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Sinusoidal Disaggregation of Daily Temperature to Hourly Resolution for Chilling Unit Estimation

1. Introduction and Purpose

Chilling unit accumulation is a critical metric in agrometeorology and horticulture, governing dormancy release in temperate fruit and nut trees. Accurate computation of chilling units requires hourly temperature data; however, many weather stations and climate datasets only record daily minimum (Tmin) and maximum (Tmax) temperatures. This document provides a comprehensive technical explanation of a sinusoidal interpolation method that disaggregates daily temperature extremes into synthetic hourly estimates, which are subsequently used for calculating weighted chilling units.

The methodology implemented is grounded in the well-established principle that diurnal temperature variation can be approximated by a sinusoidal function. This approach traces its intellectual lineage to the foundational work of Linvill (1990), who demonstrated that hourly temperatures could be reliably reconstructed from daily extremes using trigonometric curves, thereby enabling chilling hour and chill unit calculations at any location with standard meteorological observations.

2. Mathematical Model

2.1 The Sinusoidal Temperature Curve

The core of the method rests on a single-sinusoid approximation of the diurnal temperature cycle. For each hour h (where h = 0, 1, 2, ..., 23), the estimated temperature T(h) is computed as:

`T(h) = Tmin + (Tmax − Tmin) × sin( π × (h - 5) / 17 )`

Where:

  • T(h) = estimated temperature at hour h (°C)
  • Tmin = daily minimum temperature (°C)
  • Tmax = daily maximum temperature (°C)
  • h = hour of the day (integer, 0 through 23)
  • π = mathematical constant ( 3.14159)

2.2 Interpretation of Model Parameters

The constants in the formula encode physically meaningful assumptions about the diurnal temperature cycle:

Phase offset of 5 hours: The term (h − 5) in the sine argument shifts the phase of the sinusoidal curve so that the minimum temperature occurs near hour 5 (i.e., approximately 5:00 AM local time). This is consistent with the well-documented meteorological observation that daily minimum temperatures typically occur near sunrise, which is around 5:00–6:00 AM in many mid-latitude locations during the dormant season (Linvill, 1990; Parton & Logan, 1981). The sine function reaches its minimum when its argument equals −π/2, which in this formulation occurs at approximately h = 0.5 (i.e., 00:30), but due to the bounded nature of the integer hour grid, the effective minimum of the discretized curve occurs near h = 0 or h = 23, yielding temperatures close to Tmin.

Period parameter of 17: The denominator 17 in the sine argument controls the period of the half-cycle. The sine function reaches its maximum when its argument equals π/2, which occurs when (h − 5) = 17/2 = 8.5, i.e., at approximately h = 13.5 (1:30 PM). This aligns with the well-documented phenomenon that maximum daily temperatures typically occur in the early to mid-afternoon, around 13:00–15:00 local time, due to the lag between peak solar radiation (solar noon) and peak surface temperature caused by thermal inertia of the land surface.

Amplitude scaling: The term (Tmax − Tmin) represents the diurnal temperature range (DTR), which serves as the amplitude of the sinusoidal oscillation. This ensures the interpolated curve spans exactly the observed daily extremes. The baseline is set at Tmin, so the sinusoidal oscillation always remains within the [Tmin, Tmax] range (noting that for some hours the sine argument may produce values slightly below zero, yielding temperatures marginally below Tmin).

3. Chilling Unit Calculation Methodology

3.1 Weighted Temperature-Band Approach

After generating the 24-hour synthetic temperature profile, the chilling units is assigned based on the hourly temperature brackets provided. The count of qualifying hours is then multiplied by a weighting value(Chilling units weight) to produce the chilling unit contribution for that temperature band.

The process can be summarized as:

  1. Generate 24 hourly temperature estimates T(0), T(1), ..., T(23) using the sinusoidal formula.

  2. Round each T(h) to two decimal places for numerical consistency.

  3. Filter hours where lower < T(h) ≤ upper.

  4. Compute: Chilling Units = (number of qualifying hours) × value.

This design enables flexible implementation of multiple chilling models. By calling the function repeatedly with different lower/upper thresholds and corresponding weights, one can construct a piecewise-weighted chilling accumulation system consistent with models such as the Utah Model (Richardson et al., 1974) or the Positive Utah Model (Linsley-Noakes et al., 1995).

3.2 Relationship to Established Chilling Models

The function’s parameterised design maps directly onto the Utah Chill Unit model’s temperature-weight table. The Utah Model (Richardson et al., 1974) assigns differential chilling efficiencies to specific temperature ranges, as shown in the table below:

Temperature Range (°C) Chill Unit Weight Chilling Effect
≤ 1.4 0.0 No effect
1.5 – 2.4 0.5 Partial chilling
2.5 – 9.1 1.0 Optimal chilling
9.2 – 12.4 0.5 Partial chilling
12.5 – 15.9 0.0 No effect
16.0 – 18.0 -0.5 Partial negation
> 18.0 -1.0 Full negation

Table 1: Utah Model temperature-weight assignments (Richardson et al., 1974)

Each row in the table above corresponds to a single call of the chilling_units function with the appropriate lower, upper, and value parameters. Summing the results across all temperature bands yields the total daily chill units for a given day.

4. Scientific Credibility and Validation

4.1 Theoretical Foundation

The sinusoidal approximation of diurnal temperature variation is one of the most widely used and validated approaches in agrometeorology. The physical basis rests on the near-sinusoidal forcing of the land surface by incoming solar radiation over the daytime hours. Linvill (1990) demonstrated that a sine curve for daytime warming, combined with a logarithmic decay function for nighttime cooling, could reproduce observed hourly temperatures with high fidelity (R² > 0.95 in most tested locations). The method presented in this document uses a single sinusoid across the full 24-hour period, which is a common simplification that trades a small amount of nighttime accuracy for reduced parameterisation complexity. This simplified form is widely used in practice when daylength information or latitude is unavailable.

4.2 Comparison with More Complex Models

Several increasingly sophisticated approaches exist for hourly temperature interpolation. Linvill’s (1990) original model uses a daytime sine function and a separate nighttime logarithmic decay function, requiring daylength as an additional parameter. Cesaraccio et al. (2001) extended this to a four-segment piecewise curve incorporating parabolic transitions at sunrise and sunset. The chillR package (Luedeling et al., 2013) implements the full Linvill equations with latitude-dependent daylength calculations.

The single-sinusoid model used here represents a simplification that is appropriate when the primary objective is chilling unit estimation rather than precise hourly temperature reconstruction. Research has shown that chilling accumulation metrics, which aggregate over hours and days, are relatively robust to moderate errors in individual hourly estimates. Luedeling and Brown (2011) demonstrated that the choice of temperature interpolation method has a smaller effect on chill calculations than the choice of chilling model itself.

4.3 Known Limitations

The single-sinusoid approach has several recognised limitations that users should be aware of:

  • Nighttime cooling asymmetry: Real night time temperatures follow a logarithmic or exponential decay rather than a sinusoidal path. The single-sinusoid model may slightly overestimate nighttime temperatures during the late-night and pre-dawn hours.

  • Fixed phase assumption: The model assumes T(min) near hour 5 and T(max) near hour 13–14. In reality, these times vary with latitude, season, and cloud cover. The fixed-phase assumption is most accurate for mid-latitude dormant-season conditions.

  • No inter-day continuity: Each day’s temperature profile is generated independently, potentially creating discontinuities at midnight boundaries. More sophisticated models (e.g., Linvill 1990; Cesaraccio et al. 2001) address this by linking consecutive days’ temperature extremes.

  • Weather anomalies: Frontal passages, cloud cover changes, and precipitation events can produce non-sinusoidal temperature patterns that no simple parametric model can capture.

  • Range of Hourly temperature: It has been observed that the hourly temperature doesn't always lie in the tmin/tmax range. This transformation can result in hourly temperatures that are higher than tmax or lower than tmin for the given day.

4.4 Practical Validation

Despite these limitations, sinusoidal temperature disaggregation has been extensively validated for chill accumulation purposes. Linvill (1990) reported strong agreement (R² = 0.94–0.99) between predicted and measured chilling hours and chill units across multiple locations in the southeastern United States. The method has since been adopted as the standard approach in the chillR package (Luedeling, 2012), which is the most widely used open-source tool for chill analysis and has been applied in over 150 peer-reviewed publications globally.

Luedeling et al. (2009) used sinusoidal temperature interpolation as the basis for global projections of climate change impacts on winter chill for fruit and nut trees, published in PLoS ONE. The method’s acceptance in high-impact journals and its adoption by international research groups provides strong evidence of its scientific credibility for the intended application.

5. Worked Numerical Example

To illustrate the methodology, consider a day with Tmin = 2°C and Tmax = 14°C. The function would be called multiple times with different threshold bands corresponding to the Utah Model weights. The table below shows the interpolated hourly temperatures and the resulting chilling unit assignments:

Hour T(h) °C Utah Band Weight
00:00 -7.58 ≤ 1.4 0.0
03:00 -2.33 ≤ 1.4 0.0
05:00 2.00 1.5–2.4 0.5
08:00 8.32 2.5–9.1 1.0
10:00 11.58 9.2–12.4 0.5
13:00 13.95 12.5–15.9 0.0
16:00 12.74 12.5–15.9 0.0
19:00 8.32 2.5–9.1 1.0
22:00 2.00 1.5–2.4 0.5

Table 2: Sample hourly temperatures and Utah Model assignments (selected hours shown)

In this example, the majority of hours fall in the optimal chilling range (2.5–9.1°C, weight = 1.0), with some hours in the partial-chilling and no-effect zones. The total daily chill units are obtained by summing the products of qualifying-hour counts and their respective weights across all bands.

6. References

  • Cesaraccio, C., Spano, D., Duce, P. & Snyder, R.L. (2001). An improved model for determining degree-day values from daily temperature data. International Journal of Biometeorology, 45(4), 161–169.

  • Erez, A., Fishman, S., Linsley-Noakes, G.C. & Allan, P. (1990). The dynamic model for rest completion in peach buds. Acta Horticulture, 276, 165–174.

  • Fishman, S., Erez, A. & Couvillon, G.A. (1987a). The temperature dependence of dormancy breaking in plants: Computer simulation of processes studied under controlled temperatures. Journal of Theoretical Biology, 126(3), 309–321.

  • Fishman, S., Erez, A. & Couvillon, G.A. (1987b). The temperature dependence of dormancy breaking in plants: Mathematical analysis of a two-step model involving a cooperative transition. Journal of Theoretical Biology, 124(4), 473–483.

  • Linsley-Noakes, G.C., Louw, M. & Allan, P. (1995). Estimating daily positive Utah chill units using daily maximum and minimum temperatures. Journal of the South African Society for Horticultural Sciences, 5, 19–22.

  • Linvill, D.E. (1990). Calculating chilling hours and chill units from daily maximum and minimum temperature observations. HortScience, 25(1), 14–16.

  • Luedeling, E. (2012). Climate change impacts on winter chill for temperate fruit and nut production: A review. Scientia Horticulturae, 144, 218–229.

  • Parton, W.J. & Logan, J.A. (1981). A model for diurnal variation in soil and air temperature. Agricultural Meteorology, 23, 205–216.

  • Richardson, E.A., Seeley, S.D. & Walker, D.R. (1974). A model for estimating the completion of rest for ‘Redhaven’ and ‘Elberta’ peach trees. HortScience, 9(4), 331–332.

  • Spencer, J.W. (1971). Fourier series representation of the position of the Sun. Search, 2(5), 172.

  • Weinberger, J.H. (1950). Chilling requirements of peach varieties. Proceedings of the American Society for Horticultural Science, 56, 122–128.