Many downscaling techniques have been proposed to address the mismatch between the spatial and te... more Many downscaling techniques have been proposed to address the mismatch between the spatial and temporal scales of rainfall by some observational systems or numerical simulations, and the requirements of land surface hydrologic models. Most of these techniques focus on the mechanism producing the great variability in both space and time that is characteristic of rainfall. In this study the authors investigate the hydrologic consequences of the downscaling. The hydrologic scope of the study is limited to flooding and thus focuses on the scale dependence of peak flows with respect to drainage area. The authors use numerical simulations using a rainfall-runoff model based on a landscape decomposition framework that preserves the major elements of flood genesis, i.e. runoff generation at hillslopes and channel routing of the runoff. The decomposition preserves the full extent of the channel drainage network and its scaling properties. The framework allows addressing the questions of appr...
1] The goal of this study is to diagnose the manner in which radar-rainfall input affects peak fl... more 1] The goal of this study is to diagnose the manner in which radar-rainfall input affects peak flow simulation uncertainties across scales. We used the distributed physically based hydrological model CUENCAS with parameters that are estimated from available data and without fitting the model output to discharge observations. We evaluated the model's performance using (1) observed streamflow at the outlet of nested basins ranging in scale from 20 to 16,000 km 2 and (2) streamflow simulated by a well-established and extensively calibrated hydrological model used by the US National Weather Service (SAC-SMA). To mimic radar-rainfall uncertainty, we applied a recently proposed statistical model of radar-rainfall error to produce rainfall ensembles based on different expected error scenarios. We used the generated ensembles as input for the hydrological model and summarized the effects on flow sensitivities using a relative measure of the ensemble peak flow dispersion for every link in the river network. Results show that peak flow simulation uncertainty is strongly dependent on the catchment scale. Uncertainty decreases with increasing catchment drainage area due to the aggregation effect of the river network that filters out small-scale uncertainties. The rate at which uncertainty changes depends on the error structure of the input rainfall fields. We found that random errors that are uncorrelated in space produce high peak flow variability for small scale basins, but uncertainties decrease rapidly as scale increases. In contrast, spatially correlated errors produce less scatter in peak flows for small scales, but uncertainty decreases slowly with increasing catchment size. This study demonstrates the large impact of scale on uncertainty in hydrological simulations and demonstrates the need for a more robust characterization of the uncertainty structure in radar-rainfall. Our results are diagnostic and illustrate the benefits of using the calibrationfree, multiscale framework to investigate uncertainty propagation with hydrological models.
Estimation of the correlation structure of random fields has many applications in earth sciences.... more Estimation of the correlation structure of random fields has many applications in earth sciences. Using a simulation framework, the authors address sampling problems in estimating the spatial correlation function. The framework consists of generating realizations of homogeneous, isotropic Gaussian and lognormal random fields with a two-parameter power exponential spatial correlation function. The two parameters in the correlation function are the
It is well acknowledged that there are large uncertainties associated with radar-rainfall estimat... more It is well acknowledged that there are large uncertainties associated with radar-rainfall estimates. Numerous sources of these errors are due to parameter estimation, the observational system and measurement principle, and not fully understood physical processes. To describe these uncertainties, rain gauge data are usually considered as a good approximation of the true rainfall values. However, in the vast majority of
Characterization of radar-rainfall error spatial covariance requires high-density rain gauge netw... more Characterization of radar-rainfall error spatial covariance requires high-density rain gauge networks and high quality data. The authors use data from two rain gauge networks: IIHR ground validation network at Iowa City, Iowa consisting of 25 platforms, with average intergauge distance of about 5 km, and the Iowa City Municipal Airport network consisting of 9 platforms and covering approximately 1 km2.
In the large majority of studies on the evaluation of radar-rainfall estimates, rain gauge measur... more In the large majority of studies on the evaluation of radar-rainfall estimates, rain gauge measurements have been used to approximate the ground-level true rainfall. Nevertheless, as extensively documented in the literature, this approximation leads to an additional source of uncertainty due to the large sampling discrepancies between the two measuring devices: radar data are area-averaged estimates while rain gauges can
Idealized model studies have shown that space-time rainfall intermittency and its statistical sca... more Idealized model studies have shown that space-time rainfall intermittency and its statistical scaling properties have a significant impact on statistical scaling of flood peaks with respect to drainage area. In this study, simulation is used as a methodological tool to explore the effects of radar-rainfall uncertainties on the observed scaling of flood peaks. Simulated rainfall fields with varying power spectrum
Peak flows in a basin are a result of complex nonlinear interactions among rainfall and various p... more Peak flows in a basin are a result of complex nonlinear interactions among rainfall and various processes in the landscape. Studies based on simulations and data have explored the role of rainfall characteristics on the statistical structure of peak flows. However, it is widely acknowledged that the rainfall products are affected by uncertainties from various sources. In this study, we
ABSTRACT The Global Precipitation Measurement aims to provide rainfall products at scales relevan... more ABSTRACT The Global Precipitation Measurement aims to provide rainfall products at scales relevant for flood forecasting, offering effective means to predict floods in sparsely gauged and ungauged regions of the world. To achieve this goal we face two main challenges: (1) the development of hydrological models that do not rely on calibration as relevant data are mostly unavailable; and (2) the need for better understanding of the limitations imposed by the inherent features of satellite rainfall products (retrieval errors, infrequent sampling, and low spatial resolution) on flood prediction across space-time scales. In this study we applied a calibration-free multi-scale hydrological model to investigate the potential use of GPM rainfall products for flood prediction. The hydrological model used in this study is based on the faithful decomposition of the landscape into hillslopes and river channel links, and on the solution of the mass and momentum equations for each control volume. To avoid calibration we adopt parameters that are directly related to measurable properties of the basin, as provided by remote sensing techniques (e.g. topography, land cover and land use). We simulated the Iowa 2008 flood event that resulted in economical losses of billions of dollars. We ran the model using STAGE IV radar rainfall data, and we evaluated model results using observed streamflow data for 26 sites ranging in drainage area from 22 to 16,800 km2. We generated possible statistical realizations of satellite rainfall maps by modifying the radar rainfall product based on the already defined basic capabilities of GPM rainfall products. We first independently investigated different aspects of satellite rainfall products that are likely to introduce errors in the simulation of hydrological processes. We then combined multiple aspects of the errors to provide a more realistic estimation of the total uncertainty. We also included simulations with the currently available satellite rainfall products (TRMM-3B42, PERSIANN, and CMORPH). The results of this study provide insights about the aspects that current and future satellite missions should focus on to increase the value of remote sensing products for flood prediction.
This study presents a large sample evaluation of different nowcasting systems over the Southern S... more This study presents a large sample evaluation of different nowcasting systems over the Southern Swiss Alps. Radar observations are taken as a reference against which to assess the performance of the following short-term quantitative precipitation forecasting methods: -Eulerian persistence: the current radar image is taken as forecast. -Lagrangian persistence: precipitation patterns are advected following the field of storm motion (the
ABSTRACT Scaling behavior of rainfall time series is characterized using monofractal, spectral, a... more ABSTRACT Scaling behavior of rainfall time series is characterized using monofractal, spectral, and multifractal frameworks. The study analyzed temporal scale-invariance of rainfall in the tropical island of Singapore using a large dataset comprising 31 years of hourly and 3 years of 1-min rainfall measurements. First, the rainfall time series is transformed into an occurrence-non-occurrence binary series, and its scaling behavior is analyzed using box-counting analysis. The results indicated that the rainfall support displays fractal structure, but within a limited range of scales. The rainfall support has a fractal dimension (D f ) of 0.56 for scales ranging from 1 min to 1.5 h and a D f of 0.37 from 1.5 h to 1.5 days. The results further showed that the fractal dimension decreases with the increase in the threshold used to define binary series. Spectral analysis carried out on the rainfall time series and the corresponding binary series showed three distinct scaling regimes of 4 min-2 h, 2-24 h, and 24 h-1 month. In all the scaling regimes, the spectral exponents for the rainfall series were smaller than those for the binary series. The study then investigated the presence of multiscaling behavior in rainfall time series using moment scaling analysis. The results confirmed that the rainfall fluctuations display a multiscaling structure, which was modeled in the framework of universal multifractals. The results from this study would not only improve our understanding of the temporal rainfall structure in Singapore and the surrounding Maritime Continent but also help us build and parameterize parsimonious models and statistical downscaling techniques for rainfall in this region.
Quarterly Journal of the Royal Meteorological Society, 2012
The study describes the diurnal cycle of precipitation in the Alps and surrounding forelands usin... more The study describes the diurnal cycle of precipitation in the Alps and surrounding forelands using high-resolution volumetric data from a network of three groundbased radars. First, the study focused on the regional, seasonal and interannual variability in the diurnal behaviour of surface rainfall intensity, frequency and spatial correlations. Secondly, the study examined the diurnal variation in the vertical structure of summertime precipitation. Significant regional and seasonal differences are noticed in the diurnal cycle of precipitation features. The signal is strongest in the Alps, moderate in the plains north of the Alps and weak in the Po River Plain south of the Alps. The phase of the diurnal cycle is around 1630 UTC over the Alps for both spring and summer seasons. During autumn, the signal is weakest with no well-defined phase. The spatial variogram analysis showed that rainfall patterns are most variable between 1200 and 1400 UTC, approximately 3 h before the occurrence of diurnal maximum in precipitation frequency and intensity. The vertical structure of precipitation is characterized using frequency-altitude diagrams, and its diurnal behaviour is analysed using Hovmöller diagrams. The results show that the diurnal cycle of precipitation varies significantly with height.
ABSTRACT The availability of radar-based rainfall products at high space-time resolutions and ove... more ABSTRACT The availability of radar-based rainfall products at high space-time resolutions and over continental scales has greatly advanced our understanding of the rainfall process and its interactions with other hydrological processes across a wide range of scales. However, it is well known that radars provide areal estimates of the rainfall, which are affected by systematic and random errors from various sources. Some of these errors are inherent to the observation system and unavoidable. Therefore, it is important to quantify the uncertainty associated with the radar-based rainfall products and to provide a strong basis for probabilistic quantitative precipitation estimation. Literature in the last four decades abounds with several studies comparing the radar estimates with estimates from other instruments, identifying the radar-rainfall error sources, minimizing the errors, and modeling the uncertainties. This chapter reviews key literature related to the characterization of errors for different space-time scales, rainfall regimes, and geographical settings. The emphasis is on the error models that can be utilized to represent the uncertainty in the form of ensembles. The chapter also lists a few open questions and challenges concerning the statistical structure of radar-rainfall errors and the generation of the ensembles.
Recent studies have revealed the existence of power laws, or scaling, in the magnitude of peak fl... more Recent studies have revealed the existence of power laws, or scaling, in the magnitude of peak flows for individual rainfall-runoff events. These findings offer a new theoretical framework to understand the physical basis of power laws in annual flood quantiles with respect to basin areas (representing spatial scale) that arise in regional statistical analyses. Research in the last decade has
ABSTRACT In this study, a Lagrangian radar echo extrapolation scheme (MAPLE) was tested for use i... more ABSTRACT In this study, a Lagrangian radar echo extrapolation scheme (MAPLE) was tested for use in very short-term forecasting of precipitation over a complex orographic region. The high-resolution forecasts from MAPLE for lead times of 5 min-5 h are evaluated against the radar observations for 20 summer rainfall events by employing a series of categorical, continuous, and neighborhood evaluation techniques. The verification results are then compared with those from Eulerian persistence and high-resolution numerical weather prediction model [the Consortium for Small-scale Modeling model (COSMO2)] forecasts. The forecasts from the MAPLE model clearly outperformed Eulerian persistence forecasts for all the lead times, and had better skill compared to COSMO2 up to lead time of 3 h on average. The results also showed that the predictability achieved from the MAPLE model depends on the spatial structure of the precipitation patterns. This study is a first implementation of the MAPLE model over a complex Alpine region. In addition to comprehensive evaluation of precipitation forecast products, some open questions related to the nowcasting of rainfall over a complex terrain are discussed.
In this study, we report results from scaling analysis of 2.5 m spatial and 1 s temporal resoluti... more In this study, we report results from scaling analysis of 2.5 m spatial and 1 s temporal resolution lidar-rainfall data. The high resolution spatial and temporal data from the same observing system allows us to investigate the variability of rainfall at very small scales ranging from few meters to ∼1 km in space and few seconds to ∼30 min in time. The results suggest multiscaling behaviour in the lidar-rainfall with the scaling regime extending down to the resolution of the data. The results also indicate the existence of a space-time transformation of the form t∼L z at very small scales, where t is the time lag, L is the spatial averaging scale and z is the dynamic scaling exponent.
Many downscaling techniques have been proposed to address the mismatch between the spatial and te... more Many downscaling techniques have been proposed to address the mismatch between the spatial and temporal scales of rainfall by some observational systems or numerical simulations, and the requirements of land surface hydrologic models. Most of these techniques focus on the mechanism producing the great variability in both space and time that is characteristic of rainfall. In this study the authors investigate the hydrologic consequences of the downscaling. The hydrologic scope of the study is limited to flooding and thus focuses on the scale dependence of peak flows with respect to drainage area. The authors use numerical simulations using a rainfall-runoff model based on a landscape decomposition framework that preserves the major elements of flood genesis, i.e. runoff generation at hillslopes and channel routing of the runoff. The decomposition preserves the full extent of the channel drainage network and its scaling properties. The framework allows addressing the questions of appr...
1] The goal of this study is to diagnose the manner in which radar-rainfall input affects peak fl... more 1] The goal of this study is to diagnose the manner in which radar-rainfall input affects peak flow simulation uncertainties across scales. We used the distributed physically based hydrological model CUENCAS with parameters that are estimated from available data and without fitting the model output to discharge observations. We evaluated the model's performance using (1) observed streamflow at the outlet of nested basins ranging in scale from 20 to 16,000 km 2 and (2) streamflow simulated by a well-established and extensively calibrated hydrological model used by the US National Weather Service (SAC-SMA). To mimic radar-rainfall uncertainty, we applied a recently proposed statistical model of radar-rainfall error to produce rainfall ensembles based on different expected error scenarios. We used the generated ensembles as input for the hydrological model and summarized the effects on flow sensitivities using a relative measure of the ensemble peak flow dispersion for every link in the river network. Results show that peak flow simulation uncertainty is strongly dependent on the catchment scale. Uncertainty decreases with increasing catchment drainage area due to the aggregation effect of the river network that filters out small-scale uncertainties. The rate at which uncertainty changes depends on the error structure of the input rainfall fields. We found that random errors that are uncorrelated in space produce high peak flow variability for small scale basins, but uncertainties decrease rapidly as scale increases. In contrast, spatially correlated errors produce less scatter in peak flows for small scales, but uncertainty decreases slowly with increasing catchment size. This study demonstrates the large impact of scale on uncertainty in hydrological simulations and demonstrates the need for a more robust characterization of the uncertainty structure in radar-rainfall. Our results are diagnostic and illustrate the benefits of using the calibrationfree, multiscale framework to investigate uncertainty propagation with hydrological models.
Estimation of the correlation structure of random fields has many applications in earth sciences.... more Estimation of the correlation structure of random fields has many applications in earth sciences. Using a simulation framework, the authors address sampling problems in estimating the spatial correlation function. The framework consists of generating realizations of homogeneous, isotropic Gaussian and lognormal random fields with a two-parameter power exponential spatial correlation function. The two parameters in the correlation function are the
It is well acknowledged that there are large uncertainties associated with radar-rainfall estimat... more It is well acknowledged that there are large uncertainties associated with radar-rainfall estimates. Numerous sources of these errors are due to parameter estimation, the observational system and measurement principle, and not fully understood physical processes. To describe these uncertainties, rain gauge data are usually considered as a good approximation of the true rainfall values. However, in the vast majority of
Characterization of radar-rainfall error spatial covariance requires high-density rain gauge netw... more Characterization of radar-rainfall error spatial covariance requires high-density rain gauge networks and high quality data. The authors use data from two rain gauge networks: IIHR ground validation network at Iowa City, Iowa consisting of 25 platforms, with average intergauge distance of about 5 km, and the Iowa City Municipal Airport network consisting of 9 platforms and covering approximately 1 km2.
In the large majority of studies on the evaluation of radar-rainfall estimates, rain gauge measur... more In the large majority of studies on the evaluation of radar-rainfall estimates, rain gauge measurements have been used to approximate the ground-level true rainfall. Nevertheless, as extensively documented in the literature, this approximation leads to an additional source of uncertainty due to the large sampling discrepancies between the two measuring devices: radar data are area-averaged estimates while rain gauges can
Idealized model studies have shown that space-time rainfall intermittency and its statistical sca... more Idealized model studies have shown that space-time rainfall intermittency and its statistical scaling properties have a significant impact on statistical scaling of flood peaks with respect to drainage area. In this study, simulation is used as a methodological tool to explore the effects of radar-rainfall uncertainties on the observed scaling of flood peaks. Simulated rainfall fields with varying power spectrum
Peak flows in a basin are a result of complex nonlinear interactions among rainfall and various p... more Peak flows in a basin are a result of complex nonlinear interactions among rainfall and various processes in the landscape. Studies based on simulations and data have explored the role of rainfall characteristics on the statistical structure of peak flows. However, it is widely acknowledged that the rainfall products are affected by uncertainties from various sources. In this study, we
ABSTRACT The Global Precipitation Measurement aims to provide rainfall products at scales relevan... more ABSTRACT The Global Precipitation Measurement aims to provide rainfall products at scales relevant for flood forecasting, offering effective means to predict floods in sparsely gauged and ungauged regions of the world. To achieve this goal we face two main challenges: (1) the development of hydrological models that do not rely on calibration as relevant data are mostly unavailable; and (2) the need for better understanding of the limitations imposed by the inherent features of satellite rainfall products (retrieval errors, infrequent sampling, and low spatial resolution) on flood prediction across space-time scales. In this study we applied a calibration-free multi-scale hydrological model to investigate the potential use of GPM rainfall products for flood prediction. The hydrological model used in this study is based on the faithful decomposition of the landscape into hillslopes and river channel links, and on the solution of the mass and momentum equations for each control volume. To avoid calibration we adopt parameters that are directly related to measurable properties of the basin, as provided by remote sensing techniques (e.g. topography, land cover and land use). We simulated the Iowa 2008 flood event that resulted in economical losses of billions of dollars. We ran the model using STAGE IV radar rainfall data, and we evaluated model results using observed streamflow data for 26 sites ranging in drainage area from 22 to 16,800 km2. We generated possible statistical realizations of satellite rainfall maps by modifying the radar rainfall product based on the already defined basic capabilities of GPM rainfall products. We first independently investigated different aspects of satellite rainfall products that are likely to introduce errors in the simulation of hydrological processes. We then combined multiple aspects of the errors to provide a more realistic estimation of the total uncertainty. We also included simulations with the currently available satellite rainfall products (TRMM-3B42, PERSIANN, and CMORPH). The results of this study provide insights about the aspects that current and future satellite missions should focus on to increase the value of remote sensing products for flood prediction.
This study presents a large sample evaluation of different nowcasting systems over the Southern S... more This study presents a large sample evaluation of different nowcasting systems over the Southern Swiss Alps. Radar observations are taken as a reference against which to assess the performance of the following short-term quantitative precipitation forecasting methods: -Eulerian persistence: the current radar image is taken as forecast. -Lagrangian persistence: precipitation patterns are advected following the field of storm motion (the
ABSTRACT Scaling behavior of rainfall time series is characterized using monofractal, spectral, a... more ABSTRACT Scaling behavior of rainfall time series is characterized using monofractal, spectral, and multifractal frameworks. The study analyzed temporal scale-invariance of rainfall in the tropical island of Singapore using a large dataset comprising 31 years of hourly and 3 years of 1-min rainfall measurements. First, the rainfall time series is transformed into an occurrence-non-occurrence binary series, and its scaling behavior is analyzed using box-counting analysis. The results indicated that the rainfall support displays fractal structure, but within a limited range of scales. The rainfall support has a fractal dimension (D f ) of 0.56 for scales ranging from 1 min to 1.5 h and a D f of 0.37 from 1.5 h to 1.5 days. The results further showed that the fractal dimension decreases with the increase in the threshold used to define binary series. Spectral analysis carried out on the rainfall time series and the corresponding binary series showed three distinct scaling regimes of 4 min-2 h, 2-24 h, and 24 h-1 month. In all the scaling regimes, the spectral exponents for the rainfall series were smaller than those for the binary series. The study then investigated the presence of multiscaling behavior in rainfall time series using moment scaling analysis. The results confirmed that the rainfall fluctuations display a multiscaling structure, which was modeled in the framework of universal multifractals. The results from this study would not only improve our understanding of the temporal rainfall structure in Singapore and the surrounding Maritime Continent but also help us build and parameterize parsimonious models and statistical downscaling techniques for rainfall in this region.
Quarterly Journal of the Royal Meteorological Society, 2012
The study describes the diurnal cycle of precipitation in the Alps and surrounding forelands usin... more The study describes the diurnal cycle of precipitation in the Alps and surrounding forelands using high-resolution volumetric data from a network of three groundbased radars. First, the study focused on the regional, seasonal and interannual variability in the diurnal behaviour of surface rainfall intensity, frequency and spatial correlations. Secondly, the study examined the diurnal variation in the vertical structure of summertime precipitation. Significant regional and seasonal differences are noticed in the diurnal cycle of precipitation features. The signal is strongest in the Alps, moderate in the plains north of the Alps and weak in the Po River Plain south of the Alps. The phase of the diurnal cycle is around 1630 UTC over the Alps for both spring and summer seasons. During autumn, the signal is weakest with no well-defined phase. The spatial variogram analysis showed that rainfall patterns are most variable between 1200 and 1400 UTC, approximately 3 h before the occurrence of diurnal maximum in precipitation frequency and intensity. The vertical structure of precipitation is characterized using frequency-altitude diagrams, and its diurnal behaviour is analysed using Hovmöller diagrams. The results show that the diurnal cycle of precipitation varies significantly with height.
ABSTRACT The availability of radar-based rainfall products at high space-time resolutions and ove... more ABSTRACT The availability of radar-based rainfall products at high space-time resolutions and over continental scales has greatly advanced our understanding of the rainfall process and its interactions with other hydrological processes across a wide range of scales. However, it is well known that radars provide areal estimates of the rainfall, which are affected by systematic and random errors from various sources. Some of these errors are inherent to the observation system and unavoidable. Therefore, it is important to quantify the uncertainty associated with the radar-based rainfall products and to provide a strong basis for probabilistic quantitative precipitation estimation. Literature in the last four decades abounds with several studies comparing the radar estimates with estimates from other instruments, identifying the radar-rainfall error sources, minimizing the errors, and modeling the uncertainties. This chapter reviews key literature related to the characterization of errors for different space-time scales, rainfall regimes, and geographical settings. The emphasis is on the error models that can be utilized to represent the uncertainty in the form of ensembles. The chapter also lists a few open questions and challenges concerning the statistical structure of radar-rainfall errors and the generation of the ensembles.
Recent studies have revealed the existence of power laws, or scaling, in the magnitude of peak fl... more Recent studies have revealed the existence of power laws, or scaling, in the magnitude of peak flows for individual rainfall-runoff events. These findings offer a new theoretical framework to understand the physical basis of power laws in annual flood quantiles with respect to basin areas (representing spatial scale) that arise in regional statistical analyses. Research in the last decade has
ABSTRACT In this study, a Lagrangian radar echo extrapolation scheme (MAPLE) was tested for use i... more ABSTRACT In this study, a Lagrangian radar echo extrapolation scheme (MAPLE) was tested for use in very short-term forecasting of precipitation over a complex orographic region. The high-resolution forecasts from MAPLE for lead times of 5 min-5 h are evaluated against the radar observations for 20 summer rainfall events by employing a series of categorical, continuous, and neighborhood evaluation techniques. The verification results are then compared with those from Eulerian persistence and high-resolution numerical weather prediction model [the Consortium for Small-scale Modeling model (COSMO2)] forecasts. The forecasts from the MAPLE model clearly outperformed Eulerian persistence forecasts for all the lead times, and had better skill compared to COSMO2 up to lead time of 3 h on average. The results also showed that the predictability achieved from the MAPLE model depends on the spatial structure of the precipitation patterns. This study is a first implementation of the MAPLE model over a complex Alpine region. In addition to comprehensive evaluation of precipitation forecast products, some open questions related to the nowcasting of rainfall over a complex terrain are discussed.
In this study, we report results from scaling analysis of 2.5 m spatial and 1 s temporal resoluti... more In this study, we report results from scaling analysis of 2.5 m spatial and 1 s temporal resolution lidar-rainfall data. The high resolution spatial and temporal data from the same observing system allows us to investigate the variability of rainfall at very small scales ranging from few meters to ∼1 km in space and few seconds to ∼30 min in time. The results suggest multiscaling behaviour in the lidar-rainfall with the scaling regime extending down to the resolution of the data. The results also indicate the existence of a space-time transformation of the form t∼L z at very small scales, where t is the time lag, L is the spatial averaging scale and z is the dynamic scaling exponent.
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Papers by P. Mandapaka