the domain.

Is such a region boundary simply non-smooth or truly discontinuous?

-4

jonesingforozone

Reply to Philip Mulholland

August 4, 2026 11:37 pm

Yes, continuity and low residual df are independent (and unrelated).

Why would you suppose otherwise?

A low residual degrees of freedom (df = n – k – 1) means your statistical model estimates many parameters (k) relative to your total sample size (n). This leaves very few independent data points to estimate underlying error variance, reducing statistical power and inflating risk of overfitting.

-5

jonesingforozone

Reply to jonesingforozone

August 4, 2026 9:57 pm

Philip Mulholland posted, “An interpolating model does drive residual degrees of freedom to zero by construction: it fits the training points exactly.”

Thus, df is zero by definition for such a model.

The complaint, “…model has too many degrees of freedom…,” is a non-sequitur.

The df can only be zero or greater than zero.

The df increases with the sample size as the number of variables remains the same.

-8

jonesingforozone

Reply to jonesingforozone

August 4, 2026 10:59 pm

The von Neumann elephant was a different animal altogether.
See Drawing an elephant with four complex parameters by Mayer, Khairy, and Howard, Am. J. Phys. 78, 648–649 (2010), https://doi.org/10.1119/1.3254017.

-8

Andy May

Author

Reply to jonesingforozone

August 5, 2026 4:01 am

jonesforozone,
You are confusing two things. Your definition of “degrees of freedom” only applies to in the context of a single statistical calculation, not in the context of model complexity or over‑parameterization.
Those are different concepts and collapsing them together obscures the issue rather than clarifying it.

There are two different uses of “degrees of freedom.”
One applies to statistical tests; the other applies to model complexity.

The post was using the modeling definition, which is standard and widely used.

Your definition is correct in its own context but not the one relevant to the discussion.

The “simpler” explanation is only simpler because it omits the concepts needed to understand the modeling issue.

jonesingforozone

Reply to Andy May

August 5, 2026 8:34 am

An overparameterized model, or one that is overly complex, is one where the number of parameters equals or exceeds the number of data points, making the residual degrees of freedom equal to zero (df = 0).

Only one definition of degrees of freedom exists, though there is a correction for sample bias, such that df = N – B – 1.

-3

jonesingforozone

Reply to jonesingforozone

August 5, 2026 8:16 am

In recreational mathematics, von Neumann’s elephant is a problem of constructing a planar curve in the shape of an elephant from only four fixed parameters. Common examples of planar curves include circles, parabolas, and ellipses.

To parameterize an elephant, we note that its perimeter can be described as a set of points [x(t), y(t)], where t is a parameter that can be interpreted as the elapsed time along the contour. If the speed is uniform, t becomes the arc length. Mayer et al. expand x and y separately[1] as a Fourier series

         ∞
x(t) = Σ Ax[k] cos(kt) + Bx[k] sin(kt),
         k=0
         ∞
y(t) = Σ Ay[k] cos(kt) + By[k] sin(kt),
         k=0
where Ax[k], Bx[k], Ay[k], and By[k] are the expansion coefficients. The indices k apply to the kth term in the expansion, where Ax and Bx denote the x expansion, and Ay and By denote the y expansion, respectively.

With some modification, the fifth parameter simulates a wiggling trunk.

The degrees of freedom (df), a scalar value, does not apply to parameterized planar curves when the number of observations is also the number of parameters. Fitting a planar curve with N points using P parameters where N=P results in an exact interpolation rather than a statistical estimation.

On the other hand, it’s common for an objective function to have a sample population that exceeds the number of independent variables, so that df > 0.

Thus, for every value of an objective function, f(x), there is one and only one output value y, which can be expressed as

y = f(x)

where x is an independent variable or vector of variables. The N observations are implied, such that yi = f(xi) where 0 <= i < N, and df = N – number of independent variables.

Prior to the publication of Drawing an elephant with four complex parameters by Mayer et al. in 2010, Wei[2] tried this task in 1975 using a least-squares Fourier sine series, but that required about 30 terms.

References

1) F. P, Kuhl and C. R. Giardina, “Elliptic Fourier features of a closed contour,” Comput. Graph. Image Process. 18, 236-258 (1982).

2) J. Wei, “Least square fitting of an elephant,” CHEMTECH S, 128-129 (1975).

-1

jonesingforozone

Reply to jonesingforozone

August 5, 2026 10:42 am

Hi [mod]! Can we use these copyrighted images from the American Association of Physics Teachers (2010)? They are the solution to the von Neumann elephant problem! [DOI: 10.1119/1.3254017]

jonesingforozone

Reply to jonesingforozone

August 5, 2026 10:43 am

Great! Thanks!

Andy May

Author

Reply to jonesingforozone

August 5, 2026 9:25 am

This is how copilot explains the differences and similarities between the degrees of freedom in a statistical calculation and for a model. I think it is pretty good.

1. Degrees of freedom in a statistical calculation
This is the classical textbook definition:
Number of independent pieces of information free to vary after constraints are applied.
Example: linear regression with data points and fitted parameters.
Residual degrees of freedom:

This df determines:

variance estimates
t‑tests
confidence intervals
chi‑square tests
ANOVA tables

It is correct in this domain.

Degrees of freedom in a model (model complexity)
Number of adjustable parameters (or effective parameters) the model has available to fit the data.
Examples:

A polynomial of degree has degrees of freedom.
A neural network may have millions of degrees of freedom.
A climate model with many tunable parameters has high degrees of freedom.

The mathematical connection
Here’s the cleanest way to show how they relate:
Linear regression example
Model: y = Xb + e

Model degrees of freedom = number of parameters = p
Residual degrees of freedom = n-p

They satisfy: df(model) + df(residual) = n

This is the only place where the two concepts meet directly.

Why they diverge in modern modeling
In modern machine learning and physical modeling:

p can be much larger than
residual df becomes zero or negative
yet the model can still generalize (benign overfitting)

So:

model df remains meaningful (capacity, flexibility)
statistical df becomes meaningless (no variance estimate)

This is why your definition is incomplete: it only applies to classical inference, not to model complexity.

Summary:
Degrees of freedom in a statistical calculation (n–p) measure how much independent information remains after fitting a model. Degrees of freedom in a model (p) measure how many adjustable parameters the model has. They are mathematically related in simple settings, but they are conceptually different and used for different purposes. Our discussion concerns model degrees of freedom—the classical warning about over‑parameterization—not residual degrees of freedom used in t‑tests or ANOVA.

jonesingforozone

Reply to Andy May

August 5, 2026 9:47 am

The distinction between model df and statistical df involves just a dash of cognitive dissonance on Copilot’s part!

Consider that, as defined by Copilot,

model df = p
statistical df = (n-p)
statistical pf = –model pf + n

So that statistical pf is equal to the negative model pf plus the sample population! (I didn’t think so, either.)

-1

Andy May

Author

Reply to jonesingforozone

August 5, 2026 1:50 pm

What is “pf”? The bottom line is the statistical df has no meaning relative to models, it is a different thing. In the literature, universally, model df = p. This is well known among modelers.

Sparta Nova 4

Reply to jonesingforozone

August 5, 2026 9:52 am

I am not voting down a legitimate post that leads to a mature discussion.

Rud Istvan

August 4, 2026 9:00 pm

This is a much more sophisticated and understandable (than previously) physical explanation of the observationally latitude dependent effective radiation level (ERL) from which observed ‘temperature dependent’ radiative IR emission to space is observed. The satellite observed ERL varies from ~5-6 km (lower at dryer colder higher latitudes), averaging -19C but ranging -53C to -13C.

Way beyond simplistic former lapse rate explanations as in climate models, because of the complex impact of phase transition microscopic ice crystals.

The same complication as in Lindzen’s ‘adaptive iris’ hypothesis. Judith Curry and I posted paired comments on that some years ago. High cirrus clouds (comprising microscopic ice crystals) warm the surface because ice is transparent to incoming solar SW but opaque to outgoing warmed IR. Lindzen observed that less cirrus equaled more cooling, while less cirrus was also caused by more heating from more violent equatorial thunderstorms caused by equatorial warming via less cirrus resulting cumulonimbus thunderhead entrainment.

—WE, you have long also been here with the same result from the same basic reasoning!

Philip Mulholland

Reply to Rud Istvan

August 4, 2026 9:18 pm

ice is transparent to incoming solar SW but opaque to outgoing warmed IR.

Thanks Rud. Your point about Cirrus clouds being a type of one-way mirror is well put. There was a valuable contribution on WUWT a few years ago by a reader based in Florida concerning the thermal radiant refection capacities of cirrus. The commentator noted the local impact of a cirrus charged front over the Gulf of America ameliorating the effect of nighttime winter frosts on land by reflecting radiative energy from the warm sea water lying offshore to the west of his orange orchards.

Andy May

Author

Reply to Rud Istvan

August 5, 2026 4:06 am

Thanks Rud,
This critical transition from water vapor to ice has been recognized for decades and ignored by the climate establishment for just as long. It is a shame. It is critically important in understanding the atmospheric cooling process.

R.K.

Reply to Rud Istvan

August 7, 2026 4:10 am

I don’t believe the Iris effect is correct in the hypothesis that Cirris forms out of the tops of thunderstorms. I have never seen it occur in a long aviation career including with the airlines encountering thousands of thunderstorms, and other colleagues including those with significant international flying say the same. Cirris is predominantly a wispy high level cloud found above 30,000′ and associated with clear weather.
Whilst some storms around the equator may be severe from time to time they are nowhere near as violent as big storms that occur in the subtropics over land where intense heat in summer associated with fronts and geographic uplift give rise to very big storms with great height. I am talking about storms with tops around 70,000′ and these would be extremely rare in the tropics – I flew into New Guinea for a number of years in the 1970s so I am well aware of tropical thunderstorms.

Ben Wouters

August 5, 2026 3:04 am

I’m searching for a reference in these posts on the atmospheric lapse rate to the Hydrostatic Equllibrium the atmosphere is in.
At every level in the atmosphere the pressure has to equal the weight of the atmosphere above that level. Since pressure depends on the temperature this restricts the temperature at a certain level. Outside this small band the air would otherwise rise or sink.
Seems to me the HE is the main factor deciding the tropospheric lapse rate.

Philip Mulholland

Reply to Ben Wouters

August 5, 2026 3:31 am

Hydrostatic equilibrium is the force-balance statement:
dp/dz =−ρg
(or its equivalent in pressure coordinates). Combined with the ideal-gas law
ρ = p/RdT
it yields the hypsometric relation between pressure and the layer-mean temperature. That relation is a foundational mechanical constraint; radiosonde profiles (including the IGRA2 1991–2025 means used here) already satisfy it to high accuracy.
It does not, by itself, determine the temperature lapse rate dT/dz.

The dry-adiabatic lapse rate Γd = g/cp ≈ 9.8 K km^−1 follows only when the additional thermodynamic assumption of reversible adiabatic displacement of dry air is imposed. The moist-adiabatic rate is lower still, because latent-heat release must be included. The observed environmental lapse rates that enter the DPAH segment of the present work are taken directly from the radiosonde means (surface to the molar-density intersection) and are therefore empirical numbers, not predictions derived from hydrostatic balance alone.

In the post the lower-troposphere segment is fitted with three free parameters (surface temperature T0, dew-point depression, and the observed environmental lapse rate ELR). The upper segment (FPM) uses seven parameters.

The seven parameters used in the upper-troposphere (FPA / FPM) segment are those marked “FPA” in Table 3 of the original analysis. They are ranked by the variance of their optimised values across the 16 latitude slices:

strato_amp – Controls the strength of stratospheric warming above the transition pressure.
td_coeff – How dew-point responds to relative humidity (higher values produce stronger dew-point depression in dry air).
p_trans – Distance (in pressure) of the cold point above the molar-density intersection.
precip – Cooling associated with precipitation.
radiation_out – Radiative cooling of the column.
env_loss – Entrainment / mixing losses proportional to specific humidity.
latent_heat_rel – Controls how moist convection reduces the lapse rate.

These seven free parameters are optimised independently of the three DPAH parameters (T0, dpd, ELR) that are used only below the molar-density intersection. The text explicitly notes that the higher parameter count of the FPA/FPM segment makes it more of a curve-fitting exercise than a low-parameter physical model.

The resulting spliced profiles reproduce the radiosonde temperature means with the RT2 values listed in Table 2 (range 96.7 % – 99.7 %). Those R2 figures quantify residual variance after the empirical fits; they are not statements that hydrostatic equilibrium has been “solved for” the lapse rate.

Thus hydrostatic equilibrium is an essential background constraint that every realistic atmospheric profile must obey, but the tropospheric temperature structure reported here is fixed by the observed surface conditions, the measured ELR, and the location of the molar-density intersection (T=−56.4 ±5.8 ∘C , P=237.2 ±60.8 hPa, Table 1). No additional derivation of the lapse rate from hydrostatic balance alone appears in the analysis, because the data already embody that balance.

Ben Wouters

Reply to Philip Mulholland

August 6, 2026 2:33 am

No additional derivation of the lapse rate from hydrostatic balance alone appears in the analysis, because the data already embody that balance.

To me the average ELR is a representation of the temperature profile that results from a given surface temp and the requirements of the HE. No surprise that the data show this requirement on average.
I’m not sure why the DALR and MALR are in the discussion on the origin of the ELR.
DALR and MALR are only valid for air (parcels) rising or sinking within an atmosphere that is in HE. The adiabatic assumption only holds when the rising/sinking times are short, or the size of the parcel is large. I don’t see the relevance of the DALR or MALR for the determination of the ELR

Philip Mulholland

Reply to Ben Wouters

August 6, 2026 3:23 am

@Ben Wouters: Thank you for the comment. It is worth unpacking the relationships carefully, because the distinction between the observed temperature profile and the theoretical adiabatic rates is fundamental to understanding how the troposphere is structured.
Start with the dry adiabatic lapse rate (DALR). It is obtained by combining the first law of thermodynamics for an adiabatic process with the hydrostatic equation. The result is simply:
Γd = g/cp ≈ 9.8∘C km−1.
This quantity is set by gravity and the specific heat of air. In that sense the DALR is the gravity-controlled scaffold of the planetary atmosphere: it is the background rate at which a dry air parcel must cool (or warm) when it is displaced vertically in a hydrostatic field. Every other temperature gradient in the troposphere is measured or constrained relative to this scaffold.

When water vapour is present and condensation (or, higher up, deposition onto ice crystals) occurs, latent heat is released. That heat partially offsets the adiabatic cooling, producing the moist adiabatic lapse rate (MALR), which is always less steep than the DALR and varies with temperature and humidity. Conversely, evaporation or sublimation absorbs heat and can steepen the local gradient. Thus the latent-heat fluxes associated with the dominant condensing volatile—water—are the principal energy drivers (and retarders) of convection. They determine how far, and how rapidly, air parcels can rise or sink before they lose buoyancy.

The environmental lapse rate (ELR) is something different. It is the actual, measured temperature change with height at a given time and place. It is therefore a local, instantaneous observation, not a fixed physical constant. The DALR and MALR act as the atmospheric boundary constraints on that observation: they define the absolute and conditional stability limits. When the ELR exceeds the DALR the atmosphere is absolutely unstable and vigorous dry convection ensues; when it lies between the MALR and the DALR the atmosphere is conditionally unstable and moist convection can develop; when it is shallower than the MALR the layer is stable. Over time, the cumulative effect of these convective adjustments keeps the time-averaged ELR of the free troposphere close to the appropriate adiabatic rate—most often near the moist adiabat in the tropics and somewhat steeper in drier or higher-latitude regions.

Hydrostatic equilibrium alone does not select any particular value for the ELR; many different temperature profiles can satisfy hydrostatic balance for a given surface temperature. The adiabatic rates supply the additional thermodynamic constraints that explain why the observed average profile settles where it does. That is why both the DALR and the MALR remain central to any discussion of the origin and maintenance of the environmental lapse rate.

Ben Wouters

Reply to Philip Mulholland

August 7, 2026 5:49 am

Thanks for the extensive reply.

Every other temperature gradient in the troposphere is measured or constrained relative to this scaffold.

In meteorology the ELR (actual temperature profile) is the main data set against which a parcel is compared to determine whether convection can occur, how high it will reach etc. For this the DALR and past the LCL the SALR are used.
It is my understanding that in our atmosphere vertical movement of air (up or down) is less than 1% of the horizontal movement. So most of the time air is just floating around at its own density level.
It seems obvious to me that the HE determines the shape of the average ELR, with room for deviation either above or below this average ELR, before a parcel will leave its original density level and start to sink or rise.
This way we can have temperature inversions, cold and warm fronts etc. while the average ELR will still show its origin in the HE.

Philip Mulholland

Reply to Ben Wouters

August 7, 2026 7:11 am

Ben,
Thanks for the clear exposition. You are right that Hydrostatic Equilibrium (HE) is a fundamental constraint on the mean structure of the troposphere, and that vertical motions are typically a very small fraction of the horizontal flow. The environmental lapse rate (ELR) is of course assessed against the dry (DALR) and saturated adiabatic rates (SALR) precisely because those are the reference processes for parcel stability.

The Dew-Point Anchor Hypothesis (DAPH) does not dispute any of that. On the contrary, once the lifting-condensation level is fixed by the dew-point, the framework integrates the column downward using hydrostatic balance together with the appropriate adiabatic lapse rate. Hydrostatic equilibrium remains the scaffold; the dew-point height simply supplies the observable upper boundary condition from which that integration begins.
So we are in agreement on the role of HE. The distinctive claim of DPAH is only about which boundary condition is treated as independent.

David Dibbell

August 5, 2026 3:34 am

Thanks for this post. This kind of work is an important contribution to physical understanding. I read through the whole thing early yesterday when Andy posted it on X.

Inevitably, the “climate” movement (CAGW) fails because the atmosphere cannot be wrong about its own operation, which we can appreciate in the radiosonde record of observations throughout its depth.

Tim Gorman

August 5, 2026 3:46 am

“Above the molar density intersection, where water vapor is essentially zero, the FPM model has too many degrees of freedom and too many variables to be a proper model, but it is illuminating. It matches observations reasonably well by changing the variable emphasized from latitude to latitude in a reasonable way.”

The size of the degrees of freedom and the fact that there are lots of variables doesn’t mean it can’t be a proper model. The issue is being able to discern whether it is a good physical model or just a good data matching exercise. Since an unknown number of the variables are not independent but are functionally related via an unknown function, it may be a long time before such discernment can be made.

Andy May

Author

Reply to Tim Gorman

August 5, 2026 4:16 am

Good point. Our point is simply that the upper troposphere cools very differently than the lower troposphere, which is dominated by one physical process (moist adiabatic theory). The upper atmosphere has no free water vapor, thus it cools via wind, latent heat release as injected water vapor deposits on ice, and radiation, and the controls on these processes are complex and not very well understood.

Sparta Nova 4

August 5, 2026 5:43 am

Basically, there is no one size fits all model for the earth energy systems.

… meaning CAGW models and other “climate science” fails a null hypothesis.
Some are useful as tools, but do not define the greater whole.

This research explores and we gain understanding from it.
Hopefully the research continues.

One thing to pleasantly recognize is the analysis does not include “trapping heat.”

Philip Mulholland

Reply to Sparta Nova 4

August 5, 2026 3:55 pm

Basically, there is no one size fits all model for the earth energy systems.

We agree. Here is our latest Zenodo deposit that addresses this issue of zonal and temporal atmospheric model limits. Three-Cell and Six-Regime Extension of the Frost-Point Anchor Hypothesis: Markov Chain Models of Cirrus-Zone Latent Heat and Radiative Processes Across the Hadley, Ferrel and Polar Cells (with Day/Night and Seasonal Splits)

Nicholas Schroeder

August 5, 2026 6:52 am

P V = n R T is a state equation that describes the relationship between the variables.
Q = U A (Tsurf – Ttoa) is a process equation that describes how it got that way.

Sparta Nova 4

Reply to Nicholas Schroeder

August 5, 2026 9:55 am

P V = n R T defines the relationships at equilibrium defined by the specific variable values.

Tim Gorman

Reply to Sparta Nova 4

August 6, 2026 4:29 am

Yep. It doesn’t define in any way how any of the variables change over time. There is no “t” in the state equation.

Jim Gorman

August 5, 2026 7:32 am

Andy,

Wow. I must first compliment you and the other authors on doing a comprehensive analysis of the atmosphere. I will also admit, that this is beyond my depth of knowledge so criticism from me is not warranted.

I complement you and the authors for properly quoting quantitative results by using an uncertainty value. That is true science and shows you have considered variance associated with the results.

As to curve fitting, if your model reliably predicts measured values, then it is useful. Whether the parameters become actual variables as more research is done does not take away from its usefulness.

ferdberple

August 5, 2026 8:13 am

Maybe the models perform poorly because the radiative theory is wrong.

Frank from NoVA

Reply to ferdberple

August 5, 2026 9:16 am

Bingo, at least in the bulk (by mass) of the atmosphere where collisions, not absorbtion, is the predominant mechanism by which GHGs are excited.

ferdberple

August 5, 2026 8:22 am

Without convection or radiation the bottom of deep mines is observed to be warmer than the surface.

Sparta Nova 4

Reply to ferdberple

August 5, 2026 10:00 am

Some of that warmth is due to thermal conduction from the earth’s core.

Bob Armstrong

August 5, 2026 9:17 am

Hope to have time to dig into this later .
I think the most poorly appreciated fact in all this brouhaha is that at the top of the atmosphere the temperature must converge with the gray body , ie: flat spectrum , ~ 278.7+-2.3 temperature around our orbit .

Sparta Nova 4

Reply to Bob Armstrong

August 5, 2026 10:00 am

Just one little nit. The top of the atmosphere has no coherent surface, therefore black/brown/grey body calculations do not apply.

Temperature is defined in terms of molecular kinetic collisions with a sensor. The temperature is that low, primarily, due to the paucity of molecules. Thermal energy cannot transit a vacuum.

Jim Gorman

Reply to Sparta Nova 4

August 5, 2026 11:53 am

Let’s talk a bit. Kinetic energy is what defines temperature, that is, its velocity. The low density doesn’t mean low temperature, it just means you need a large sensor to obtain a measurable average velocity.

Thermal energy in the form of an EM wave cat transit a vacuum. A vacuum can’t capture kinetic energy, only gravity can do that thank God.

Tim Gorman

Reply to Bob Armstrong

August 5, 2026 3:51 pm

“the temperature must converge with the gray body , ie: flat spectrum , ~ 278.7+-2.3 temperature around our orbit .”

The surface of a grey body is not required to be isothermal. Only emission is constant, (actual emission)/(black body at the same temperature).

Since the surface of the grey body doesn’t have to be isothermal the radiative flux from any point on the surface can be different. Thus there is no requirement that the temperature “must* converge at TOA.

dh-mtl

August 5, 2026 11:21 am

Excellent paper Andy May.

From what I can understand from your paper, below the ‘molar-density intersection’, the most important variable for heat transfer from the earth’s surface to the upper atmosphere is the rate of water evaporation, and this is captured by your DPAH model.

If this is the case then it raises a couple of questions in my mind.

Might increased atmospheric CO2 actually increase the rate of heat transfer from the atmosphere to space, as it would increase the emissivity of the atmosphere above the MDI?
Since most water evaporation is from the surface of the tropical oceans, then one of the most important variables related to the rate of water evaporation would be the strength of phenomena such as ENSO and tropical storms, which are highly sensitive to water temperatures (above 25C). If this is the case, do we pay enough attention, in climate modeling, to (largely solar based) heat accumulation within the oceans, that, after being transported along the ocean currents, is eventually released, via water evaporation, to the atmosphere in the tropical oceans?

Andy May

Author

Reply to dh-mtl

August 5, 2026 11:41 am

Point #1: That is the consensus idea. Additional cooling of the stratosphere is observed and could be due in part to additional CO2. However, as noted in the post, current CMIP models extend their equivalent of the DPAH too high and as a result overestimate temperatures in the upper troposphere. They need to divide the troposphere into upper and lower as we did in this study to correct that error.

#2: Heat storage in the oceans is almost entirely from solar radiation, since GHG radiation cannot penetrate the ocean surface. I do not think modern models pay enough attention to variations in solar radiation absorbed by the oceans. It is modulated by clouds, Earth’s orbital variations, and by changes in the Sun itself. Unfortunately, modern models uniformly assume a constant Sun over climatic periods of time, which is an error IMHO.

Tim Gorman

Reply to Andy May

August 6, 2026 4:37 am

There are LOTS of factors involved with absorption of insolation by the oceans. Some of ones you didn’t list are “contaminants” in the water such as plankton, plastics, etc; frothing during wave action (a function of wind?), and salinity. Some may be small effects but lots of small effects add up sooner or later.

Once again, climate science just ignores the physical realities and tries to parameterize everything into some “guessed” at average.

Philip Mulholland

Reply to Tim Gorman

August 6, 2026 5:12 am

There are LOTS of factors involved with absorption of insolation by the oceans.

Tim,
You are right that a long list of real physical factors (plankton, plastics, wave frothing, salinity, surface films, etc.) affect how the ocean absorbs and partitions insolation. Most of those effects ultimately show up as changes in the surface energy budget and, critically, in the rate of evaporation that supplies water vapour to the atmosphere.

The Dew-Point Anchor Hypothesis (DPAH) framework was developed precisely to avoid burying those processes inside opaque “average” parameterisations. Instead of starting from a top-down radiative forcing and then tuning surface fluxes, DPAH treats the observable lifting-condensation level (the dew-point height) as the primary thermodynamic boundary condition. Surface temperature, pressure and the vertical structure of the troposphere then emerge as dependent variables constrained by moist-adiabatic and hydrostatic physics.

A concrete illustration is the Markovian state-space model documented here:
DPAH Markovian Matrix: Dew-Point Anchor Hypothesis – Stochastic Modeling of Tropospheric Attractors (Runs 301–311)

That deposit constructs a large discrete state space (surface pressure × temperature × dew point × environmental lapse rate) and computes stationary distributions for tropical ascent and subtropical descent regimes. One of the explicit sensitivity experiments systematically suppresses ocean-surface evaporation (simulating oil films or other surface contaminants). The result is a measurable, physically consistent shift in the location and strength of the warm-moist tropospheric attractor. In other words, the model does not ignore the surface physics; it lets those physics alter the boundary condition that then organises the entire column.

The same philosophy is being extended to the upper troposphere with the Frost-Point Anchor Hypothesis (FPAH) and to latitude-dependent regimes (Hadley / Ferrel / Polar). The goal is not another guessed average, but a set of observationally anchored, regime-specific stochastic models that can accept the very real surface complexities you list and still produce testable stationary distributions.
Happy to discuss further if you take a look at the Zenodo material.

Tim Gorman

Reply to Philip Mulholland

August 6, 2026 9:16 am

“That deposit constructs a large discrete state space (surface pressure × temperature × dew point × environmental lapse rate) and computes stationary distributions for tropical ascent and subtropical descent regimes.“

My main concerns here is with the measurement uncertainty of the data used in the state space. It’s going to propagate into your state matrix. It won’t limit the usefulness of the model as a research tool but if results don’t quite match what you expect they may still be covered by the uncertainty interval.

Philip Mulholland

Reply to Tim Gorman

August 6, 2026 9:48 am

Tim,
You raise a fair and important point. Measurement uncertainty in the four state variables (surface pressure, temperature, dew point and environmental lapse rate) will inevitably propagate into the transition matrix and the derived stationary distributions.

In the present implementation the state space is discretised on a relatively coarse grid. That choice was deliberate: the primary goal of the Runs 301–311 series is to test whether a clean, physics-kernel Markov model, anchored at the dew-point LCL, can produce coherent and physically interpretable attractors for the ascent and descent regimes. At this stage the model is a research and scoping tool rather than a high-precision forecasting instrument. Consequently the absolute numerical values of the stationary probabilities should be read with the understanding that they sit inside uncertainty envelopes that have not yet been formally quantified.

There are several practical ways the uncertainty can be addressed in future work:

Monte-Carlo perturbation of the input fields – drawing the four state variables from distributions that reflect instrument and sampling uncertainty and then recomputing the stationary distributions for each realisation.Coarser versus finer discretisations – checking how sensitive the location of the attractor is to the bin widths used for pressure, temperature, dew point and lapse rate.Observational filtering – restricting the analysis to radiosonde or reanalysis subsets that meet stricter quality-control criteria, thereby reducing the input variance.
All of them increase the computational demand. The current code base was written to remain fully usable on my standard consumer laptop (the Dell Inspiron class of machine that has been the target platform throughout this series). That hardware constraint sets a practical upper bound on grid resolution and ensemble size for the moment, but it does not change the architectural claim of DPAH: that the dew-point height functions as an observable thermodynamic boundary condition from which the rest of the tropospheric structure can be integrated.

So I agree with you — the uncertainty interval is real and will need to be characterised more rigorously. For the present the model’s usefulness lies in demonstrating that a low-parameter, phase-change-anchored stochastic framework can generate stable, regime-dependent tropospheric attractors and can respond in a physically consistent way to changes in surface evaporative flux. Quantifying the uncertainty envelope is a natural next refinement once the computational resources allow it.
Thanks for the constructive critique.

Tim Gorman

Reply to Philip Mulholland

August 7, 2026 4:14 am

You have to walk before you run. You are in the toddling stage right now. I’m sure you and Andy will move to the next stage soon enough.

I am not a modeling expert but I am always leery of using MC in physical science studies like this. Determining the distribution profiles of the data and the uncertainty is typically a guess limited by the measurement data. You can’t just assume Gaussian for everything like climate science does.Thus the MC runs inherit the same. In addition, different MC profile runs can produce similar overall results. Deciding which profiles actually match reality becomes a challenge all of its own. Good luck.

Philip Mulholland

Reply to Tim Gorman

August 7, 2026 5:10 am

Tim,
Thank you — “toddling” is a fair description and I am happy to own it. The present Markov experiments are deliberately scoped as first steps: can a low-parameter, phase-change-anchored state-space model produce coherent tropospheric attractors at all, and do those attractors respond in a physically sensible way when surface evaporation is altered? So far the answer appears to be yes, but it is still early days.
Your caution about Monte Carlo (MC) is well taken. In physical problems the input uncertainty distributions are rarely known with precision, and they are often non-Gaussian. Simply drawing from assumed normal distributions can therefore embed the same optimistic assumptions that appear elsewhere in climate modelling. Different plausible input distributions can also map onto similar-looking stationary distributions, which then leaves the difficult inverse problem of deciding which input profile is actually closest to reality. This is where Andy brings his heavy-weight petrophysical expertise to our collaboration. During my professional career as an oil industry geoscientist the input of a skilled petrophysicist always helped to keep me grounded 😉

For those reasons we have not yet moved to a full Monte-Carlo uncertainty quantification. The current results rest on a fixed, relatively coarse discretisation and a deterministic physics kernel. When we do introduce stochastic sampling it will have to be done carefully — starting with the actual empirical distributions of the radiosonde and reanalysis data rather than with convenient parametric assumptions, and with clear diagnostics that show how sensitive the attractors are to those choices.
In the final analysis modelling is always driven by observational data, it can never be simply an exercise in pseudo-data generation.
I appreciate the realistic perspective. Walking comes before running, and we will try to keep the steps solid.

Jim Gorman

Reply to Philip Mulholland

August 7, 2026 6:12 am

I would be remiss if I didn’t congratulate you and Andy on the “modelling is always driven by observational data,” goal of your research. That is what science is, creating a hypothesis (model) that predicts accurately.

So much of climate science is based upon time series trends being the proof of the hypothesis of CAGW, yet that hypothesis does not predict accurate results from predictor variables.

Keep up the good work!

Tim Gorman

Reply to Philip Mulholland

August 7, 2026 9:36 am

“ deterministic physics kernel”

Your model has a grounded base. Current climate science doesn’t. You two are already one step ahead.

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Tropospheric structure

The Molar Density Intersection

The modeling results

Methods

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