[1] Rainfall varies in space and time in a highly irregular manner and is described naturally in terms of a stochastic process. A characteristic feature of rainfall statistics is that they depend strongly on the space-time scales over which rain data are averaged. A spectral model of precipitation has been developed based on a stochastic differential equation of fractional order for the point rain rate, which allows a concise description of the second moment statistics of rain at any prescribed space-time averaging scale. The model is thus capable of providing a unified description of the statistics of both radar and rain gauge data. The underlying dynamical equation can be expressed in terms of space-time derivatives of fractional orders that are adjusted together with other model parameters to fit the data. The form of the resulting spectrum gives the model adequate flexibility to capture the subtle interplay between the spatial and temporal scales of variability of rain but strongly constrains the predicted statistical behavior as a function of the averaging length and time scales. We test the model with radar and gauge data collected contemporaneously at the NASA TRMM ground validation sites located near Melbourne, Florida and on the Kwajalein Atoll, Marshall Islands in the tropical Pacific. We estimate the parameters by tuning them to fit the second moment statistics of radar data at the smaller spatiotemporal scales. The model predictions are then found to fit the second moment statistics of the gauge data reasonably well at these scales without any further adjustment.Citation: Kundu, P. K., and J. E. Travis (2013), A stochastic fractional dynamics model of space-time variability of rain,

A statistical method is developed for comparing precipitation data from measurements performed by (hypothetical) perfect instruments using a recently developed stochastic model of rainfall. The stochastic dynamical equation that describes the underlying random process naturally leads to a consistent spectrum and incorporates the subtle interdependence of the length and time scales governing the statistical fluctuations of the rain rate field. The main attraction of such a model is that the complete set of second-moment statistics embodied in the space-time covariance of both the area-averaged instantaneous rain rate (represented by radar or passive microwave data near the ground) and the time-averaged point rain rate (represented by rain gauge data) can be expressed as suitable integrals over the spectrum. With the help of this framework, the model allows one to carry out a faithful intercomparison of precipitation estimates derived from radar or passive microwave remote sensing over an area with direct observations by rain gauges or disdrometers, assuming all the measuring instruments to be ideal. A standard linear regression analysis approach to the intercomparison of radar and gauge rain rate estimates is formulated in terms of the appropriate observed and model-derived quantities. We also estimate the relative sampling error as well as separate absolute sampling errors for radar and gauge measurements of rainfall from the spectral model. Key Points: • Radar and gauge rain estimates are compared using a stochastic rainfall model • The model is based on a stochastic differential equation of fractional order • Estimates of radar-gauge regression parameters are computable from the model Correspondence to: (2014), Statistical intercomparison of idealized rainfall measurements using a stochastic fractional dynamics model,In this paper we apply the stochastic fractional dynamics model formulated in KT13 to a typical precipitation intercomparison scenario that is encountered in the course of ground validation, namely, an intercomparison between radar and gauge estimates of rain. Such an intercomparison involves taking into account two A major caveat in the above argument is the assumption of the retrieval errors being uncorrelated with true rain. An improved treatment would require a more elaborate error model. Fuller [1987] gives examples of such measurement error models. Such an analysis is beyond the scope of our study.

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