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Keywords:

  • autocorrelation;
  • intermittency;
  • rainfall;
  • statistical analysis

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Modeling the Intermittency
  5. 3. Data
  6. 4. Zero-Rainfall Probability
  7. 5. Spatial Autocorrelation Structure
  8. 6. Temporal Autocorrelation Structure
  9. 7. Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[1] Rainfall intermittency is analyzed and quantified at small spatial and temporal scales using 2 years of radar and disdrometer data collected in Switzerland. Analytical models are fitted and used to describe the intermittency for spatial scales between 0 and 30 km and temporal resolutions between 30 s and 6 h, providing climatological parameterizations for efficient and accurate upscaling/downscaling of intermittent rainfall fields. First, the zero rainfall probability is analyzed with respect to the considered spatial resolution. Second, the spatial autocorrelation of rainfall intermittency is quantified with respect to the temporal resolution. Finally, the temporal autocorrelation is analyzed with respect to the spatial resolution. The results show that all these different aspects of rainfall intermittency can be accurately described by a scaled exponential function with a fixed shape parameter and a variable scale parameter. Models describing this variability are provided.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Modeling the Intermittency
  5. 3. Data
  6. 4. Zero-Rainfall Probability
  7. 5. Spatial Autocorrelation Structure
  8. 6. Temporal Autocorrelation Structure
  9. 7. Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[2] Precipitation is a highly variable non-continuous process in space and time. It is characterized by relatively long dry periods punctuated by shorter rain events with complex spatial and temporal structures. This constant alternating between dry and rainy periods, called rainfall intermittency, significantly affects the environment and the ecosystems. For example, vegetation cover in arid or semi-arid regions is highly sensitive to rainfall intermittency which limits the available resources (water and nutrient) and controls their abundance in time [Kletter et al., 2009]. In more temperate regions, the soil and surface hydrology are known to be strongly sensitive to rainfall intermittency which influences important natural processes like stream flow, runoff, soil moisture and soil erosion [Pitman et al., 1990].

[3] The major difficulty with rainfall intermittency is the fact that it varies significantly with respect to the considered spatial and temporal scales. Short time periods are more likely to be completely dry than long ones and small areas are more likely to be dry than large ones. Yet the ability to quantify the probability of zero rainfall at multiple space and timescales is crucial for many practical applications in hydrology, meteorology and remote sensing of precipitation. For example, it plays a major role in rainfall interpolation and disaggregation techniques and must be taken into account when upscaling/downscaling the outputs of numerical weather models or weather radar data [e.g., Seo, 1998; Lanza et al., 2001]. Other examples can be found in the field of stochastic rainfall simulation [Kang and Ramirez, 2010], partial beam filling and path-integrated attenuation using ground based weather radars or satellites.

[4] Several studies have investigated rainfall intermittency at different spatial or temporal resolutions. Among the studies focusing on spatial intermittency, Braud et al. [1993] and Jeannin et al. [2008] analyzed the relations between the mean areal rainfall and the fractional area where it rains above a fixed threshold. Using mathematical morphology, Kumar and Foufoula-Georgiou [1994] proposed different algorithms for downscaling/upscaling intermittent rainfall fields. Using radar data, Pavlopoulos and Gritsis [1999] and Pavlopoulos and Gupta [2003] analyzed the durations and scaling of wet and dry periods. Using time series from tipping bucket rain gauges, Molini et al. [2009] and Rigby and Porporato [2010] investigated different relations between rainfall intermittency and turbulence across a large range of time scales and climatic regimes. Their study, however, did not include any information on the spatial structure of rainfall intermittency. Using a different approach based on the maximum entropy principle, Koutsoyiannis [2006] noted that, under some circumstances, the probability that a time interval is dry, follows a scaled exponential function of timescale. Using disdrometer measurements and high temporal resolutions, Lavergnat and Golé [1998] analyzed the inter-arrival times of raindrops and proposed to model the time intervals between raindrops using a Bi-Pareto law. Finally, recent work by Kundu and Siddani [2011] shows that dry probabilities (both spatial and temporal) can be modeled using a scaled exponential function.

[5] This letter revisits rainfall intermittency and extends previous results by providing a statistical analysis of rainfall intermittency (including spatial and temporal structures) for a large range of scales. Section 2 provides some definitions and section 3 presents the data used for the analysis. In section 4, the zero-rainfall probability is quantified using 2 years of data collected in Switzerland. In sections 5 and 6, the spatial and temporal autocorrelation of rainfall intermittency are analyzed. The conclusions and perspectives are given in section 7.

2. Modeling the Intermittency

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Modeling the Intermittency
  5. 3. Data
  6. 4. Zero-Rainfall Probability
  7. 5. Spatial Autocorrelation Structure
  8. 6. Temporal Autocorrelation Structure
  9. 7. Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[6] Rainfall intermittency, i.e., the presence or absence of rainfall, can be described by an indicator function

  • equation image

where R(x, t) [mmh−1] represents the instantaneous rain rate at location x and time t. By extension, we define

  • equation image

where X is a given area and T a given time period. For simplicity, only square areas of size k × k and continuous time periods [t0, t0 + τ] of duration τ are considered. For convenience, k is chosen to be expressed in kilometers and τ in hours. The probability that a randomly chosen area of size k remains dry for at least τ hours is denoted by p(k, τ). For consistency, it is supposed that p(k, τ) is well-defined and does not depend on the location nor on the timing, i.e., that I(X, T) is stationary over the considered period and area. Note that if this is not the case, the analysis can always be confined into appropriate areas and periods (e.g., months or seasons) for which I(X, T) can be considered stationary.

3. Data

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Modeling the Intermittency
  5. 3. Data
  6. 4. Zero-Rainfall Probability
  7. 5. Spatial Autocorrelation Structure
  8. 6. Temporal Autocorrelation Structure
  9. 7. Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[7] The results in this letter are mostly based on the analysis of nearly 2 years of operational radar rain-rate data provided by MeteoSwiss. These data are available from 30th April 2009 to 2nd February 2011, at a spatial resolution of 1 × 1 km2 and a temporal resolution of 5 min. There are almost no missing data for the considered time period. The estimated rain rates are obtained by combining the measurements of 3 C-band weather radars at different elevations, correcting for the main sources of errors (ground clutter, beam shielding, vertical variability) according to a procedure described in Germann et al. [2006]. For the purpose of this study, only the rain/no-rain information is retrieved from the estimated rain-rate maps, which limits the influence of the uncertainties associated with rain rate retrieval from radar measurements. Note that the minimum detectable rain rate is 0.16 mmh−1 which defines the threshold for rain/no-rain detection. Two 30 × 30 km2 areas located in the North-East and in the South-West of Geneva are selected for statistical analysis. These are the biggest square areas with no evident signs of data contamination (e.g., ground clutter or beam shielding) the authors could find for the selected time period. Note that both areas are relatively flat and close (less than 60 km) to one of the 3 C-band radars. It is therefore assumed that false rain and dry detections are negligible over the selected areas and do not affect significantly the statistical and structural analysis.

[8] At very small spatial scales (less than 1 km2), rainfall intermittency is analyzed using the data from a network of disdrometers deployed over EPFL campus, Lausanne, Switzerland [Jaffrain et al., 2011]. The distance from the network to the 2 areas defined above is less than 100 km. For comparison with radar data, the same time period between 30th April 2009 and 2nd February 2011 is considered. During this period, a total of 6 disdrometers sampling at 30 s temporal resolution and separated by 80 m to 800 m were available. For consistency with radar data, the same rain/no-rain detection threshold of 0.16 mmh−1 (corresponding to the 20% rain-rate quantile) is applied to the disdrometer data.

4. Zero-Rainfall Probability

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Modeling the Intermittency
  5. 3. Data
  6. 4. Zero-Rainfall Probability
  7. 5. Spatial Autocorrelation Structure
  8. 6. Temporal Autocorrelation Structure
  9. 7. Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[9] In this section, the data presented in section 3 are used to investigate the probability p(k, τ) that a randomly chosen area of size k remains completely dry for at least τ hours. Because they are very similar for the 2 considered areas, the results are only shown and discussed for the first area. For details about the second area, see Table 1. During this analysis, the largest spatial scale that can be considered is given by the size of the studied area, i.e., k = 30 km. The smallest possible scale is given by the radar resolution (k = 1 km). For smaller scales, the rain/no-rain information derived from the 6 disdrometers are used to estimate p(k, τ) at the point-scale, i.e., k = 0.

Table 1. Estimated Parameter Values
 ParameterArea 1Area 2
Zero rainfall probabilityp00.9250.936
 p1−0.005−0.007
 a50.845.1
 b−0.096−0.173
 β0.570.64
Spatial autocorrelationaS161.5147.2
 bS0.3760.362
 βS0.5790.527
Temporal autocorrelationaT2.9772.958
 bT0.1560.138
 βT0.5140.501

[10] The values of p(k, τ) for different spatial and temporal scales are displayed in Figure 1. It shows that, for a fixed spatial scale k, p(k, τ) can be described by a scaled exponential function:

  • equation image

where λ(k) > 0 [h−1] is a scale parameter, β(k) > 0 [−] a shape parameter and p(k) ∈ [0, 1] represents the dry probability for τ = 0. The fact that scaled exponential functions can be used to describe zero-rainfall probabilities has been pointed out previously by Kundu and Siddani [2011]. Another parallel can be found in the field of survival analysis, where scaled exponentials are used to express the probability that the time of “death” occurs later than some specified time. Hence another way of seeing rainfall intermittency is through a marked point process characterized by the “birth” and “death” of individual dry periods.

image

Figure 1. Estimated values of p(k, τ) at k = 0, 1, 5, 10 and 15 km spatial resolution. All estimates are obtained using radar data except k = 0 which represents the average value computed from the 6 disdrometers. The dashed lines represent the fitted model from equation (3).

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[11] In order to investigate the dependence of p(k), λ(k) and β(k) to the considered spatial scale, the radar estimates of p(k, τ) corresponding to each value of k between 1 and 15 km, were fitted using non-linear least squares. The scatter plots of p(k), λ(k) and β(k) with respect to k suggest that p(k) is a linear function, λ(k) a power law and that β(k) can be assumed constant.

  • equation image

[12] Figure 2 shows the fitted scatter plots of p(k) and λ(k) with respect to k. The triangles at k = 0 represent the average parameter estimates from the 6 disdrometers. Note that these were not used to fit the relations in equation (4) but are in very good agreement with their values predicted using radar measurements only. The fitted values of p0, p1, a, b and β are given in Table 1. Note that they are likely to be specific to the considered region and must therefore be adapted to the local climatology. It is conjectured, however, that the functional forms for p(k) and λ(k) are generally valid (at least for the considered range of spatial scales) and can be used to describe rainfall intermittency for other regions and climatologies. The sensitivity of p0, p1, a, b and β to the rain/no-rain detection threshold has been investigated by considering the next possible radar rain detection threshold, i.e., 0.25 mmh−1 (corresponding to the 30% rain-rate quantile at the point-scale). Higher thresholds would not be reasonable as too many rain-rate values would be below this threshold. The results show that the model parameters are not very sensitive to the rain threshold (less than 2% variation for p0, a and β and 15% for b and p1). Moreover, the functional forms in equations (3) and (4) remained unchanged. Finally, the stationarity hypothesis was tested by splitting the data into 4 periods (summer 2009, winter 2009–2010, summer 2010 and winter 2010–2011) and by computing p(k, τ) for each season. No statistically significant differences between these periods could be observed.

image

Figure 2. Estimated values of (top) p(k) and (bottom) λ(k) for spatial resolutions between 1 km and 15 km. The triangles at k = 0 represent the average parameter estimates from the 6 disdrometers. The dotted lines represent the fitted models given in equation (4).

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5. Spatial Autocorrelation Structure

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Modeling the Intermittency
  5. 3. Data
  6. 4. Zero-Rainfall Probability
  7. 5. Spatial Autocorrelation Structure
  8. 6. Temporal Autocorrelation Structure
  9. 7. Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[13] In this section, the spatial autocorrelation structure ρS(h, τ) of the rainfall intermittency is investigated at different distance lags h and temporal resolutions τ using the data presented in section 3. In order to ensure the highest possible resolution in the spatial domain, this analysis is performed at the highest available spatial resolution, i.e., 1 × 1 km2.

[14] Figure 3 shows the spatial autocorrelation for τ = 5 min, 15 min, 45 min, 120 min and 360 min. The data show that, for each temporal resolution τ, the spatial autocorrelation at lag h can also be described by a scaled exponential function:

  • equation image

where λS(τ) > 0 [km−1] is a scale parameter and βS(τ) > 0 [−] a shape parameter. The scatter plots of λS(τ) and βS(τ) versus τ show that λS(τ) can be described as a power law of τ and that βS(τ) can be assumed constant for all temporal scales:

  • equation image

where aS, bS and βS depend on the local climatology. The fitted values of aS, bS and βS for the considered areas are given in Table 1. The fitted model for λS(τ) is shown in Figure 4. The sensitivity of aS, bS and βS to the radar rain detection threshold (see section 4) has been investigated and is in the order of 3% for aS and βS and 15% for bS. The functional forms in equations (5) and (6) were also preserved.

image

Figure 3. Estimated spatial autocorrelation ρS(h, τ) of rainfall intermittency at 5, 15, 45, 120 and 360 min temporal resolutions. For each temporal resolution, a scaled exponential function as given in equation (5) has been fitted.

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image

Figure 4. Estimated values of λS(τ) for τ between 5 min and 6 h. The dotted line represents the fitted power law given in equation (6).

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6. Temporal Autocorrelation Structure

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Modeling the Intermittency
  5. 3. Data
  6. 4. Zero-Rainfall Probability
  7. 5. Spatial Autocorrelation Structure
  8. 6. Temporal Autocorrelation Structure
  9. 7. Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[15] In this section, the temporal autocorrelation structure ρT(τ, k) of the rainfall intermittency is investigated at different spatial resolutions using the data presented in section 3. The approach is similar to the previous section and, for conciseness, only the results without any further illustrations are given. In order to ensure the highest possible resolution in the temporal domain, this analysis is performed at the highest available temporal resolution, i.e., 5 min. The results show that, for each fixed spatial resolution k, the temporal autocorrelation ρT(τ, k) of rainfall intermittency can be described by a scaled exponential function:

  • equation image

for a given scale parameter λT(k) > 0 [h−1] and shape parameter β(k) > 0 [−]. In this case, the scatter plots of λT(k) and βT(k) versus k show that λT(k) is a linear function of k and that βT(k) can be assumed constant at all spatial scales.

  • equation image

The fitted values of aT, bT and βT are provided in Table 1. Similarly to the zero-rainfall probability, the values of λT(0) and βT(0) obtained from the disdrometer data (i.e., k = 0) are in very good agreement with their predicted values using radar data only. The sensitivity of aT, bT and βT to the rain detection threshold (similarly to the previous sections) is lower than 2% for bT and βT and in the order of 15% for aT.

7. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Modeling the Intermittency
  5. 3. Data
  6. 4. Zero-Rainfall Probability
  7. 5. Spatial Autocorrelation Structure
  8. 6. Temporal Autocorrelation Structure
  9. 7. Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[16] Rainfall intermittency significantly varies with respect to the spatial and temporal scale at which it is considered. It is larger at finer scales and its spatial correlation is larger at lower temporal resolutions (i.e., longer time periods). In this letter, a statistical analysis of rainfall intermittency at different spatial and temporal resolutions has been performed. It shows that zero-rainfall probabilities can be adequately described by a scaled exponential function (of the timescale) with parameters depending on the considered spatial scale. Models describing this dependency have been provided. The estimated parameters are likely to depend on the considered region and climatology but the functional forms and models are assumed to be generally valid. The choice of models is supported by the fact that independent disdrometer measurements (at scales that can not be observed by the radar) are in very good agreement with their radar-based predictions. Using the same approach, the spatial and temporal autocorrelations of rainfall intermittency are also quantified. The model parameters are not very sensitive to the rain/no-rain detection threshold (up to 15% for the considered thresholds). The functional forms describing p(k, τ), ρS(h, τ) and ρT(τ, k) are not influenced by this threshold and can be used to improve the modeling of rainfall intermittency (including its spatial and temporal structure) in hydroclimatological models, to upscale and downscale the outputs of numerical weather prediction models and to parameterize stochastic rainfall simulators.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Modeling the Intermittency
  5. 3. Data
  6. 4. Zero-Rainfall Probability
  7. 5. Spatial Autocorrelation Structure
  8. 6. Temporal Autocorrelation Structure
  9. 7. Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

[17] The first two authors acknowledge the financial support from the Swiss National Science Foundation, grant 200020-232002/1. The authors also thank MeteoSwiss for providing the radar data necessary for this analysis.

[18] The Editor thanks the two anonymous reviewers.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Modeling the Intermittency
  5. 3. Data
  6. 4. Zero-Rainfall Probability
  7. 5. Spatial Autocorrelation Structure
  8. 6. Temporal Autocorrelation Structure
  9. 7. Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Modeling the Intermittency
  5. 3. Data
  6. 4. Zero-Rainfall Probability
  7. 5. Spatial Autocorrelation Structure
  8. 6. Temporal Autocorrelation Structure
  9. 7. Conclusions
  10. Acknowledgments
  11. References
  12. Supporting Information
FilenameFormatSizeDescription
grl28503-sup-0001-t01.txtplain text document0KTab-delimited Table 1.

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