Net radiation estimation using the Brunt equation for clear sky emissivity and air and canopy temperatures for longwave radiation in well watered crops

This study demonstrates that for well-watered crops, net radiation can be accurately estimated at daily time scales using regionally calibrated Brunt equations for clear-sky emissivity and air temperature as a practical substitute for canopy temperature, outperforming the standard Allen/FAO method.

The Gist

Imagine you are trying to figure out how much "energy money" a field of crops has in its bank account at any given moment. This energy money is called Net Radiation. It's the total energy the plants get from the sun minus the energy they lose back to the sky.

Why do we care? Because this energy is the fuel that drives everything the plants do: drinking water, growing leaves, and cooling themselves down. If you want to know how much water a farmer needs to irrigate, you first need to know how much energy the plants are using.

The Problem:
Measuring this "energy bank account" directly is like trying to weigh a ghost. You need expensive, delicate sensors that are hard to keep working perfectly. So, scientists usually try to calculate it using math formulas.

The Two Big Guesses:
To do the math, scientists have to make two big guesses:

1. The Sky's Blanket: How much heat does the sky send down to the earth? (This is called downward longwave radiation).
2. The Plant's Body Temperature: How hot is the plant actually running? (This is called canopy temperature).

Usually, scientists use a standard "rule of thumb" (the Allen/FAO method) to guess these numbers. But this paper asks: Can we do better? And can we simplify the process?

The Experiment: The "Thermometer Swap"

The researchers set up a giant experiment in cotton and sesame fields in Texas over two years. They treated the fields like a laboratory to test two main ideas:

1. The "Regional Recipe" vs. The "Universal Recipe"
Think of the standard math formula as a generic cake recipe that works okay everywhere but isn't perfect.

• The researchers tried two "specialized recipes" (the Brunt equation) that had been pre-tuned for specific regions (one for Texas, one for the whole US).
• The Result: Just like using a recipe written specifically for your local oven, these specialized recipes baked a much better cake (more accurate energy estimates) than the generic one. The best part? You don't need to be a master chef (or a local scientist) to tweak the recipe; you can just use the pre-tuned one and get great results.

2. The "Air vs. Leaf" Thermometer Swap
This is the most interesting part. To calculate how much heat the plants radiate back up, you technically need to measure the temperature of the leaves (Canopy Temperature). But leaf thermometers are expensive and tricky to install.

• The Question: Can we just use the temperature of the air around the plants (Air Temperature) instead? It's like asking, "If I know the room temperature, can I guess the body temperature of a healthy person sitting in that room?"
• The Catch: If the person is sick or dehydrated, their body temp might be very different from the room. But if they are healthy and well-hydrated (well-watered crops), their body temp usually matches the room temp pretty closely.
• The Result:
  • Hourly Scale (Minute-by-minute): The air and leaf temperatures were a bit different, like a person shivering slightly in a warm room. The math was a little off if you swapped them.
  • Daily Scale (The whole day): When you average it out over 24 hours, the difference almost vanished. The "room temperature" was a perfect stand-in for the "body temperature."

The Big Takeaway (In Plain English)

1. Stop using the old, generic math. If you want to know how much energy crops are using, use the newer "regional" formulas (the Brunt equation). They are more accurate and don't require you to do extra homework to calibrate them for your specific farm.
2. You don't need expensive leaf sensors for daily planning. If you are a farmer or a water manager trying to figure out how much to water your crops for the day, you can safely use the standard air temperature from your local weather station. You don't need to measure the actual temperature of the leaves, as long as the crops are getting enough water.
3. Timing matters. If you need to know the energy balance every single hour (like for a very precise scientific study), you do need the leaf temperature. But for the big picture (daily water needs), the air temperature is good enough.

The Analogy:
Think of the crops as a car engine.

• Net Radiation is the fuel gauge.
• The Old Method was like guessing the fuel level based on the color of the sky.
• The New Method is like using a GPS that knows the local traffic patterns (the regional calibration).
• The Temperature Swap is like checking the engine temperature. If you need to know the exact temperature right now to fix a broken part, you need a probe in the engine. But if you just want to know if the car is running cool enough for a long trip today, checking the temperature of the air coming out of the radiator is close enough to tell you everything you need to know.

Conclusion:
This paper gives farmers and scientists a "cheat code." It says, "You can use simpler, cheaper tools (air temperature) and better pre-made formulas (regional Brunt models) to get the job done without losing much accuracy." It saves money and time while keeping the science solid.

Technical Summary

1. Problem Statement

Net radiation ($R_n$) is a critical input for biophysical models (e.g., Penman-Monteith, Crop Water Stress Index) as it represents the energy available for evapotranspiration and heat fluxes. However, direct measurement of $R_n$ is often expensive and logistically difficult. Consequently, $R_n$ is frequently estimated using theoretical models.

Two primary challenges exist in estimating $R_n$:

1. Incoming Longwave Radiation ($L_{\downarrow}$): Calculating this requires estimating sky emissivity ($\epsilon_{sky}$). While the Brunt equation is a standard reference, many models require site-specific calibration, limiting their broader applicability.
2. Outgoing Longwave Radiation ($L_{\uparrow}$): This requires surface temperature ($T_s$). In many practical applications, $T_s$ is approximated by air temperature ($T_{air}$). However, this substitution is generally considered valid only for "reference" (well-watered) conditions. It remains unclear if $T_{air}$ is a sufficient substitute for canopy temperature ($T_c$) in specific crops like cotton without introducing significant errors, particularly at short time scales.

2. Methodology

The study was conducted over two years (2015 and 2025) at the Texas A&M AgriLife Research and Extension Center in Uvalde, Texas, under well-watered (irrigated) conditions.

• Experimental Setup:
  • 2015: Measurements in sesame and cotton fields (no canopy temperature sensors).
  • 2025: Measurements in a cotton field with added infrared radiometers to measure canopy temperature ($T_c$).
  • Data: Net radiation was measured using NR-Lite2 sensors at 1-minute intervals. Air temperature, humidity, and solar radiation were recorded at 5-minute intervals.
• Modeling Approaches:
  • Standard Method: The FAO-Allen (1994) model was used as the baseline.
  • Brunt Models: Two previously calibrated Brunt-type models for clear-sky emissivity were tested without further site-specific calibration:
    1. Formetta et al. (2016): Regionally calibrated for Texas/U.S. states.
    2. Li et al. (2017): A universal model calibrated using a large dataset across the U.S.
  • Cloud Correction: The Crawford and Duchon (1999) model was applied to adjust clear-sky emissivity for cloud cover.
  • Temperature Substitution: In 2025, $R_n$ was calculated twice: once using measured $T_c$ and once using $T_{air}$ to evaluate the substitution hypothesis.
• Statistical Analysis:
  • Performance was evaluated at hourly and daily time scales.
  • Metrics included Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Bias, and Kling-Gupta Efficiency (KGE).
  • Statistical differences between $T_{air}$ and $T_c$ methods were assessed using the Wilcoxon signed-rank test, linear regression, and Bland-Altman analysis.

3. Key Contributions

• Validation of Regional/Universal Calibration: The study demonstrates that Brunt equations calibrated by Formetta (regional) and Li (universal) can be applied to cotton and sesame crops without local calibration, outperforming the original FAO-Allen parameters.
• Temperature Substitution Threshold: It provides empirical evidence regarding the interchangeability of $T_{air}$ and $T_c$ for $R_n$ estimation in well-watered cotton, distinguishing between hourly and daily time scales.
• Error Source Identification: The paper identifies that errors in short-time-scale estimations are driven by assumptions regarding constant albedo and cloudiness functions, rather than just emissivity models.

4. Results

• Time Scale Dependency: Errors were significantly higher at the hourly scale compared to the daily scale for all methods.
  • Daily Scale (Year 1): RMSE = 11.88 W m⁻², KGE = 0.91.
  • Daily Scale (Year 2): RMSE = 13.45 W m⁻², KGE = 0.74.
  • Hourly Scale: Errors were roughly 3–4 times higher (e.g., RMSE ~39–56 W m⁻²).
• Model Performance: Both the Formetta and Li Brunt models outperformed the standard FAO-Allen method. The differences between the two Brunt models were modest, though the Li model showed slightly lower bias.
• $T_{air}$ vs. $T_c$ Substitution:
  • Statistical Significance: The Wilcoxon test indicated a statistically significant difference between $R_n$ calculated with $T_{air}$ and $T_c$.
  • Practical Significance: Despite statistical differences, the error metrics (RMSE, MAE) were very similar.
  • Bias: Using $T_{air}$ resulted in a systematic underestimation of approximately -5.0 W m⁻² compared to $T_c$.
  • Time Scale Effect: At the daily scale, the regression slope was close to 1 with an intercept not significantly different from zero, indicating negligible differences. At the hourly scale, a systematic bias was observed (intercept significantly different from zero), with $T_{air}$ generally higher than $T_c$ (average difference of -0.82°C).
• Year 2 Discrepancy: Higher errors in 2025 were attributed to lower Leaf Area Index (LAI), leading to increased soil exposure. The fixed albedo assumption (0.21) likely underestimated $R_n$ because exposed soil has a lower albedo than the crop canopy.

5. Significance and Conclusion

• Elimination of Site-Specific Calibration: The study confirms that regionally or universally calibrated Brunt equations are robust enough for estimating net radiation in well-watered crops, removing the need for expensive, site-specific calibration campaigns.
• Practical Application for $T_{air}$: For daily time scale applications in well-watered crops, air temperature ($T_{air}$) is a valid and practical substitute for canopy temperature ($T_c$). The modest improvement gained by measuring $T_c$ does not justify the added complexity and cost for daily energy balance calculations.
• Limitations: The substitution of $T_{air}$ for $T_c$ is less accurate at hourly scales and likely invalid under water-stressed conditions where canopy temperatures can rise significantly above air temperatures. Future research is needed to define the critical temperature difference threshold where this substitution fails.

In summary, the paper provides a simplified, robust framework for estimating net radiation in agricultural settings, validating the use of pre-calibrated emissivity models and air temperature proxies for daily energy balance assessments in irrigated systems.