# Irradiance Variability Quantification and Small-Scale Averaging in Space and Time: A Short Review

## Abstract

**:**

## 1. Introduction

^{2}and 80 km

^{2}, some of which are readily available online [72,73,74,75,76].

## 2. Irradiance Normalization and Time-Scale-Specific Changes

^{−2}[26,50]. In order to study the stochastic nature of weather-induced variability on short time scales, most authors take measures to remove deterministic trends from GHI time series G by estimating either the clearness index (a.k.a. transmission coefficient, e.g., [20])

^{6}s worth of 1 s clear-sky index data).

## 3. Variability Quantification

^{−2}can be used to normalize to STC [26]). The variability score (VS) [50] is another versatile measure, which can be defined for any time step $\tau $ as

## 4. Spatial Averaging

^{2}, and (b) 1 min increments are shown for both the single sensor and the spatial average. Pronounced spatial smoothing is evident in both panels.

^{−1}, based on several proposed models [10,13,52,68,83,103,106]. Isotropic structures obtained without regards to the direction of cloud motion and structures calculated along the main wind direction are respectively considered in panels (a) and (b). While each isotropic curve in panel (a) decreases monotonously from 1 towards 0 for increasing distances, the rate of decrease differs considerably between models. For example, at a pair distance of ${d}_{ij}\approx 100\mathrm{m}$, model results range between $0\lesssim {\rho}_{ij}^{\u2206{k}_{\tau}^{\ast}}\lesssim 0.5$. Similarly, all along-wind correlation structures in panel (b) decrease with distinct rates for increasing distances up to ${d}_{ij}=100\mathrm{m}$, where differences between models range between $-0.5\lesssim {\rho}_{ij}^{\u2206{k}_{\tau}^{\ast}}\lesssim 0$. For further increasing distances ${d}_{ij}>100\mathrm{m}$, each curve transitions to its own unique character of constant or increasing correlation coefficients, approaching 0.

## 5. Temporal Averaging

- determining instantaneous irradiance variations for each of a few hundred days in spring and summer by calculating the second temporal derivative of each observation, considering the minimum (i.e., negative) value of a day’s derivatives to represent the day’s most severe fluctuation, and then computing an ideal averaging time by assuming the variations to feature parabolic shapes and accepting an error of 10 Wm
^{−2}in the measurements [17]; - assessing the reduction of the standard deviation of an irradiance time series (measured during 7 h on a single summer’s day) as a function of increasing averaging time scales [63];
- separately studying the variability index and the variability score of irradiance for seven selected days as functions of increasing averaging time scales [16]; and
- characterizing the changes of ${k}^{\ast}$ and $\u2206{k}_{\tau}^{\ast}$ standard deviations as a function of averaging time using thousands of hours worth of irradiance observations with raw temporal resolutions ranging from 0.01 through 1 s [111].

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) An example of a diurnal cycle of measured global horizontal irradiance (GHI) along with simulated clear-sky irradiance derived by a basic model [140]; (

**b**) A corresponding clear-sky index estimate ${k}^{\ast}$ according to Equation (2). The irradiance data were measured during the HD(CP)2 Observational Prototype Experiment (HOPE) campaign [138,139] with 1 s temporal resolution near Jülich, Germany, on 25 April 2013, and have been conditioned to exclude times of low solar elevation angles $\alpha <15\xb0$ after sunrise and before sunset.

**Figure 2.**Probability density estimates of clear-sky index ${k}^{\ast}$ based on the single-day example time series previously shown in Figure 1 (histogram, black lines), as well as a considerably longer series containing a total of about $4.5\xb7{10}^{6}$ s worth of single-sensor 1 s measurements collected during the HOPE campaign [138,139] (kernel density estimate, KDE, dashed blue line). The KDE was derived using Gaussian kernels with a smoothing bandwidth (i.e., smoothing kernel standard deviation) of 0.05.

**Figure 3.**A five-minute subset of the time series previously shown in Figure 1 (there indicated by a vertical gray line). (

**a**) Clear-sky index ${k}^{\ast}$ time series; (

**b**) Corresponding clear-sky index increments $\u2206{k}_{\tau}^{\ast}(t)$ as per Equation (5) for three distinct short-term increment time steps $\tau =5\mathrm{s}$, $\tau =10\mathrm{s}$, and $\tau =60\mathrm{s}$. The underlying irradiance data were collected during the HOPE campaign [138,139].

**Figure 4.**Kernel density estimates of clear-sky index increments, using about $4.5\xb7{10}^{6}$ s worth of single-sensor 1 s data collected during the HOPE campaign [138,139]. The estimates were derived for increment time steps $\tau =5\mathrm{s}$, $\tau =10\mathrm{s}$, and $\tau =60\mathrm{s}$ using Gaussian kernels with a smoothing bandwidth of 0.01.

**Figure 5.**A comparison between daily values of the clear-sky index variability score ${\mathrm{VS}}_{\tau}^{{k}^{\ast}}$ (cf. Equation (8)) and the clear-sky index increment standard deviation ${\sigma}^{\u2206{k}_{\tau}^{\ast}}$ (cf. Equation (9)) for three time steps (

**a**) $\tau =5\mathrm{s}$, (

**b**) $\tau =10\mathrm{s}$, and (

**c**) $\tau =60\mathrm{s}$, using 105 days worth of single-sensor 1 s clear-sky index estimates derived from irradiance data collected during the HOPE campaign [138,139] (i.e., the same data as in Figure 4). Least-square fits are included as dashed lines, and the corresponding coefficients of determination ${R}^{2}$ are quoted in each panel.

**Figure 6.**An example of spatial averaging: (

**a**) the same 1 s single-sensor clear-sky index ${k}^{\ast}$ time series as in Figure 1 along with a corresponding spatially averaged 1 s clear-sky index computed over as many as 99 pyranometers dispersed over an area of about 80 km

^{2}; (

**b**) 1 min clear-sky index increments $\u2206{k}_{\tau}^{\ast}$ derived according to Equation (5) for both the single-sensor time series and the spatial average. The underlying irradiance data were collected during the HOPE campaign [138,139].

**Figure 7.**Some examples of previously proposed models to estimate the two-point correlation coefficient of clear-sky index increments ${\rho}_{ij}^{\u2206{k}_{\tau}^{\ast}}$ as a function of distance ${d}_{ij}$ for increment time scale $\tau =10$ s and cloud speed $v=10$ ms

^{−1}. Panel (

**a**) shows examples of isotropic (i.e., without considering the direction of cloud motion) correlation structures proposed by Hoff and Perez [10], Perez et al. [106], and Lave et al. [83] (each derived from simple equations implying a linear relationship between ${d}_{ij}$, $\tau $ and v), as well as Lohmann et al. [52] (based on 3 h worth of 1 s ${k}^{\ast}$ field data simulated by a fractal cloud model with an assumed average cloud cover of 50%). Panel (

**b**) presents along-wind structures according to models proposed by Lonij et al. [103], Arias-Castro et al. [68], and Widén [13] (the presented curves were extracted from Elsinga and van Sark [71]; cf. their Figure 2), as well as the aforementioned fractal model [52].

**Figure 8.**An example of temporal averaging: (

**a**) Averages of 1 min and 10 min temporal resolutions based on the 1 s single-sensor clear-sky index ${k}^{\ast}$ time series shown in Figure 1 (collected during the HOPE campaign [138,139]); (

**b**) Corresponding 10 min clear-sky index increments $\u2206{k}_{\tau}^{\ast}$ derived according to Equation (5).

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Lohmann, G.M. Irradiance Variability Quantification and Small-Scale Averaging in Space and Time: A Short Review. *Atmosphere* **2018**, *9*, 264.
https://doi.org/10.3390/atmos9070264

**AMA Style**

Lohmann GM. Irradiance Variability Quantification and Small-Scale Averaging in Space and Time: A Short Review. *Atmosphere*. 2018; 9(7):264.
https://doi.org/10.3390/atmos9070264

**Chicago/Turabian Style**

Lohmann, Gerald M. 2018. "Irradiance Variability Quantification and Small-Scale Averaging in Space and Time: A Short Review" *Atmosphere* 9, no. 7: 264.
https://doi.org/10.3390/atmos9070264