Spatial-temporal statistics of rainrate random fields



[1] Rainrate near the ground can be interpreted as a collection of random variables forming a random field parameterized by three coordinates: two spatial coordinates and one temporal. Interest among the radio propagation community has been focused on the first order statistics of these random variables, principally on the average annual probability distribution of rainrate as a function of location. However, increasingly interest is turning to second order statistics describing high-resolution rainrate variation in both time and space. This information is important for the design of fade countermeasures and for efficient spectrum management. In this paper radar data from a widespread, slow moving and intense, stratiform rain event experienced by the southeastern UK on May 1, 2001, are used to calculate the second order, spectral, statistics of spatial-temporal log rainrate variation. It is demonstrated that log rainrate is self-similar in all three coordinates and that symmetry exists between the spatial and temporal variation. These data are used to develop an isotropic, spatial-temporal log rainrate model assuming that log rainrate is a homogeneous, Gaussian, random field. This model is developed further to yield closed form expressions for the mean and covariance of the random fields associated with rainrate, with specific attenuation and the logarithm of specific attenuation. Furthermore, expressions for the temporal covariance of the rain attenuation experienced by pairs of radio links are developed, and these expressions are tested against measured rain attenuation data from an experimental link.

1. Introduction

[2] Currently, the ITU-R provides Rec. ITU-R P.837-2 for calculating the average annual, one-minute averaged, rainrate distribution for any point on Earth. This may be used to calculate the average annual rain-fade distribution experienced by a terrestrial link. However, telecommunications operators are increasing interested in the second order statistics of rain-fade. Such statistics include the temporal power spectrum, rain-fade duration statistics, rain-fade slope statistics and rain-fade covariance for pairs of links, or the same link at different times. These statistics are important for the design of fading countermeasures, such as time/path diversity, and for interference coordination. The second order statistics of rain-fade depend directly upon the second order statistics of rainrate variation.

[3] Although there exists a large number of pulse models describing rainrate variation, including the model by Capsoni et al. [1987] designed for propagation studies, these models do not contain information on the high resolution stochastic variation of rainrate in time and space. The Synthetic Storm model of Matricciani [1996] does provide high resolution, temporal, rain-fade variation but relies of simultaneous raingauge measurements. Veneziano et al. [1996] have postulated that log rainrate, while raining, may be modeled as a stationary, Gaussian stochastic process. Furthermore, theoretical models of rain as a passive tracer in a turbulent, two-dimensional flow [Kraichnan and Montgomery, 1980; Lovejoy and Schertzer, 1992] predict that the temporal and spatial spectral density of log rainrate follows a segmented power law form. These fluid dynamical models often lead to spectral density power laws with exponents expressed as ratios of small integers. Over ranges of scales where the spectral density follows a simple power law, there is no special scale and the random variable exhibits self-similarity. Models of rainrate variation that assume power law spectra are often termed “fractal models.” Crane and Shieh [1989] identify one-dimensional (1-D) spatial log rainrate spectra with two power law segments. Below the scale of energy injection, typically the size of a front, the power spectrum has the form |f|−3, where f is spatial or temporal frequency, reflecting a direct entropy cascade toward larger wave numbers. Above the scale of energy injection an inverse energy cascade toward smaller wave numbers leads to spectra of the form |f|−5/3. Based on the observation of eight storms, Veneziano et al. suggest that spatial or temporal 1-D log rainrate has a spectral density function (s.d.f.) with four power law segments, see Figure 1. The corner frequencies are associated with the scale of convective cells and cell clusters. A log-log plot of such a spectral density is segmented-linear with the gradient of each segment equal to the power law exponent.

Figure 1.

One-dimensional temporal or spatial spectral density of log rainrate postulated by Veneziano et al. [1996].

[4] Assuming that log rainrate statistics exhibit self-similarity allows the low resolution available data to be used to develop models of rainrate variation at scales of interest to the radio propagation community. Paulson and Gibbins [2000] have verified the self-similarity of log rainrate variation at a point over scales from 10 s to one day. Other authors, such as Venugopal et al. [1999], have used courser meteorological radar data to investigate spatial statistical scaling and symmetries between temporal and spatial variation. In this paper, radar data are used to verify self-similarity of log rainrate variation in both time and space. Extrapolating this model to higher resolution than the scales measured by the radar, a three-dimensional model for the second order statistics of log rainrate variation is proposed. From this model, complete descriptions of the first order and second order statistics of rainrate and specific attenuation variation within rain events is developed. In section 4, this model is extended to predict the second order statistics of the rain attenuation experienced by pairs of radio links within a rain event. This model is verified using data from an experimental 38 GHz link.

2. Statistics of Random Fields

[5] In this section a review is made of the important statistical properties of random fields. The notation and the concept of quadrant symmetry come from Vanmarcke [1983]. In subsequent sections we will be investigating the properties of the log rainrate random field X(t) where t is a vector of all or some of the variables {x,y,t} describing position on a horizontal plain close to ground level and time respectively. The real, scalar random field X is a collection of random variables associated with the coordinates t. We will assume that X is homogeneous (a 1-D homogeneous random field is called “stationary”), meaning that the statistics of X do not depend upon t. In this case, each random variable is a sample from a cumulative distribution function (c.d.f) F(x). As X is homogeneous, the mean and variance of X(t) are the same as the mean and variance of F:

equation image

where E[·] is the expected value. The autocovariance function of X describes the relationship between values of the random field at two coordinates t1 and t2:

equation image

When X is homogeneous, the autocovariance depends only upon the lag τ = t1t2 and the autocovariance for lags located symmetrically with respect to the origin are identical. The autocovariance of a 1-D stationary field is an even function. A random process is known as quadrant symmetric if the autocovariance is even with respect to every component of the lag vector:

equation image

The autocorrelation is the normalized autocovariance:

equation image

which satisfies ρ(0) = 1. The s.d.f. of the n-dimensional random field X is related to the autocovariance by the Wiener-Khinchine relations:

equation image
equation image

where integrals over vector parameters are understood to be multidimensional. Note that equation image. If X is quadrant symmetric then the spectral density is completely defined by the one sided spectral density equation image, and equation image. A special case of quadrant symmetry occurs when all directions in lag space are equivalent. For an isotropic, homogeneous random field, the autocovariance is a scalar function of lag magnitude only: B(τ) ≡ BR(|τ|). The spectral density is, similarly, a function only of frequency magnitude: G(ω) ≡ GR (|ω|).

[6] For a homogeneous, Gaussian, random field, the random variable associated with each point in the coordinate space is a sample from the same Gaussian distribution and the autocovariance is translation independent. Such a random field is completely described by the first and second order statistics. In this paper it is assumed that log rainrate and log specific attenuation are homogeneous, Gaussian, random fields and so they are completely described by their mean, variance and autocovariance (or spectral density).

[7] When one of the coordinates is fixed, e.g. tn = tn*, X, B and S become functions of one fewer dimension and the restricted s.d.f. may be found by integration:

equation image

where ωn = (ω1, ω2,…, ωn). If the n-dimensional spectral density follows a power law i.e. for Sn(ωn) ∝ |ωn|β for |ωn| > ω0, then the spectral density of the (n-1)-dimensional random field with one coordinate fixed follows a power law with coefficient β + 1:

equation image

3. Analyses of Radar Data

[8] The Chilbolton Advanced Meteorological Radar, CAMRa, is a 25 m steerable antenna equipped with a 3 GHz, Doppler-Polarization radar, located in Hampshire, in the south of England, at latitude 51° 9′ North, longitude 1° 26′ West. The climate is temperate maritime with an average annual rainrate exceeded 0.01% of the time of approximately 22.5 mm/hr. The radar has an operational range of 100 km and a beam width of 0.25 degrees. By scanning with an inclination of 1.5 degrees, to avoid reflections from ground clutter, maps of the rainrate field near the ground can be produced on a polar grid with an angular resolution of 0.3 degrees and a range resolution of 300 m. Depending upon the total angle scanned, these maps may be produced every few minutes. Typically, horizontal scans over the area of interest are interspersed with scans in a vertical plane and 360 degree scans, leading to irregular temporal sampling.

[9] On May 1, 2001, a slow moving cold front system, associated with a low-pressure area over central France, crossed the southern UK bringing widespread stratiform rain containing convective cells of very heavy rain. Over the 4 hour period from 8AM to noon GMT, 124 near-horizontal radar scans were measured over an angle of 80 degrees approximately southeast of Chilbolton over Surrey and Hampshire. These radar data were interpolated onto a square Cartesian grid, 56.2 km along each edge and with a grid spacing of 300 m and radar reflectivities were transformed into log rainrates. Each grid contains 1882 data points spanning more than 3100 km2. 97.4% of the 1882 × 124 rainrate samples were higher than the measurement threshold of 10−1.5 mm/hr.

3.1. Temporal Statistics

[10] Each of the 1882 data points yields a time series of 124 rainrate measurements, approximately regularly sampled over the four-hour event. Following the work of Veneziano et al. [1996] and Paulson and Gibbins [2000], the spectral density of each log rainrate time series was calculated and a mean spectral density calculated, Figure 2. The temporal spectral density is closely fitted by a power law with a least squares gradient of −1.61. This is close to the theoretical figure of −5/3 cited by Veneziano et al. It is also consistent with the mean temporal s.d.f. derived from a fast-response rain gauge at Chilbolton. Figure 3 illustrates the mean temporal s.d.f. derived from 320 rain events monitored with a Scientific Technology Inc. ORG700 optical rain gauge over the period June 1998 to November 1999. The instantaneous rainrate measurements were passed through a pseudo-integrator with a 10 s time constant and samples are recorded every 10 s. The dotted line indicates a slope of −5/3. Both these results appear consistent with the temporal variation of log rainrate possessing a spectral density well approximated by a −5/3 power law over the range of frequencies measured. A more rigorous hypothesis test is impracticable due to the number of unquantified sources or error in the radar derived rain rates e.g. variation in drop-size-distribution within the rain event.

Figure 2.

Temporal spectral density of log rainrate for event on May 1, 2001, derived from radar scans (solid line) compared with |f|−5/3 model (dotted line) and best fit power law |f|−1.61 model (dashed line).

Figure 3.

Average spectral density of log rainrate from 320 rain events (solid line), measured by optical rain gauge, compared with |f|−5/3 model (dotted) for slope comparison.

3.2. Spatial Statistics

[11] If each near-horizontal radar scan is treated as an instantaneous snapshot of the rainrate field then the spatial s.d.f. may be calculated using a 2-D Fourier transform. Figure 4 illustrates the two-sided spectral density of spatial log rainrate variation, averaged over the 124 scans. The near circular contours are consistent with a rotationally symmetric, and hence quadrant symmetric, spectral density and spatial autocorrelation. Assuming rotational symmetry, the radial s.d.f. may be found by averaging the spatial spectral density around circular contours. Figure 5 illustrates the rotationally averaged, radial spatial s.d.f. and the least squares best-fit power law with an exponent of β = −2.74. This is consistent with a power law 1-D spatial spectral density with an exponent of β + 1 = −1.74 which is in good agreement with the theoretical value of −5/3, see (2).

Figure 4.

Two-dimensional spatial spectral density of log rainrate for event on May 1, 2001, averaged over 124 scans. Contours increase toward DC in half order of magnitude steps.

Figure 5.

Radial spatial spectral density of log rainrate for event on May 1, 2001, averaged over 124 scans assuming rotational symmetry, compared with best fit power law |f|−2.74 (dashed line).

3.3. Spatial-Temporal Statistics

[12] A similar process may be followed with mixed spatial-temporal data. Fixing one spatial coordinate yields 376 two-dimensional data sets with coordinates (x,t) or (y,t). Each data set is a 188 × 124 array of log rainrate measurements acquired along a line over the duration of the experiment. Figure 6 illustrates the two-sided spectral density of spatial-temporal log rainrate variation, averaged over the 376 data sets. Allowing for the distortion around the Nyquist frequency introduced by aliasing in the time dimension, the elliptical contours are consistent with a rotationally symmetric spectral density once the spatial and temporal units have been normalised. The contours imply that the radial spectral density is a function of the norm |ω|2 = ωx2 + η2 ωt2 where η ≈ 55 s/km is the scaling factor required to transform between 1-D spatial and temporal autocorrelation functions. Assuming rotational symmetry, the radial, spatial-temporal, s.d.f. may be found by averaging around equi-norm elliptical contours. Figure 7 illustrates the rotationally averaged, radial, spatial-temporal, spectral density and the least squares best-fit power law with an exponent of β = −2.61. Once again, this is consistent with a power law 1-D, spatial, spectral density with a coefficient of −5/3.

Figure 6.

Two-dimensional spatial-temporal spectral density of log rainrate for event on May 1, 2001, averaged over 376 data sets. Contours increase toward DC in order of magnitude steps.

Figure 7.

Spatial-temporal spectral density of log rainrate for event on May 1, 2001, averaged over 376 scans assuming rotational symmetry, compared with best fit power law |f|−2.6 (dashed line).

4. Link Rain-Fade Statistics

[13] Combining the results of the previous section allows the n-D spatial-temporal s.d.f. of log rainrate to be written:

equation image

where ω is a vector of 1,2 or 3 of the coordinates ωi ∈ {ωx, ωy, ωt}, ω0 is the frequency below which the spectrum must flatten for the power to be finite and can be estimated from knowledge of the physical size or temporal duration of the largest rain events. K is a normalizing constant which ensures that equation image where σX2 is the variance of log rain rate estimated from experimentally measured distributions of rainrate. The weighted Euclidian norm must be used when both space and time coordinates are present. The associated autocovariance, BX(τ), is the n-D inverse Fourier transform of this function (2a). Where the rainrate random field, R, is well approximated as homogeneous and isotropic then so may the associated random fields of log rainrate, X, specific attenuation, γ, and log specific attenuation Γ. For a specific frequency and polarization, Rec. ITU-R 838 provides a power law relationship between rainrate and specific attenuation:

equation image
equation image

Therefore, if rainrate is a lognormal random field then so is specific attenuation. The mean and variance of the random fields for rainrate, specific attenuation and log specific attenuation can be written using the mean and variance of the log rainrate field, see Table 1. For convenience define the function equation image. The autocovariance of the lognormal fields R and γ may be written:

equation image
equation image

where μX, σX2 and BX(τ) are the mean, variance and autocovariance of log rainrate respectively. Knowledge of the spatial-temporal statistics of the specific attenuation random field allows the calculation of the temporal statistics of rain attenuation. The rain attenuation, as a function of time, experienced by a radio link along a path may be approximated by the path integral of specific attenuation:

equation image

where x(l1) is the position at distance l1 along the link. The temporal autocovariance of rain attenuation on this link can be written in terms of the autocovariance of the specific attenuation:

equation image

If the cross-covariance of the rain attenuation on two links using different frequencies or polarizations is required, then the different specific attenuation functions need to be taken into account and (9) becomes:

equation image


equation image

As an example of the application of these results, consider the passage of the rain event on May 1, 2001, over an experimental 5 km, 38 GHz vertically polarized link operated from Sparsholt, near Chilbolton, in the southern UK. The theoretical autocorrelation function equation image obtained from equation 1, may be calculated from the 2-D, spatial-temporal, log rainrate s.d.f., given by (4), using (2a) to calculate the autocovariance of log rainrate BX, then (7) to calculate the autocovariance of specific attenuation Bγ, using the k and alpha appropriate for the frequency and polarization of the link. Finally, the link length is used in (9) to yield the autocorrelation of link attenuation. Figures 8a and 8b compare the theoretical autocorrelation and s.d.f.s with those measured from six hours of 10 s integrated attenuation measurements. The lowest frequencies have the highest amplitudes and statistical variation in these lead to long wavelength differences between the measured and predicted autocorrelation. However, the predicted spectral density follows the envelope of the measured spectral density down to the noise level at 0.02 Hz. It is also similar to the −20 dB/decade power law spectrum used by many workers to simulate link rain-attenuation [e.g., Castanet et al., 2000].

Figure 8.

Comparison of measured (solid) and theoretical (dashed) rain attenuation: (a) autocorrelation function and (b) s.d.f., for a 5 km, 38 GHz, vertically polarized terrestrial link in the southern UK.

Figure 8.


Table 1. Statistics of Log Rainrate X, Log Specific Attenuation Γ, Rainrate R, and Specific Attenuation γ in Terms of the Mean μX and Variance σX2 of Log Rainrate and the Specific Attenuation Power Law Coefficients k and α
Process ZE(Z)E(Z2)Variance
XμXμX2 + σX2σX2
Γln(k) + αμX(ln(k) + αμX)2 + α2σX2α2σX2
Rexp(ψ(1))exp(ψ(2))(exp(σX2) − 1)exp(2ψ(1))
γk exp(ψ(α))k2 exp(ψ(2α))(exp(α2σX2) − 1)k2 exp(2ψ(α))

5. Conclusions

[14] In the case study described in section 3 it was demonstrated that a wide spread, stratiform rain event may be modeled as an isotropic, 3-D, homogeneous Gaussian random field where the parameters are the x and y location near the ground and time. A model for the n-dimensional s.d.f. has been proposed which is consistent with fluid dynamic theory and many observations. Spatial and temporal s.d.f.'s are all well described by a single power law, contrary to the finding of other authors. This model allows the prediction of all the first and second order statistics of individual or pairs of radio links while entirely embedded in a rain event. The model does not account for the cases where any part of either link is outside the region of the rain event not well modeled as a homogeneous Gaussian field. The restriction means that average annual link attenuation statistics cannot be derived from this model due to the incidence of events such as small but intense convective storms passing over part of a link network. However, where the restriction conditions are satisfied, the model has a large number of applications:

  1. Diversity studies: knowledge of the mean and covariance of rain-fade on pairs of links allows the joint c.d.f of rain-fade to be calculated explicitly. The mean rain attenuation experienced by a link of length Li is Liμγ where μγ is the mean specific attenuation for the link frequency and polarization, given by the formula in Table 1. The covariance of the link attenuations is given by equations 9, 10 and 11. These parameters determine the joint rain attenuation while raining c.d.f, which is bi-lognormal. This distribution completely determines the diversity improvement and diversity gain for links with arbitrary frequency, polarization, geometry, fade margins and measurement integration time.
  2. Duration statistics: knowledge of the mean, and autocovariance of rain-fade on an individual link allows the fade duration distributions to be calculated by simulation [see Paulson and Gibbins, 2000]. Furthermore, knowledge of the cross covariance function of pairs of links allows duration statistics on route diverse networks to be calculated by simulation.
  3. Fade slope statistics: the fade slope distribution of log rain-fade is Gaussian with zero mean and variance given by:
    equation image
    where SA(ω) is the temporal s.d.f. of log rain-fade. It is clear from (12) that the rain-fade slope distribution is very sensitive to the high frequency tail of the spectral density. The distribution is therefore very sensitive to the smoothing and sampling used to acquire the rain attenuation data. In the limit of no smoothing and continuous sampling the variance may not converge and the distribution may not exist. However, where the smoothing and sampling characteristics of the particular link are known, the rain-fade slope distributions can be calculated from the model.


[15] I would like to acknowledge the support of the UK Radiocommunications Agency, in particular Dave Eden and David Bacon, the staff of Chilbolton who acquired the radar data, and other members of the RCRU for their helpful contributions.