Figure 4.5 Flood prediction uncertainty for hourly, 3-, and 6-h PM retrievals for .... Figure 7B.1 NOAH-LSM vegetation parameter and bias adjustment for soil ..... In this chapter, an attempt was made to answer the following question: what is.
INVESTIGATING ERROR PROPAGATION IN FLOOD PREDICTION BASED ON REMOTELY-SENSED RAINFALL
Faisal Hossain
B.Tech, Institute of Technology, Banaras Hindu University, 1996 M.Eng, National University of Singapore, 1999
A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy at the University of Connecticut 2004
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APPROVAL PAGE
Doctor of Philosophy Dissertation
Investigating Error Propagation in Flood Prediction based on Remotelysensed Rainfall
Presented by
Faisal Hossain, B.Tech, M.Eng.
Major Advisor_____________________________________________________ Emmanouil N. Anagnostou
Associate Advisor__________________________________________________ Fred L. Ogden
Associate Advisor___________________________________________________ Amvrossios C. Bagtzoglou
University of Connecticut 2004
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ACKNOWLEDGEMENTS
I must first thank my major advisor, Dr. E.N. Anagnostou (Manos) for making the whole duration of my doctoral candidature a very meaningful and enjoyable experience. It was Manos who taught me that the journey was as important as the destination. During this passage, I have never, for a single moment, felt what it was like to be under a state of stress or unhappiness. The list of publications on my resume that has accumulated over the last 5 years is no doubt a testament to this nurturing I have received from my advisor-cum-bestfriend on campus. I must also thank Dr. Fred L Ogden for his kindness in providing me with very constructive comments, particularly on the pertinent aspects of hydrologic modeling. Gratitude is also extended to Ross – or Dr. Amvrossios C. Bagtzoglou. I have learned very valuable tools for scientific analyses of the environment during the last 2 years that I had the opportunity of interacting with him closely. But more importantly, Ross taught me something that is not yet offered as a graduate course in any university curriculum – the importance of intellectual modesty. Dr. Marco Borga of the University of Padova, Italy, provided me with a significant portion of the data used in this study. Without Marco’s generosity, half of the scientific inquiries that I wished to pursue would have remained unexplored. I must also thank the NASA Earth System Science Student Fellowship Program that had funded the last two and most fruitful years of my graduate studies. Last but not least, I wish to remember my mother, father and my adorable sister for standing by me throughout my doctoral work.
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DEDICATION PAGE
To The Human Spirit of Scientific Inquiry: May Sacrifice, Tolerance and Understanding continue to make this world a better place than what it is today
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TABLE OF CONTENTS
1. Introduction ………………………………………………………………………1 2. Hydrological Model Sensitivity to Parameter and Radar Rainfall Estimation Uncertainty…………………………………………………………14 3. Assessment of a Probabilistic Scheme for Flood Prediction………………….45 4. Sensitivity Analyses of Satellite Rainfall Retrieval and Sampling Error on Flood Prediction Uncertainty………………………………………. 69 5. Assessment of Current Passive Microwave and Infrared based Satellite Rainfall Remote Sensing for Flood Prediction………………….…102 6. On the Dependency of Soil Moisture Prediction Accuracy to Satellite Rainfall Estimation and Land Surface Modeling Uncertainties …………………………………………………………………142 7. Improving Computational Efficiency of Error Propagation Studies A. On Latin Hypercube Sampling for Efficient Uncertainty Estimation of Satellite Rainfall Observations in Flood Prediction……….182 B. A Stochastic Response Surface Method for Bayesian Estimation Of Uncertainty in Soil Moisture Simulation from a Land Surface Model………………………………………………………………………...209 8. Conclusions and Recommendations for Further Study…………………...246 9. References……………………………………………………………………253
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LIST OF FIGURES Figure 2.1 Left panel: Relief map of the Posina basin. Right panel: Locations, relative to radar position, of the rain gauge stations used for mean-field radar bias estimation (*), error statistics computation during optimisation of the radar rainfall algorithm (+), and independent stations close to the Posina basin used in this study (o). The dotted semi-circles represent 10-km radar ranges. Figure 2.2 Cumulative rain gauge and radar hyetographs for basin-averaged rainfall for OCT92 flood event. Figure 2.3 Dotty plots of the Nash-Sutcliffe model efficiency measure for the four GLUE parameters. Each dot represents one run of the model with parameter values chosen randomly by uniform sampling across the range of each parameter in Table 2.4. Figure 2.4 Propagated uncertainty bounds for OCT92 flood event with rainfall input from rain gauge data. Figure 2.5 Propagated uncertainty bounds for OCT92 flood event with rainfall input from radar rainfall estimates. Figure 2.6 Runoff prediction uncertainty assessment in terms of Uncertainty Ratio and of Exceedance Ratio for various behavioral thresholds. Figure 3.1 Generalized Sensitivity Analyses (GSA: Spear and Hornberger 1980) for the four most sensitive TOPMODEL parameters. Upper left panel -overland flow velocity parameter RV (m hr-1); Upper right panel – exponential decay rate parameter SZM (m); Lower left panel – time delay parameter TD (hr); Lower right panel – lateral transmissivity parameter T0 (ln(m2 hr-1)). Three different curves represent the three different cumulative probability density functions achieved with behavioral threshold Eexp > 0.2, Eexp>0.4 and Eexp>0.6. Figure 3.2 Flow chart for the probabilistic discharge prediction scheme. Figure 3.3 Observed (solid line) versus predicted runoff hydrographs derived from the deterministic (thin dashed line) and probabilistic (thick dashed) scheme for the chronological sequence of 15 storm events. Figure 3.4 The Probabilistic prediction with the uncertainty limits corresponding to 90% confidence bounds. The thick solid line represents observed discharge. Dashed lines represent the 5th and 95th percentiles of runoff prediction. The vertically drawn lines with shading represent the 5th and 95th percentiles of time to peak.
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Figure 3.5 Runoff and time to peak exceedance probability (upper and middle panels) versus width of prediction limit (upper-lower quantile range). Lower panel: uncertainty in time to peak versus width of prediction limit. Uncertainty in time to peak is expressed as a fraction of total storm duration. Figure 3.6 Mean and one standard deviation (vertical bars) of the norm of optimum parameter set (upper panel) and optimum quantile (lower panel) versus prediction instance. Figure 4.1 Geographic location of (right panel) Posina watershed and (left panel) watershed elevation map overlaid by the rain gauge network locations (in solid circles). The inner box represents a typical 10X10 km2 satellite foot print. Figure 4.2 October 1992 storm event hydrograph and hyetograph in Posina, Italy. Figure 4.3 Probability of rain detection by PM (TMI) retrievals over Southern Africa, Amazon and the Southern United States determined using TRMM Precipitation Radar (PR) surface rainfall estimates as reference. The data products used for TRMM PR and TMI rain rates were 2A25 and 2A12, respectively. The period of matched PR/TMI data was January-July 2002. Figure 4.4 Algorithmic structure of the RSREM. rn is a randomly generated uniform [0,1] deviate. Figure 4.5 Flood prediction uncertainty for hourly, 3-, and 6-h PM retrievals for October 1992 storm event in Posina. The quantiles shown are the 50% (median) and the confidence bounds for 5% and 95% percentiles (90% area). Figure 4.6 Effect of PM retrieval bias (multiplicative) on flood prediction uncertainty presented in terms of ERs. (Top) Time ERs: ratio of runoff error for a given sampling scenario (3 or 6h) normalized by the 1-h runoff error derived from the correspondingly varied retrieval error parameters. (Bottom) Default ER: ratio of runoff error for a given sampling scenario (3 or 6h) normalized by the 1-h runoff error with the default retrieval error parameters. The upper panels show the effect of sampling, and the lower panels show the combined effect of sampling and retrieval error on flood prediction uncertainty. Figure 4.7 Effect of PM retrieval standard deviation (σ) on flood prediction uncertainty presented in terms of ERs. (Top) Time ERs: ratio of runoff error for a given sampling scenario (3 or 6h) normalized by the 1-h runoff error derived from the correspondingly varied retrieval error parameters. (Bottom) Default ER: ratio of runoff error for a given sampling scenario (3 or 6h) normalized by the 1-h runoff error with the default retrieval error parameters. The upper panels
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show the effect of sampling, and the lower panels show the combined effect of sampling and retrieval error on flood prediction uncertainty. Figure 4.8 Effect of PM retrieval error lag-one correlation (ρ2) on flood prediction uncertainty presented in terms of ERs. (Top) Time ERs: ratio of runoff error for a given sampling scenario (3 or 6h) normalized by the 1-h runoff error derived from the correspondingly varied retrieval error parameters. (Bottom) Default ER: ratio of runoff error for a given sampling scenario (3 or 6h) normalized by the 1-h runoff error with the default retrieval error parameters. The upper panels show the effect of sampling, and the lower panels show the combined effect of sampling and retrieval error on flood prediction uncertainty. Figure 4.9 Contours of peak runoff Time ERs in peak runoff as a function of PM retrieval log-error’s lag-one hourly correlation (ρ2) and standard deviation (σ) for 3-h (left panel) and 6-h (right panel) sampling scenarios. Figure 4.10 Merging ER values of runoff volume presented as a function of PM retrieval error standard deviation (σPM) for two PM sampling scenarios (left panel: 3 h; right panel: 6 h) and three levels of IR retrieval error standard deviation (σIR). Figure 5.1 An example of the assumed complement of four PM sensor overpasses (sampling) for Storm 9 (October, 1992). The assumed constellation comprises SSM/I-F14, SSM/I-F15, TMI and AMSR-E. Figure 5.3 Probability of Detection of instantaneous reference rainrate PODINST , as a function of accumulated reference rainrate : Prob( RREF-INST >0.0 | RREF-ACCU >0.0). The dashed line are the best fits to the data, which determine the error model parameters of RSREM. Figure 5.4 An example of matched TMI (2A12), IR (3B41RT) and TRMM-PR (2A25) retrieved rain maps over Central Africa. Left panel is TMI (Orbit No. 24225); Middle Panel is PR (Orbit No. 24225); Right panel is IR (1300 hrs UTC), on February 13, 2002. PR and IR are shown at the 25 km X 25 km2 resolution. Figure 5.5 Probabilities of rain detection (right panels) and probability distributions of rain rates in false alarms (left panels) for PM (upper panels) and IR (lower panels) retrievals. These experimental results are determined on the basis of coincident TRMM PR overland rainfall data around the Globe. The solid lines are best fits to the data, which determine the error model parameters presented in the paper. Figure 5.6 Flood prediction uncertainty (5% and 95% upper/lower quantiles) of Storm 9 (October, 1992) for the GPM based 3 hourly sampling (left), and viii
current PM sampling (right). Upper panels are based on PM retrievals, and lower panels on the merged IR-PM rain products. Figure 5.7a Effect of storm morphology on the uncertainty of flood prediction driven by current PM retrieval sampling scenarios. Upper left panel: Error in Time to Peak vs. storm duration; Upper right panel: Error in Runoff Volume vs. rain fraction; Lower left panel: Error in Runoff Volume vs. mean conditional rain rate; Lower right panel: Error in Runoff Volume vs. standard deviation of conditional rain rate. Figure 5.7b Same as in Figure 5.7a, but for flood predictions driven by the GPM based 3-hourly PM sampling scenario. Figure 5.8a Contours of Merging ER values representing the relative runoff prediction error of the combined PM-IR rain products to PM retrieval alone (current PM sampling scenarios), for various levels of IR rain estimation accuracy (SIR vs. maximum IR rain detection probability). Left panel: Merging ERs in Peak Runoff; Middle Panel: Merging ERs in Time to Peak Runoff; Right Panel: Merging ERs in Runoff Volume. The solid circle represents the performance level of current IR algorithm. Figure 5.8b Same as in Figure 5.8a, but for the GPM based 3-hourly sampling scenario. Figure 6.1 Schematic diagram indicating the two major sources of uncertainty and their interaction in the soil moisture simulation by a LSM. Dotted lines indicate uncertainty. Figure 6.2a NOAH-LSM simulation of soil moisture (with adjustment for vegetation parameters) at 5cm depth for CHAMP_ILL. Upper panel is rainfall measurement; lower panel – soil moisture simulation. Figure 6.2b NOAH-LSM simulation of soil moisture (with adjustment for vegetation parameters) for PERK_OK. Uppermost panel – observed rainfall from Mesonet; From bottom panel and up – soil moisture simulation at 60 cm, 25 cm and 5 cm depth, respectively. Figure 6.3a Rainfall estimation by Infra-red technique (3B41RT and RSREM) with simulated ranges of uncertainty over PERK_OK for 2002. Figure 6.3b Rainfall estimation by the assumed Infra-red technique (RSREM) over CHAMP_ILL for 1998. Figure 6.4 Schematic representation of partial and total uncertainty in soil moisture simulation. Dotted wave-like lines represent uncertainty in the form of random realizations.
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Figure 6.5 Comparisons of soil moisture prediction uncertainty (partial and total) for CHAMP_ILL. Figure 6.6a Partial uncertainty in soil moisture prediction due to precipitation uncertainty for PERK_OK. Figure 6.6b Partial uncertainty in soil moisture simulation due to modeling uncertainty for PERK_OK. Figure 6.6c Total uncertainty in soil moisture prediction due to uncertainties in precipitation measurement and modeling for PERK_OK. Figure 6.7a The sensitivity of soil moisture prediction uncertainty to precipitation uncertainty as the regime transforms from a CHAMP_ILL-like environment (left-hand side, regime scale=0) to a more OK_PERK-like environment (right-hand side, regime scale=1). The y-axis represents the relative increase of UR (%) in soil moisture prediction (5 cm depth) at the 90% quantile width. The meso-forcing data pertained to the CHAMP_ILL regime. Figure 6.7b Error propagation from rainfall to soil moisture for two regimes when model performs at optimal level. The soil moisture simulations are at the 5 cm depth. Figure 6.8a The Generalized Sensitivity Analyses (GSA) of NOAH-LSM model parameters for CHAMP_ILL for three different behavioral thresholds (ENS >0.4, ENS>0.5 and ENS>0.7). Figure 6.8b The Generalized Sensitivity Analyses (GSA) of NOAH-LSM model parameters for PERK_OK for three different behavioral thresholds. Figure 6.9a The categorization of model parameter sets in HIGH (0.75 0.2, 0.4, and 0.6), indicating differences in the shapes of the CDF functions. The CDFs of the three non-sensitive parameters were almost identical (not shown here). Following the GLUE methodology, multiple simulations were conducted by sampling randomly from the specified ranges of the sensitive parameters (Table 3.2). Discharge simulations were carried out for each randomly generated parameter set and the likelihood measure calculated on the basis of Equation 3.1. A large number of random Monte Carlo (MC) simulations were conducted to allow the selection of 500 behavioral parameter sets characterized by simulation efficiency greater than an assigned threshold of 0.4. Ensemble runs based on the selected behavioral parameter sets characterize the range of uncertainty in the model prediction. Beven and Binley (1992) have argued that this procedure of GLUE would typically reflect all sources of uncertainty collectively in discharge simulation and allow the uncertainty to be carried
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forward into the predictions. For further details about the GLUE method, the reader is referred to the original work of Beven and Binley (1992) and subsequent follow-ups in Freer et al. (1996) and Beven and Freer (2001).
b)
The Prediction Scheme
As discussed above the behavioral ensemble parameter sets can be used to generate a range of discharge predictions that can be weighted by the corresponding likelihood weights of the parameter sets (hereafter named ‘likelihood-weighted discharges’). The likelihood-weighted discharges are then ranked to determine discharge prediction quantiles (ranging from 1 to 99th percentile). Comparison with observed runoff is used to evaluate the likelihood weight (based on Equation 3.1) of each quantile. The hydrograph resulting from the quantile associated with the highest likelihood value (hereafter referred to as ‘optimum quantile’) is termed as the ‘most probable hydrograph’. Thus, by virtue of the a priori evaluated model parameter sets and prediction quantiles’ likelihood, the most probable discharge can be predicted for a subsequent storm along with an estimate of its uncertainty. When new observations become available, each of the behavioral parameter sets and prediction quantiles’ likelihood are updated on the basis of Bayes’ theorem as follows,
LP (θ i | Y ) =
L(θ i | Y ) Lo (θ i ) C
(3.2)
where θi refers either to the ith parameter set or the ith prediction quantile, L0(θi) is the prior likelihood weight and L(θ|Y) is the likelihood weight calculated with the set of
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observed (current) data Y; LP(θ|Y) is the posterior likelihood weight for the simulation by θ given Y; and C is a scaling constant calculated such that the posterior weights add up to one. For successive storm events, the calculated posterior likelihood distribution is used to project the uncertainty associated with the predictions to future events, thus becoming the prior distribution in Equation 3.2. The posterior likelihood distribution is also used directly to evaluate the uncertainty limits for future events for which observed streamflow data is not available to validate model predictions. In successive prediction instances, the most probable hydrograph may not necessarily correspond to the same quantile. Nevertheless, the optimum quantile may be expected to eventually converge to a certain percentile depending on the worth of model, additional data and stationarity of the hydrologic process. The Bayesian updating described above may also have the effect of reducing the size of the sample of behavioral parameter sets, i.e., sets that were behavioral a priori may become non-behavioral a posteriori due to the multiplicative updating process of Equation 3.2. Thus, it may be necessary to resample the parameter distribution to reflect the regions of high likelihood values in parameter space, which can be expected to vary from storm to storm (Beven and Binley, 1992). In resampling, it is ensured that the sample size remains constant (500 sets) to maintain consistency. In this study, the re-sampling procedure of nearest neighborhood search proposed by Beven and Binley (1992) has been employed. The probabilistic discharge prediction scheme is summarized with a flowchart shown in Figure 3.2.
The scheme starts with a historical storm-runoff dataset to
determine 500 behavioral parameter sets and the derived optimum quantile. In a
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subsequent storm event the above information of the optimum quantile is used to evaluate its most probable hydrograph. The associated uncertainty is characterized by the ensemble discharge prediction generated by the behavioral parameter sets. When runoff observations for the new event become available, the likelihood weights of the parameter sets and prediction quantiles are updated using Bayes’ equation. The procedure is repeated for successive storm events.
3.4 Assessment Framework
Assessment of the probabilistic discharge prediction scheme is performed on series of 15 flood inducing storm events shown in Table 3.1. Comparisons were made against discharge predictions derived from a best parameter set (hereafter referred to as ‘deterministic scheme’) derived based on the optimization algorithm of Duan et al. (1992). The first storm of the series was used as a priori dataset for generating the ensemble of behavioral parameters and deriving the optimal parameter set for the deterministic scheme.
For consistency in comparison, the deterministic scheme
employed Equation 3.1 for the cost function and the same parameter search space shown in Table 3.2. The comparison between the two schemes is performed as following. On the basis of the chronological series of 15 flood events one can evaluate 14 prediction/update instances. To generate statistically sufficient cases for comparison, the original chronological storm series was shuffled to synthesize more hypothetical storm event sequences. As is it impossible to account for all possible combinations of storm sequences (in the order of 1012), limited shuffling was performed to generate 14 more
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distinct storm sequences shown in Table 3.3. This led to a total of 210 prediction/update cases for assessment. No two sequences in the selected set had a common initial storm event in order to minimize potential dependency on the initialization of the hydrologic model. The deterministic scheme was recalibrated after prediction for every new storm (i.e., updated) in a sequence considering all prior storm cases using as first guess of the parameter set the values used in prediction. For the probabilistic scheme, the likelihood weights of the ensemble of behavioral parameter sets and prediction quantiles were updated with each new observation as described in section 3.3 and summarized in the flow-chart of Figure 3.2. For single event simulations, it is recognized that the initial moisture conditions of the watershed can be a sensitive issue and that the model parameter associated with this initial condition may be uncertain. To ensure consistency in comparison, we constrained the initial moisture condition for each event to physically representative values common to both schemes.
The relative error (ε) in three runoff parameters: Runoff Volume (RV), Peak Runoff (PR) and Time to Peak (TP), are defined as:
εX =
X sim - X obs X obs
(3.3)
where X is defined as one of the runoff parameters (i.e., RV, PR, TP).
For the
probabilistic scheme, the simulated runoff parameters were derived from the most probable hydrograph, and for the deterministic scheme these were obtained from the optimum parameter set based hydrograph. The probabilistic scheme’s prediction limits 54
were evaluated (derived on the basis of the different quantiles) in terms of its ability to envelope the observed peak runoff and time to peak parameters. For this purpose, an analysis of the sensitivity of exceedance probability in peak runoff and time to peak with respect to different quantile ranges was carried out. Consistent with the arguments of Beven and Binley (1992), the aim herein was to assess whether the probabilistic scheme predicted uncertainty bounds that were consistent with validation statistics, which represent both modeling and input uncertainty.
3.5 Results and Discussion
Figure 3.3 shows comparisons between deterministic and probabilistic scheme predictions for the chronological storm sequence (1st column in Table 3.3). The most probable hydrograph is shown by the thick dotted line while the deterministic prediction is shown by the thin dotted line. It is observed that both schemes perform with the same accuracy. At certain storm cases (such as storms 2, 3 and 13) the most probable hydrograph appear superior in prediction than the deterministic scheme, while at other cases (such as storms 5, 9 and 14) peak runoff is moderately overestimated while the deterministic scheme underestimates it. In Figure 3.4, the prediction limits associated with the 90% confidence bounds (5th and 95th percentile) are presented for prediction of runoff and time to peak. The observed peak runoff is bounded within the 90% confidence bounds for 10 out of 14 prediction instances (storms 4,5,7,8,9,10,11,13,14,15). This indicates that the upper and lower uncertainty limits corresponding to the 90% confidence bounds of the probabilistic scheme have sufficient value to the decision makers for
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assessment of the probable level of river stages during a flood event. Similarly, for predicting the arrival time of the peak flood wave (i.e., time to peak), 13 storm events out of 14 are sufficiently bounded by the 90% prediction limits (compare the vertical lines with shading with the thick solid line in Figure 3.4). The probabilistic scheme can therefore adequately predict the lower and upper limits of the two most important runoff parameters during a flood event: time to peak and peak runoff. In Table 3.4a, a summary of the statistics (mean and standard deviation) of the prediction accuracy of the two schemes on the basis of the 210 prediction cases are provided. The mean prediction efficiency (Equation 3.1) for the probabilistic scheme (corresponding to the most probable hydrograph) is slightly higher (0.57) than the deterministic scheme (0.55). The standard deviation of the probabilistic scheme’s efficiency of prediction is about 15% less than that of the deterministic scheme. For mean simulation error of runoff volume, both schemes have very low bias (around 6%), while peak runoff is underestimated by about 13% by the probabilistic scheme compared to the 6% overestimation by the deterministic scheme. The most noticeable difference is observed in the prediction of time to peak. The probabilistic scheme has 50% less variability than the deterministic scheme in time to peak error, even though the mean errors are comparable. In Table 3.4b, the reliability of the probabilistic scheme’s confidence bounds in terms of runoff and time to peak predictions are shown. The Runoff Exceedance Probability is defined as the number of hours the observed discharge exceeds the 90% confidence bounds derived from the probabilistic scheme normalized by the total number of hours with runoff. The Time to Peak Exceedance Probability is evaluated by the
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number of prediction cases where the observed time to peak is beyond the 90% confidence bounds of the predicted time to peak normalized by the total number cases (i.e., 210). A high exceedance probability would indicate the probabilistic scheme’s inability to provide useful estimates of uncertainty in runoff prediction. In Table 3.4b, it is observed that the probabilistic scheme has a low Time to Peak Exceedance Probability (12.4%). The Runoff Exceedance Probability is moderately low (47%) indicating that the runoff (including the peak runoff) error bounds can capture 53% of the variability in true runoff. In Figure 3.5 the sensitivity of runoff and time to peak exceedance probabilities and uncertainty range in time to peak (in hours) is evaluated with respect to error bounds varied from 10% (45th to 55th percentile) to 90% (5th to 95th percentile) quantile ranges. We define this range as the ‘width of prediction limits’. It is observed that a monotonic decrease of the Runoff Exceedance Probability for an increasing width of prediction limits (uppermost panel). The middle panel shows a similar pattern (rate of increase) in the sensitivity of time to peak uncertainty (expressed in relative terms as fraction of the storm duration) to the width. The lowermost panel shows the variation of Time to Peak Exceedance Probability with respect to the width of prediction limits. When the middle and uppermost panels are assessed jointly, 70% confidence bounds yield the best prediction uncertainty scenario, which gives an optimal combination of low exceedance probability with relatively low uncertainty range in time to peak. Analyses of all three panels combined can potentially assist decision-makers in identifying the optimum uncertainty quantile range, which is not too wide yet reliable.
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In Figure 3.6, the impact of additional data on the convergence towards the optimum parameter set for the deterministic scheme (upper panel) and optimum prediction quantile for the probabilistic scheme (lower panel) is shown. The purpose of this analysis is to identify whether the schemes (probabilistic and deterministic) are capable of reaching steady-state prediction conditions. In the upper panel, the convergence towards the global parameter set is evaluated using the norm of the optimum parameter set. The norm was computed by first normalizing the four optimum parameter values by their respective means and standard deviations evaluated from all the 210 prediction cases. This transformed each parameter to a consistent 0-1 scale. The norm was then computed as the square root of the sum of squares of the normalized optimum parameter values. For the lower panel, the optimum quantile is essentially the prediction quantile with the highest posteriori likelihood weight after each Bayesian updating is applied. For both schemes, convergence is noted. The norm gradually minimizes and then stabilizes to a near-zero value as additional storm data is incorporated in the optimization process (upper panel). The error bars representing one standard deviation gradually reduce with successive prediction cases. This suggests that the space of the parameter set that contains the optimum set is becoming increasingly constrained as more data is taken into account. For the probabilistic scheme, the mean optimum quantile appears to stabilize at the 70th percentile after the 14th Bayesian updating (uppermost right panel). The error bars, however, do not decrease in a similar fashion as that for the deterministic scheme. This indicates that there may be more than one region of high likelihood values in the parameter space, which may be an indicator of the non-stationarity of the hydrologic system and the inherent uncertainty in model formulation.
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3.6 Conclusions
This study assessed a probabilistic discharge prediction scheme based on the GLUE uncertainty framework In prediction, the probabilistic scheme simulated the most probable hydrograph with the upper and lower uncertainty limits associated with a given confidence bound. The Bayes’ theorem was used to update the posterior likelihood weights of the parameter sets and prediction quantiles. Upon comparison with the conventional optimum parameter set deterministic prediction it was observed that the probabilistic scheme was subject to nearly 50% less variability in time to peak prediction error. In terms of the most probable hydrograph, the scheme has similar levels of prediction accuracy as the deterministic scheme. The probabilistic scheme has an added value to decision-making and risk assessment due to the uncertainty predicted for the arrival time of peak runoff and magnitude of the flood wave. The procedure is simple in design, model independent, and can be easily implemented in a real-time operational scenario for computationally efficient rainfall-runoff models. Currently a major limitation of the scheme is its requirement for multiple model runs, which is computationally costly for use with physically-based distributed rainfallrunoff models. However, such models are increasingly used to predict consequences of land-use and climatic change in catchments. Their incorporation in a GLUE framework to support the probabilistic scheme formulated herein would be inefficient due to the large number of Monte Carlo runs needed for both model prediction and parameter sampling. An improved parameter-sampling scheme that accelerates both parameter search and reduces the total number of model runs is desirable. The uniform sampling procedure
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usually recommended for GLUE (Beven and Binley, 1992; Freer et al., 1996; Beven and Freer, 2001) warrants the exploration of more efficient sampling schemes. Methods such as the guided Monte Carlo scheme of Shorter and Rabitz (1997), tree-structured search of Spear et al. (1994), Latin Hypercube Sampling (McKay et al., 1979), and the Monte Carlo Markov Chain method (Kuczera and Parent, 1998; Bates and Campbell, 2001) are worth considering as future extensions to this study.
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Table 3.1 Statistics of the flood events used in this study. Storm No
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Date
Month Year 08 1987 10 1987 07 1989 11 1990 12 1990 03 1991 10 1991 04 1992 10 1992 12 1992 09 1993 11 1994 10 1996 11 1996 12 1997
Rainfall Statistics Duration Rainfall Maximum (hrs) Volume Rainrate (mm) (mm/hr) 72 128.8 26.7 96 137.3 11.2 96 125.2 10.9 96 120.8 8.7 108 191.4 13.9 72 77.4 8.3 84 185.2 16.9 120 113.0 8.7 120 440.3 18.0 144 118.6 6.8 132 286.2 8.4 72 146.0 11.7 96 299.8 12.9 120 179.9 10.2 84 126.1 9.3
Discharge (m3/s) Peak Discharge (m3/s) 54.40 75.72 31.49 64.37 76.41 32.59 117.40 56.89 192.50 41.60 49.40 106.90 156.50 70.80 70.26
Table 3.2 Parameter value ranges used in GLUE sampling.
SZM TD T0 RV
(m) (hr m-1) (ln(m2hr-1)) (m hr-1)
Minimum Value 0.001 0.001 0.001 200.000
Maximum Value 0.25 15.0 10.0 2000.0
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Sampling Strategy Uniform Uniform Uniform Uniform
Table 3.3 The 15 storm sequences (first column is the original chronological series). The numbers refer to the Storm Numbers in Table 3.1.
Storm No.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
2 1 3 4 5 6 7 8 9 10 11 15 14 12 13
3 1 8 7 9 4 7 2 9 10 11 12 13 14 15
4 6 13 3 11 1 10 8 9 7 5 15 2 14 12
5 1 2 3 12 6 7 8 9 10 11 4 13 14 15
Storm Sequences 6 7 8 9 1 1 1 10 2 2 2 11 3 3 3 15 4 4 9 4 5 5 5 5 7 6 6 6 14 8 7 13 9 9 4 8 10 10 10 1 11 11 13 2 12 12 12 12 13 13 11 7 8 14 14 14 15 15 15 3
10 1 6 3 4 5 2 7 8 9 11 12 13 14 15
11 13 14 15 4 5 6 7 8 9 10 12 1 2 3
12 1 2 3 4 5 6 10 8 9 7 11 13 14 15
13 14 1 1 2 2 3 3 4 4 5 5 6 11 7 7 8 8 9 9 10 10 11 6 12 12 14 13 15 15
15 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Table 3.4a Comparison of prediction accuracy of probabilistic scheme with deterministic scheme. Prediction Efficiency is based on Equation 3.1. For the Probabilistic scheme it is computed from the most probable hydrograph.
Prediction Efficiency Error in Runoff Volume Error in Peak Runoff Error in Time to Peak
Mean 0.552 -0.063 0.064 -0.043
Deterministic Scheme Standard Deviation 0.318 0.502 0.515 0.428
Probabilistic Scheme Mean Standard Deviation 0.571 0.280 0.063 0.591 -0.134 0.584 0.094 0.205
Table 3.4b Exceedance probabilities for runoff and time to peak for the probabilistic scheme.
Runoff Exceedance Probability (mean) 0.467
Time to Peak Exceedance Probability (mean) 0.124
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Figure 3.1 Generalized sensitivity analyses (GSA: Spear and Hornberger 1980) for the four most sensitive TOPMODEL parameters. Upper left panel -overland flow velocity parameter RV (m hr-1); Upper right panel – exponential decay rate parameter SZM (m); Lower left panel – time delay parameter TD (hr); Lower right panel – lateral transmissivity parameter T0 (ln(m2 hr-1)). Three different curves represent the three different cumulative probability density functions achieved with behavioral threshold Eexp > 0.2, Eexp >0.4 and Eexp > 0.6.
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CALIBRATION 1) Generate 500 behavioral parameter sets from historical data 2) Evaluate the prior likelihood weights of these parameter sets 3) Evaluate the prior likelihood weights of prediction quantiles 4) Identify the optimum quantile
PREDICTION 5) Propagate the parameter sets using GLUE for a new event 6) Determine the uncertainty limits (e.g. 5th – 95th percentiles) 7) Determine the most probable hydrograph based on the optimum quantile
UPDATING 8) Update for posterior likelihood weights of parameter sets (Eqn. 6.2) 9) Update for posterior likelihood weights of prediction quantiles (Eqn. 6.2) 10) Update optimum quantile for the most probable hydrograph 11) Resample parameter sets if necessary following Beven and Binley (1992)
Figure 3.2 Flow chart for the probabilistic discharge prediction scheme.
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Figure 3.3 Observed (solid line) versus predicted runoff hydrographs derived from the deterministic (thin dashed line) and probabilistic (thick dashed) scheme for the chronological sequence of 15 storm events.
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Figure 3.4 The Probabilistic prediction with the uncertainty limits corresponding to 90% confidence bounds. The thick solid line represents observed discharge. Dashed lines represent the 5th and 95th percentiles of runoff prediction. The vertically drawn lines with shading represent the 5th and 95th percentiles of time to peak.
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0.8 0.6 0.4 0.2
Uncertainty in Time to Peak (%)
Time to Peak Exceedance Probability
Runoff Exceedance Probability
1
0
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
30 25 20 15 10 5 0 0
20
40 60 Width of Prediction Limits (%)
80
100
Figure 3.5 Runoff and time to peak exceedance probability (uppermost and middle panels) versus width of prediction limit (upper-lower quantile range). Lowermost panel: uncertainty in time to peak versus width of prediction limit. Uncertainty in time to peak is expressed as a fraction of total storm duration.
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Figure 3.6 Mean and one standard deviation (vertical bars) of the norm of optimum parameter set (upper panel) and optimum quantile (lower panel) versus prediction instance.
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4
SENSITIVITY
ANALYSES
OF
SATELLITE
RAINFALL
RETRIEVAL AND SAMPLING ERROR ON FLOOD PREDICTION UNCERTAINTY
Abstract The Global Precipitation Measurement (GPM) mission planned jointly by US, Japanese and European space agencies envisions providing global rainfall products from a constellation of Passive Microwave (PM) satellite sensors at time scales ranging from 3 to 6 hours.
In this study, a sensitivity analysis was carried out to understand the
implication of satellite PM rainfall retrieval and sampling errors on flood prediction uncertainty for medium-sized (~100 km2) watersheds. The 3-hourly rainfall sampling gave comparable flood prediction uncertainties with respect to the hourly sampling, typically used in runoff modeling, for a major flood event in Northern Italy. The runoff prediction error, though, was magnified up to a factor of 3 when rainfall estimates were derived from 6-hourly PM sampling intervals. The systematic and random error components in PM retrieval are shown to interact with PM sampling introducing added uncertainty in runoff simulation. The temporal correlation in the PM retrieval error was found to have a negligible effect in runoff prediction. It is shown that merging rain retrievals from hourly IR and PM observations generally reduces flood prediction uncertainty. The error reduction varied between 50% (0%) and 80% (50%) for the 6hourly (3-hourly) PM sampling scenarios, depending on the relative magnitudes of PM and Infra-red (IR) retrieval errors. Findings from this study are potentially useful for the
69
design, planning, and application assessment of satellite remote sensing in flood and flash flood forecasting.
4.1 Introduction
Passive Microwave (PM) radiometers for remote sensing of rainfall have shown great promise because of the direct interaction between hydrometeors and the radiation field. Unlike IR measurements, which are sensitive only to the uppermost layer of clouds, PM radiation has the ability to penetrate the clouds offering insight into the physical structure of rainfall. PM sensors have flown on a number of space-borne platforms. In 1987, the first Special Sensor Microwave/Imager (SSM/I) was launched on the Defense Meteorological Satellite Program (DMSP) F-8 satellite. Currently, there are three SSM/I spacecrafts (F13, F14 and F15) providing PM rainfall retrievals in sunsynchronous orbits. In 1997, the Tropical Rainfall Measuring Mission (TRMM) was launched. TRMM carries a Microwave Imager (TMI) similar to the SSM/I (Simpson et al., 1996). Very recently, another PM sensor for rainfall retrieval, the Advanced Microwave Scanning Radiometer (AMSR) was launched in 2002 as part of the AQUA mission (NASA, ‘AQUA Brochure’, 2002). The success of TRMM in improving our understanding on Tropical and Subtropical rainfall distribution and precipitation structures has now spurred a larger scale mission aimed at the study of global water cycle. This mission, named the Global Precipitation Measurement (GPM), envisions a constellation of PM sensors that will provide global rainfall products at scales ranging from 3 to 6 hours over regions as small
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as 100 km2 (Smith, 2001; Bidwell et al., 2002; Flaming, 2002; Yuter et al., 2003). GPM also envisions the extension of ‘scientific and societal applications’ of this highresolution global rainfall data as one of its major objectives (Bidwell et al., 2002). Previous uses of PM rainfall retrievals include weather forecasting, climate analysis, and large-scale hydrologic studies (Petty and Krajewski, 1996; Xiao et al., 2000; Grecu and Anagnostou, 2001; Hou et al., 2001; Kummerow et al., 2001). The enhanced revisit frequency and global coverage of PM sensors as planned in GPM will make for the first time PM retrievals attractive for the prediction of floods over unguaged watershed (i.e., watersheds lacking in rain and stream-flow measurements from a surface network). This is extremely important, as flood is one of the deadliest and economically most destructive natural hazard; more than 2000 lives are lost and at least 10 million people are displaced annually since 1991 (see www.dartmouth.edu/~floods). Most importantly, floods are more frequent in regions that lack financial resources to employ networks of surface weather stations necessary for flood monitoring. Thus, observations from satellite sensors stand to offer tremendous benefit to such ungauged areas. However, PM rainfall retrieval is subject to errors caused by various factors ranging from instrument issues (e.g., calibration, measurement noise) to the high complexity and variability in the relationship of brightness temperatures to precipitation parameters. Past studies have revealed that the presence of errors in remote sensing of rainfall can potentially lead to high uncertainties in the simulation of runoff at the watershed scale (Winchell et al., 1998; Borga et al., 2000; Hossain et al., 2004a). Guetter et al. (1996) have found that satellite retrieval errors propagated through a hydrologic
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model forced with satellite data can yield significant uncertainty in the prediction of hydrologic parameters. In a study by Hossain et al. (2004a), it was shown that the systematic and random components of a rain retrieval algorithm would interact nonlinearly with the hydrologic modeling uncertainty, leading to a high uncertainty in the resulting flood forecasts. Hence, there is a need to understand the sensitivity of flood prediction uncertainty to the error characteristics of the satellite rainfall retrievals that are used as input to the hydrological model. Furthermore, as space based sampling from PM sensors is less frequent than the hourly scale typically used in most types of flood prediction (Smith, 2001), it would be important to study the effect of infrequent sampling (hereafter named ‘sampling error’ ranging from 6-hourly (current conditions) to 3-hourly (planned GPM constellation) on the flood prediction uncertainty. In contrast to PM sensors, IR radiometers on geosynchronous satellites provide excellent time and space sampling, but the remotely sensed parameter (primarily cloud-top brightness temperature) is connected loosely to the physics of rainfall below. Hence, it is also worthwhile investigating the utility of the less definitive IR rainfall retrievals in conjunction with PM retrievals for flood prediction. The specific questions that this study seeks to address are: (1) How do factors of PM sampling (3 hourly and 6 hourly) and retrieval error characteristics interact in the runoff transformation process at the watershed scale and contribute to flood prediction uncertainty? And (2) what is the impact of using combined IR and PM retrievals at hourly time scales on flood prediction uncertainty? With the anticipated wider availability of satellite rainfall products from GPM, this study is probably the first of the many required to answer bigger questions facing the hydrologic community today.
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In the next section (Section 4.2) a description is provided of the data, the watershed and hydrologic model used for flood prediction. Section 4.3 describes the formulation of a satellite rainfall retrieval error model. This error model is necessary in making multiple simulated realizations of the rainfall process as would have been typically observed by a constellation of satellite sensors (PM and IR) during a storm event. In section 4.4, the simulation framework used to investigate the sensitivity of sampling and retrieval error characteristics in flood prediction uncertainty are described. Results and discussion of the implications of this study are also provided in this section. Section 4.5 describes the conclusions and discusses extensions of this study.
4.2 Watershed, Data and Hydrologic Model
The watershed chosen for this study (named Posina) is located in northern Italy, close to Venice (Figure 4.1, right panel). A detailed description of the watershed has already been provided in section 2.1. Within a radius of 10 km from the center of the watershed there is a network of 7 rain gauges providing representative estimates of the basin-averaged hourly rainfall (hereafter referred to as ‘reference rainfall’). This is considered a relatively dense network considering that a previous hydrologic application study of satellite data by (Guetter et al., 1996) involved much less dense network of 29 guages over watersheds larger than 2000 km2. The estimation of basin-averaged rainfall was based on an inverse distance weighting technique that had earlier proved to be a reliable method for similar hydrologic studies over Posina (Borga et al., 2000; Hossain et al., 2004a; Dinku et al., 2002). A major storm event that took place in October 1992 and
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was associated with catastrophic flooding in the area was selected for this study. Figure 4.2 shows the storm hydrograph (lower axis) and the corresponding hourly basinaveraged gauge rainfall (upper axis). The rainfall event lasted 120 hours (5 days), while the flood wave receded to the base flow level on the 7th day (after 168 hours from the beginning of the storm. TOPMODEL (Beven and Kirkby, 1979) was chosen to simulate the rainfallrunoff processes of the Posina watershed. This model is a semi distributed watershed model that can simulate the variable source area mechanism of storm-runoff generation and incorporates the effect of topography on flow paths. More details on the hydrologic model can be found in section 2.4.
4.3 The Remote Sensing Rainfall Error Model (RSREM)
A Remote Sensing Rainfall Error Model (RSREM) is formulated to simulate PM and IR satellite rainfall retrievals by corrupting more accurate sources of “reference” surface rainfall, in this study provided by a dense rain gauge network. The utility of this model was then in its ability to mimic basin-averaged rainfall retrievals from hypothetical satellite observations over the Posina basin. Rain retrieval from a satellite observation may exhibit the following possible outcomes:
1) It retrieves non-zero rainfall when it actually rains (successful rain detection). 2) It retrieves zero rainfall when it actually rains (false no-rain detection).
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3) It retrieves zero rainfall when reference rainfall is zero (successful no-rain detection). 4) It retrieves non-zero rainfall when reference rainfall is zero (false rain detection).
The
successful
rain
detection
probability
(first
outcome),
P1=Prob(RSAT>0|RREF>0) is defined as a function of the reference rainfall:
P1 = 1.0 - λO exp ( - λR REF )
(4.1)
where, λ0 and λ are parameters to be derived from boundary condition values of the rain detection probability.
The false no-rain detection probability (second outcome) is
consequently defined as 1-P1. The choice for an exponential-type function for P1 was on the basis of real data as shown in Figure 4.3, derived from TRMM radar/TMI radiometer rain product comparisons over different sites on the Globe. Figure 3 shows that the rain detection probability of TMI retrieval converges to near 1 when the reference surface rain rate (in this case surface rainfall is inferred for TRMM precipitation radar) exceeds 5 mm/h. However, for the IR rain retrieval this threshold rain rate can take a wide range of values depending on the retrieval algorithm and resolution (in this study we assume the value being twice the PM threshold value, i.e., 10 mm/hr). A very low probability of detection (0.001) is given to the PM rain detection probability, P1PM, when reference surface rainfall is below 0.2 mm/hr, while the corresponding value for IR rain algorithms was set to 0.8 mm/hr. The values of λ0 (λ) satisfying the above boundary conditions were therefore found to be 1.332 (1.439) and 1.821 (0.751) for PM and IR retrievals,
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respectively. In low rain rates (below 1 mm/h), there is a slight discrepancy between detection probabilities derived from data versus the model of Equation 4.1. Since the storm of this study is associated with rain rates significantly higher than the above threshold (see Figure 4.2, upper axis) this discrepancy is expected to have little on the flood prediction uncertainty assessment. The justification for assigning a significantly less accurate IR rain detection capability is based on a number of reasons. Firstly IR retrievals tend to suffer from a spatial and temporal offset that often lowers its rain detection probability at scales relevant to this study (Huffman et al., 2003). Typically IR retrievals are not always suited to detection of very light rain (Negri et al., 2002). An algorithm inter-comparison study by Negri and Adler (1993) found that IR rain retrievals have poor performance at hourly time scales. Another study by (Grecu et al., 2000) reported that IR retrievals can have a rain detection probability of 0.64 for rain rates ranging between 0–2 mm/hr, which is quantitatively consistent with the detection probability derived from Equation 4.1 using upper and lower IR rain detection thresholds of 0.2 mm/hr and 10 mm/hr, respectively. Overall, the parameter values chosen for the RSREM are representative of warm-rain systems (i.e., widespread events) similar to the one studied herein. The successful no-rain detection probability (third outcome) of the satellite retrieval is defined as P0=Prob (RSAT=0 | RREF=0), while 1-P0 defines the false rain detection probability (fourth outcome). The probability P0 was defined to be high for PM retrievals (0.98) as shown in (Grecu and Anagnostou, 2001), while for IR it was given a lower value (0.90). The lower value of no rain detection for IR was based on results from
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a previous hydrologic study (Tsonis et al., 1996) that reported a value of 0.92 on the basis of comparisons with area-averaged gauge rainfall over a basin in northern USA. The next step is to assign rainfall rate values in the first and fourth outcome where the satellite retrieval is non-zero. In the first outcome the retrieved satellite rain rate, RSAT, is statistically related to the reference surface rainfall, RREF, as:
RSAT = RREF .ε S
(4.2)
where the multiplicative satellite error parameter, εs, is assumed log-normally distributed. A log transformation of the log(RSAT)-log(RREF) statistical relationship transforms the error εs to a Gaussian deviate ε (hereafter named ‘log-error’) with N(µ,σ2) statistics (µ mean; σ2- variance). To compute the mean (hereafter named ‘multiplicative bias’ - mu) and variance (S2) of the multiplicative error, εs, the following conversion is used in terms of µ and σ2,
mu = exp ( µ + 0.5σ 2 )
(4.3)
S 2 = [exp(σ 2 ) - 1] exp(2µ + σ 2 )
(4.4)
The log-error can be space and time correlated. Only time correlation was considered due to the nature of this study, i.e., the basin-averaged rainfall is represented by a single PM rain retrieval pixel, which is expected to be of the order of about 10x10 km2 resolution for GPM (Yuter el al., 2003). A lag-one autocorrelation function was used to
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model the correlated error sequence, which for a Gaussian random variable leads to the following equations for the propagation of µ and σ2,
µ i = µ + (ρ ) (ε i −1 − µ )
(
σ 2 i = σ 2 1 - (ρ 2 )
(4.5)
)
where time index i represents discrete hourly time step while ρ2 is the lag-one autocorrelation of ε. In the fourth outcome (false rain detection), the satellite (PM and IR) rain retrieval is statistically generated from a uniform distribution, U[0, RL], representing an acceptable range of low intensity rainfall estimates usually occurring in false rain detection. The RL parameter was assigned a low value of 0.5 mm/h for both PM and IR sensor retrievals, which are similar to the false alarm rain rates suggested by Guetter et al. (1996). As discussed earlier in this section, RSREM is used to statistically simulate multiple realizations of satellite rainfall retrievals over Posina basin using observations from a dense rain gauge network as the basis for reference basin-averaged rainfall. The algorithm structure is shown in Figure 4.4. The algorithm is applied at discrete hourly time steps (defined by time index i). The probabilities of rain detection (P1) and no rain detection (P0) are modeled through Bernoulli distributions B(1,P1) and B(1,P0) respectively in a fashion similar to (Guetter et al., 1996). Table 4.1 summarizes the PM and IR retrieval error characteristics considered in this study. The satellite rain and norain detection probabilities and the false alarm rain rate parameter values are shown in
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Table 4.1a. The satellite’s multiplicative bias (mu), log-error standard deviation (σ) and auto-correlation parameter (ρ2) were varied across a range of values shown in Table 4.1b. A set of “default” values was also assigned to facilitate relative comparisons between the varied error parameter values.
These default values are deemed error parameters
associated with realistic levels of satellite retrieval accuracies. On the basis of past studies, the default values of mu(σ)(ρ2) for PM and IR retrievals were assigned as 1.15(0.40)(0.40) and 1.15(0.70)(0.40), respectively. From equations 4.3 and 4.4, it can be shown that the selected default values are in fact equivalent to a standard error of 45% for PM and 98% for IR retrievals. Kummerow et al. (2001) have shown that PM retrievals can be biased in the ranges of 15-20% overland, while Negri and Adler (1993) have reported rms error for IR retrievals at the range of 100% to 200% at the hourly time scale. The primary distinction between PM and IR retrievals is represented in this error model through P1, the satellite rain estimation error statistics (in particular the conditional error variance), and to a lesser effect by P0. By assigning higher upper/lower rain thresholds for IR in (1), the probability of IR rain (false no-rain) detection becomes consistently lower (higher) than that of the PM retrieval. The lower P0 will lead to rate of higher false alarms, which is typical in IR retrievals. Furthermore, the five times higher standard error adequately characterizes the higher degree of IR rain estimation uncertainty with respect to PM retrieval.
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4.4 The Simulation Framework
The hydrologic model parameters were calibrated based on the optimization algorithm of Duan et al. (1992) for three time resolutions (hourly, 3-hourly, and 6-hourly time steps) using the rain gauge basin-averaged rainfall as input to TOPMODEL. The simulated hydrograph based on the hourly TOPMODEL parameters and reference rainfall input was considered to represent the most accurate rainfall-runoff transformation for the basin, and was used as reference for the subsequent error analysis (henceforth called the “reference runoff’”). Because the correlation with observed runoff was very high (0.92), comparisons of reference runoff with runoff predictions driven by different synthetic satellite retrieval inputs would indicate the adequacies of the retrieval scheme and its impact on flood prediction uncertainty. The simulation exercise consisted of making multiple runs (i.e., realizations) of the hydrologic model, each with a random realization of synthetic satellite rain retrievals derived from RSREM, and evaluating error statistics at the level of flood prediction. Three errors that adequately characterize uncertainty in flood prediction are evaluated: 1) Mean Absolute Error in Peak Runoff; 2) Mean Absolute Error in Time to Peak and 3) Mean Absolute Error in Runoff Volume. The error in these three flood hydrograph parameters is defined as,
Error in Peak Runoff =
1 N sim
Nsim
∑ i =1
Peak Runoff i − Peak Runoff ref Peak Runoff ref
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Error in Time to Peak =
1 N sim
Error in Runoff Volume =
N sim
∑ i =1
1 N sim
Time to Peak i − Time to Peak ref Time to Peak ref
Nsim
∑ i =1
(4.6)
Runoff Volumei - Runoff Volumeref Runoff Volumeref
where ‘Nsim’ is the total number of simulation runs (satellite realization of retrievals); subscript ‘i’ indicates the simulation index; and subscript “ref” signifies the hydrologic parameter was derived from the reference runoff.
a. Assessment of PM retrievals The above simulation framework is used to investigate how PM sampling and retrieval error characteristics interact in the runoff transformation process (characterized by the three runoff error parameters in Equation 4.6). For each investigated scenario, 20,000 Nsim realizations were performed, and synthetic satellite retrievals propagated through the calibrated hydrologic model to derive ensemble runoff simulations.
A
preliminary study with synthetic satellite retrievals indicated 20,000 model runs were adequate for the Monte Carlo sample. For both the 3- and 6-hourly scenarios, a polynomial interpolation was performed to interpolate the runoff at hourly scales. A point to note is that a satellite may not overpass a watershed exactly at the start of a storm event, but may initiate sampling with a delay. Thus, considering delays rounded off to the nearest hour, there may be a maximum of 2 hours delay for the 3-hourly sampling, while for the 6-hourly sampling the delay can be up to 5 hours. To account for this effect the simulation exercise was repeated for these possible sampling-initiation patterns (3
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patterns for the 3-hourly and 6 patterns for 6-hourly sampling scenarios). The errors in runoff parameters for the 3- and 6-hourly sampling scenarios were normalized by the corresponding runoff simulation error of the hourly sampling scenario to derive Error Ratios (ER) for the different error parameters as follows,
E.R 3− hourly =
Error in Runoff 3-hourly
E.R 6− hourly =
Error in Runoff 6-hourly
Error in Runoff hourly
(4.7)
Error in Runoff hourly
The Error Ratios shown above allow comparison of 3- and 6-hourly sampling to a common reference of hourly sampling, which is typically used in hydrologic forecasting of floods. The ER sensitivity to satellite retrieval error parameters (mu, σ and ρ2) for the 3- and 6-hourly sampling scenarios is evaluated to study the effect of sampling and retrieval error interaction in the runoff transformation process. Two measures are used, both of which represent the ratio of runoff simulation errors for the 3 and 6 hourly sampling scenarios to those from the hourly sampling scenarios.
The Default ER
statistics use the varying values of the error measures in Table 4.1b for the 3 and 6 hourly scenarios, but the default values for the hourly scenario. This allows the evaluation of the combined effect of retrieval and sampling error on flood prediction uncertainty. In order to isolate the effect of the sampling error, the Time ER statistics use the same (varying) values of the error measures from Table 4.1b for both 3 or 6 and hourly scenarios. The impact of PM sampling frequency on flood prediction uncertainty is presented in Figure 4.5, which shows the “reference” hydrograph of the catastrophic
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flood event with the 5%, 50% and 95% quantiles of the satellite predicted hydrographs derived from the simulation exercise run with the default PM retrieval error parameters shown in Tables 4.1a and 4.1b. Compared to the 6-hourly scenario, the 3-hourly has significantly (about half) lower flood prediction uncertainty, which appears to be comparable to the hourly sampling scenario. The 6-hourly sampling scenario would overestimate peak runoff by up to 1500 m3/s, a magnification of about 8 times the observed value, and underestimate the time to peak by 15 hours. This tremendous overestimation of the flood wave by 6 hourly sampling further underscores the importance of more frequent sampling for storms of such magnitude. In the subsequent three figures (4.6, 4.7, and 4.8) we demonstrate the sensitivity of flood prediction uncertainty (Default ER-lower panels; Time ER-upper panels) to PM retrieval error characterized by varying values of multiplicative bias (mu), standard deviation of log-error (σ), and lag-one correlation (ρ2), respectively. In varying one of the error parameters in this sensitivity experiment the other PM error parameters are set to their default value shown in Table 4.1b. Several features are worth noting from these figures, starting with Figure 4.6. It is observed that ER generally reaches a minimum at moderate retrieval bias conditions. The observation that the minimum ER does not occur at no bias (mu=1.0), but at moderate bias condition, is expectable, as random error in rainfall retrieval may introduce biases in runoff simulation (Hossain et al., 2004a). The 3-hourly sampling exhibits minimum ERs at bias values closer to 1 than the 6-hourly counterpart, indicating that sampling can magnify the effect of retrieval error in runoff. The Time ER in runoff volume is observed to be nearly insensitive to the retrieval bias, which is expected as Time ER represents the
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effect of sampling alone and runoff volume is a time-integrated hydrologic parameter with a tendency to compensate for random input errors in time. This indicates that water balance studies for a medium-sized watershed can be well represented by 3- and 6-hourly PM observations provided the retrieval scheme performs at the levels assumed here. Sampling error, though, can significantly affect peak runoff error (and to a lesser effecttime-to-peak error) as Time ER is shown to reach 2.5 (2.0) for the peak runoff (time-topeak) in the 6-hourly sampling scenario. For the 3-hourly scenario this effect is moderate to negligible. Another observation is that the rate of increase of ER (both Time and Default) with increasing bias is stronger in overestimation (mu>1), and again more so for the 6-hourly sampling scenario, as indicated by the steeper gradients and the widening difference between 3- and 6-hourly ER plots. This indicates that there exists a higher degree of error interaction with sampling when the sensor overestimates rainfall, which is consistent with past findings (Anagnostou and Hossain, 2001). In Figure 4.7 (upper panels), Time ER indicates that increasing the retrieval error standard deviation obscures the error due to sampling on peak runoff prediction. However, this effect is not apparent at the time-to-peak and runoff volume Time ER plots, which seem to only slightly increase relative to the retrieval random error. As with Figure 4.6, the 6-hourly sampling is associated with higher values of Time ER values than those of the 3-hourly, which range from 1 to 1.5 for the time-to-peak and runoff volume parameters. In the Default ER plots (lower panels of Figure 4.7), which show the combined effect of sampling and retrieval error, it is observed that the 6-hourly sampling is consistently higher than 3-hourly sampling error by a factor of 1.2 or higher. The
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Default ER plot for peak runoff exhibits a steeper rate of increase in the 6-hourly sampling scenario relative to the corresponding 3-hourly scenario.
In Figure 4.8 the Default ERs (lower panels) for all three runoff parameters are shown to be nearly insensitive to the temporal correlation of the retrieval error (ρ2) for both 3- and 6-hourly sampling scenarios. This indicates that the time lag (minimum of 3 hours) between PM observations is long enough to minimize potential effects from temporally correlated errors. At the hourly time scale, though, the rain retrieval error correlation can play a role in runoff prediction error. This is because correlation would indirectly increase the systematic error in the retrieval, which would consequently lead to an increase in runoff error parameters. This effect is apparent in the Time ER plots of the upper panels of Figure 4.8, which are shown to decrease (peak runoff has the steepest descent) with increasing the temporal correlation of the retrieval error. Figure 4.9 presents contour plots of the Time ER values for co-variations of the retrieval error temporal correlation and standard deviation for 3- and 6-hourly sampling scenarios (left and right panels of Figure 4.9 respectively). The patterns of ER values are the same in both contour plots, but the magnitudes of the Time ER values associated with the 6 hourly sampling are almost double those of the 3 hourly sampling. The nearly circular nature of contours indicates that the temporal correlation and standard deviation contribute roughly equally to the magnitude of the Time ER. A maximum point (Time ER > 2.00 for 3 hourly and >5.00 for 6 hourly) is observed at standard deviation (σ) of 0.3 and lag one correlation (ρ2) of 0.1. At high retrieval errors (σ>0.6, and ρ2>0.6) it is observed the error ratios reaching lower values (1.1 in 3-hourly and 2.0 in 6-hourly).
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These lower values are caused by relatively higher increase in errors derived from the hourly sampling. This indicates that the error due to sampling alone tends to become obscured by the higher error in rainfall retrieval for both 3- and 6-hourly sampling scenarios. Consequently, at such high levels of retrieval error, it would not matter what the sampling frequency is–-the algorithm performs too poorly to have any benefit in runoff simulation accuracy by increased sampling. It is therefore important to ensure that the PM retrieval algorithms perform at adequate levels of retrieval accuracy to maximize the benefit achieved in flood prediction by any increase in sampling frequency.
b. Assessment of Combined PM-IR retrievals The simulation framework was also used to assess the impact of combining IR and PM retrievals in flood prediction. Considering three different levels of IR retrieval error (σIR: 0.5, 0.7 and 0.9), the utility of IR merging with PM was studied for varying levels of PM retrieval error (σPM: 0.1, 0.2, 0.3, 0.4 and 0.5). In all cases, the retrieval biases and lag-one correlation for both IR and PM were fixed at their Default value (Table 4.1b). The IR rainfall was used as input to the hydrologic model at hours with no PM measurement, while the simulation framework was as mentioned above. A statistical assessment of the merged IR and PM rainfall input relative to the PM-only scenario is defined based on the following error ratios (named Merging ER) for the 3- and 6-hourly sampling,
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Merging E.R 3hourly =
Error in runoff merged IR -PM
Merging E.R 6 hourly =
Error in runoff merged IR -PM
Error in runoff 3-hourly PM
(4.8)
Error in runoff 6-hourly PM
The Error in Runoff indicates the error statistics of each of the three runoff-parameters, as defined in Equation 4.6. A Merging ER value of less than one would indicate that the use of IR rainfall during hours with no PM observations reduces uncertainty in flood prediction. Figure 4.10 shows the effect of combining IR with PM retrievals on reduction of flood prediction uncertainty. Merging ER (Equation 4.8) values shown for runoff volume are presented as a function of PM retrieval error standard deviation for three levels of IR retrieval error standard deviations as discussed earlier. The rest of the parameters of the IR and PM retrieval error model are set to the default values in Table 4.1a, Table 4.1b. At low PM retrieval errors (standard deviation (σPM): 0.20 - 0.35) the use of the more erroneous IR retrieval (standard deviation (σIR): 0.5, 0.7 and 0.9) can actually increase flood prediction uncertainty for the 3-hourly sampling scenario (Merging ERs ranging from 1.25 to 2.25). This indicates that when the difference between IR-PM retrieval error standard deviations is within the range of 0.30 to 0.55, merging with IR retrievals would generally fail to achieve any beneficial effect on flood prediction accuracy for a 3-hourly PM sampling. As the difference between PM and IR retrieval error variances narrows, the use of IR retrievals becomes effective in reduction of flood uncertainty for the 3 hourly case. In contrast, it is observed that the use of IR rainfall jointly with 6-hourly PM rainfall retrievals (right panel, Figure 4.10) invariably reduces flood prediction
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uncertainty at all ranges of IR and PM retrieval errors considered in this study. This is explained from the fact that the error due to infrequent sampling is so high in the 6-hourly case (as shown in Figure 4.4) that any additional rainfall information from an IR sensor at the hourly scale becomes always beneficial in constraining that uncertainty in flood prediction. The relative reduction in uncertainty for the 6-hourly sampling scenario ranges from 50% to 80% (Merging ERs: 0.5 to 0.2 respectively), while for the 3-hourly scenario the reduction of uncertainty can be up to 50% (Merging ERs: 0.5) for high PM retrieval error. This reduction for 3 hourly PM sampling with the use of hourly IR retrievals draws the uncertainty level closer to that achieved by hourly PM sampling when compared with that derived from 3 hourly PM sampling alone.
4.5 Summary and Conclusion
This paper studied the sensitivity of satellite retrieval and sampling error on flood prediction uncertainty for a week-lasting rainfall event over a medium-sized watershed of a typical sensor footprint size (~100 km2). It was shown that a 3-hourly PM sampling is comparable to the hourly sampling in terms of flood prediction uncertainty for the major storm event, while uncertainty was shown to increase by a factor of 2 to 3 times for a 6hourly sampling. Runoff uncertainty was found sensitive to both, systematic and random error components of PM retrievals. Particularly, the 6-hourly scenario was found to exhibit much higher sensitivity, especially when rainfall is overestimated. The effects of temporal correlation of retrieval error effects alone were not found to be significant for either 3- or 6-hourly sampling, but their strong interaction with error variance was
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evident. For estimation of runoff volume or water balance studies at the medium-sized watershed scale from overpassing PM sensors, the loss in accuracy due to sampling was found to be minimal. However, this was not the case for estimation of other flood hydrograph parameters like time-to-peak and peak runoff. At high errors of PM rainfall retrievals, the runoff error due effects of sampling alone was shown to become obscured by the retrieval error. Application of Infra-red (IR) retrieved rainfall jointly with PM retrievals was shown to reduce flood prediction uncertainty consistently for the 6-hourly PM sampling scenario. In the 3-hourly sampling scenario the potential runoff error reduction was shown to depend on the relative accuracy of IR retrieval with respect to the PM. The reduction in flood prediction uncertainty was found to range from 50%-80% and up to 50% for 6-hourly and 3-hourly sampling scenarios, respectively. This study was limited to PM sampling scenarios of fixed revisit times (3 and 6 hourly) between successive overpasses. While this is an expected sampling scenario for the GPM in the timeframe of 2009 onwards, current constellation of PM sensors (comprising a smaller array of satellites) sample rainfall events with variable revisit times ranging from 1 to 10 hours (Smith, 2001). As un-gauged watersheds affected by flood problems are spread wide within the tropics, it would be necessary to calibrate the retrieval error parameters based on observed PM and IR sensor data matched at a more global scale. The current study has been conditioned on sensor rain retrieval assumptions based on previous studies/results. Calibration with real sensor data is important to verify several of the key assumptions made during the formulation of RSREM including (1) the probability distribution of PM rainfall retrievals during false alarms, (2) the functional form of rain detection probability, and (3) the magnitudes of retrieval bias, error
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variances, and temporal correlation. There is also need to understand the effect of storm morphology (storm duration, fractional rain coverage, and rain rate variability) on satellite based flood prediction uncertainty. Certainly, this information would be useful in identifying the potential subset of storm systems not suitable for flood prediction by satellite data. Future research should therefore address those issues to make a better assessment of satellite rainfall remote sensing for flood prediction of ungauged watersheds.
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Table 4.1a Rain detection and No-Rain detection probabilities for IR and PM retrievals. Probability of Rain Detection (P1)
Probability of No-Rain Detection (P0)
Upper boundary
Lower boundary
(P1=1.0)
(P1=0.001)
PM
5 mm/hr
0.2 mm/hr
0.98
IR
10 mm/hr
0.8 mm/hr
0.90
Table 4.1b Ranges and Default values of error model parameters (RSREM) for multiplicative error bias (mu), log-error standard deviation (σ), and log-error temporal correlation (ρ2) for PM and IR sensor retrievals. The range, RL, of the uniform distribution U[0,RL], for sampling false alarms was set at RL=0.5 mm/hr for PM and IR.
Multiplicative Bias
Standard Deviation
Correlation (log-error)
(mu)
(log-error)
(ρ2)
(σ) Default
Range
Default
Range
Default
Range
PM
1.15
0.25–1.75
0.40
0.10–1.30
0.40
0.10–0.90
IR
1.15
-
0.70
0.5–0.90
0.40
-
The range RL of the uniform distribution U[0,RL] for sampling false alarms were set at RL=0.5 mm/hr for PM and IR.
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Figure 4.1 Geographic location of (right panel) Posina watershed and (left panel) watershed elevation map overlaid by the rain gauge network locations (in solid circles). The inner box represents a typical 10X10 km2 PM satellite foot print.
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Figure 4.2 October 1992 storm event hydrograph and hyetograph in Posina, Italy.
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Figure 4.3 Probability of rain detection by PM (TMI) retrievals over Southern Africa, Amazon and the Southern United States determined using TRMM Precipitation Radar (PR) surface rainfall estimates as reference. The data products used for TRMM-PR and TMI rain rates were 2A25 and 2A12, respectively. The period of matched PR/TMI data was January-July 2002.
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i= time index RREF, i
i=1
i = i +1 Is RREF, i > 0.0?
YES
NO
Compute P1 from Eqn. 4.1 Generate uniform rn (0 –1)
Generate uniform rn (0-1)
Is rn ≤ P1? YES
Is rn ≤ P0 ? NO
YES
NO
RSAT,i RSAT, i =Eqn. 4.2
RSAT, i = 0.0
RSAT,i =0.0
from U[0,RL]
Figure 4.4 Algorithmic structure of the RSREM. rn is a randomly generated uniform [0,1] deviate.
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Figure 4.5 Flood prediction uncertainty for hourly, 3-, and 6-h PM retrievals for October 1992 storm event in Posina. The quantiles shown are the 50% (median) and the confidence bounds for 5% and 95% percentiles (90% area).
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Figure 4.6 Effect of PM retrieval bias (multiplicative) on flood prediction uncertainty presented in terms of ERs. (Top) Time ERs: ratio of runoff error for a given sampling scenario (3 or 6h) normalized by the 1-h runoff error derived from the correspondingly varied retrieval error parameters. (Bottom) Default ER: ratio of runoff error for a given sampling scenario (3 or 6h) normalized by the 1-h runoff error with the default retrieval error parameters. The upper panels show the effect of sampling, and the lower panels show the combined effect of sampling and retrieval error on flood prediction uncertainty.
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Figure 4.7 Effect of PM retrieval standard deviation (σ) on flood prediction uncertainty presented in terms of ERs. (Top) Time ERs: ratio of runoff error for a given sampling scenario (3 or 6h) normalized by the 1-h runoff error derived from the correspondingly varied retrieval error parameters. (Bottom) Default ER: ratio of runoff error for a given sampling scenario (3 or 6h) normalized by the 1-h runoff error with the default retrieval error parameters. The upper panels show the effect of sampling, and the lower panels show the combined effect of sampling and retrieval error on flood prediction uncertainty.
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Figure 4.8 Effect of PM retrieval error lag-one correlation (ρ2) on flood prediction uncertainty presented in terms of ERs. (Top) Time ERs: ratio of runoff error for a given sampling scenario (3 or 6h) normalized by the 1-h runoff error derived from the correspondingly varied retrieval error parameters. (Bottom) Default ER: ratio of runoff error for a given sampling scenario (3 or 6h) normalized by the 1-h runoff error with the default retrieval error parameters. The upper panels show the effect of sampling, and the lower panels show the combined effect of sampling and retrieval error on flood prediction uncertainty.
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Figure 4.9 Contours of peak runoff Time ERs in peak runoff as a function of PM retrieval log-error’s lag-one hourly correlation (ρ2) and standard deviation (σ) for 3-h (left panel) and 6-h (right panel) sampling scenarios.
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Figure 4.10 Merging ER values of runoff volume presented as a function of PM retrieval error standard deviation (σPM) for two PM sampling scenarios (left panel: 3 h; right panel: 6 h) and three levels of IR retrieval error standard deviation (σIR).
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5 ASSESSMENT OF CURRENT PASSIVE MICROWAVE AND INFRARED BASED SATELLITE RAINFALL REMOTE SENSING FOR FLOOD PREDICTION Abstract The adequacy of current Passive Microwave (PM) and Infra-red (IR) based satellite rainfall retrieval and sampling for flood prediction of a medium sized watershed is investigated. On the basis of Tropical Rainfall Measuring Mission (TRMM) Precipitation Radar rainfall measurements, rain retrieval error parameters for PM and IR sensors are derived. PM rain retrievals are inferred from the overland component of the TRMM Microwave Imager (TMI) rain estimation algorithm, while IR retrievals are obtained from hourly PMcalibrated IR rain fields, which are part of a variable rainfall product (VAR) array produced at NASA/GSFC. A probabilistic error model is developed for satellite-based precipitation measurements based on retrieval error parameters in this simulation study. The PM rain detection ability was found to be significantly more sensitive than that of IR while the successful non-rain detection probabilities were found to be 93% and 88% respectively. The IR retrieval was found to give false alarm rain rates about twice as large as that of PM. The PM sensor constellation comprised two Special Sensor Microwave Imagers (SSM/I) F14 and F15, the TMI, and the Advanced Microwave Sensing Radiometer (AMSR-E). It was found that current PM sampling is associated with flood prediction uncertainty approximately 50-100% higher than that of a canonical 3-hourly sampling planned for the Global Precipitation Measurement (GPM). The comparatively greater limitation in capturing the correct space-time rain structure by IR retrievals had the effect of increasing
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the error in predicting the time of peak runoff when merging was performed with PM retrievals. It was found that a reduced standard error (0.90) can make IR retrievals useful in reducing uncertainty in the prediction of peak runoff. To reduce the error in time to peak, further improvement such as the reduction in the IR retrieval’s false alarm rates coupled with an even higher POD may be necessary. In terms of overall runoff volume, combined moderate improvements in POD and error variance of current IR retrieval algorithms are sufficient for the reduction of prediction uncertainty.
5.1 Introduction
Advancements in space-based precipitation observation systems that originated nearly three decades ago (Griffith et al., 1978) have enabled us to track the fate of precipitation in the hydrologic cycle through improved understanding of its variability. With the increased availability and quality of precipitation observations from space it is now possible to assimilate these estimates in hydrologic models to provide flood prediction over regions with poor in-situ data. This is important as a substantial portion of floods takes place in regions remote or lacking the financial resources to be adequately covered by ground stations. In such cases space-based rainfall estimates are the sole source of rain input to hydrological models. Since precipitation is the single most important determinant of the state of surface runoff, it is therefore important to understand how errors in satellite retrieval manifest as flood prediction uncertainty.
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The Global Precipitation Measurement (GPM), which is a mission to be launched by the international community by 2009, envisions a large constellation of Passive Microwave (PM) sensors to provide global rainfall products at scales ranging from 3 to 6 hours, and spatial resolution of 100 km2 (Smith, 2001; Bidwell et al., 2002; Flaming, 2002; Yuter et al., 2003). These resolutions offer tremendous opportunities to address the flood prediction problem of local un-gauged watersheds over the globe. Nevertheless, satellite rainfall retrieval is subject to errors caused by various factors ranging from infrequent sampling to the high complexity and variability in the relationship of the measurement to precipitation parameters. The presence of such errors in remote sensing of rainfall can potentially lead to high uncertainties in runoff simulation (Winchell et al., 1998; Borga et al., 2000; Hossain et al., 2004a,b). A study by Smith (2001) has revealed that the current (pre-GPM era) PM sensors collectively have their maximum revisit times exceeding 9 hours over the tropics. Unfortunately, this is a region where a large number of un-gauged watersheds exist. Half-hourly Infra-red (IR) rainfall estimates from Geostationary platforms could potentially reduce the flood prediction uncertainty during such infrequent PM revisit intervals. However, the weak physical connection of IR cloud top observations to precipitation processes offers indirect relationships to surface rainfall variability, which are associated with significant uncertainty at high spatial and temporal resolutions (Arkin and Meisner, 1987). Consequently, this study seeks to assess how current satellite rainfall retrievals and sampling frequencies affect flood prediction uncertainty. This understanding is important in the identification of aspects that could make future satellite missions such as the GPM useful for flood prediction applications.
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To understand the error propagation, it is important to realize that the error due to the three major sources of uncertainty - retrieval, sampling and hydrological modeling system - are all intimately linked and can not be decomposed into simple additive components in flood prediction uncertainty (Borga et al., 2000; Borga 2002; Hossain et al., 2004a,b).
Past satellite rainfall studies have concentrated on the rain retrieval
uncertainty issue for large spatial scales and temporal accumulations (daily, monthly and yearly) using limited number of error statistics (Griffith et al., 1978; Arkin and Meisner, 1987; Negri and Adler, 1993; Tsonis et al., 1996; Huffman 1997; Xu et al., 1999a,b; Todd et al., 2001, among others). These statistics are quite useful in assessing the use of satellite data for large-scale climatological or water management studies (Guetter et al., 1996; Nijssen et al., 2001). However, the averaging introduced at coarse resolution smooths the small-scale variability of measurement error, which can have non-linear effects in runoff simulation parameters of a flood event (e.g., the time and magnitude of the peak runoff).
A recent study by Nijssen and Lettenmaier (2004) attempted to
quantify the sole effect of precipitation sampling error on the prediction of land surface processes at the scale of large continental river basins. They reported a strong sensitivity of streamflow prediction error to the size of the drainage area. However, their study provides insufficient assessment towards the evaluation of the utility of current satellite retrievals for flood prediction due the non-representation of the retrieval uncertainty, which is known to interact intimately with the sampling error. It also appears that the macroscale hydrologic models that have been developed to study land-atmospheric processes and interactions in this regard have not yet been fully assessed in terms of their sensitivity to the detailed structural properties of satellite rainfall estimation error
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(O’Donnell et al., 2000; Nijssen et al., 2001; Rhoads et al., 2001; Nijssen and Lettenmaier, 2004). This study aims at assessing the use of current PM and IR satellite rainfall remote sensing in flood prediction. The study builds upon the recent work by Hossain et al. (2004b) (hereafter referred to as HAD04) who investigated the adequacy of a canonical 3 hourly and 6 hourly PM sampling for flood prediction using simulated satellite retrievals. The objectives of the present study are to: (1) compare the effect of current PM sampling frequencies to the planned canonical 3-hourly GPM era in terms of flood prediction uncertainty for a mid-sized watershed; (2) study the effect of storm morphology on the uncertainty of flood prediction driven by satellite rainfall data, and (3) investigate potential improvements associated with the combination of IR rain estimates with PM retrievals. In the next section (Section 5.2), the study area, hydrologic model, and data are described. In section 5.3 description of the formulation and tuning to actual data is provided of the statistical model used to simulate PM and IR rain retrievals from hypothetically true rain processes. Section 5.4 presents the simulation experiment, results and discussion of findings. Section 5.5 contains the conclusions and discusses proposed extensions of this study.
5.2 Study Area, Hydrologic Model, and Data
The watershed chosen for this study (named Posina) is located in northern Italy, close to Venice (Figure 4.1, right panel). Posina is described in detail in section 2.1.
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Within a radius of 12 km from the center of the watershed there is a network of 11 rain gauges reporting hourly rain accumulations, 7 of which are closer to the watershed, providing representative estimates of the basin-averaged hourly rainfall rates (hereafter referred to as ‘reference accumulated rainrate’). The rainfall runoff model TOPMODEL (Beven and Kirkby, 1979) was chosen to simulate the rainfall-runoff processes of floods in the Posina watershed. More details on the model can also be found in section 2.4. A series of 15 widespread storm events (3-7 days long) that took place from 1987 to 1997 were studied (Table 5.1). Shorter duration intense precipitation events (0 | RREF-ACCU >0}
(5.1)
where RREF-INST represents realizations of an instantaneous area-averaged reference rain rate value and RREF-ACCU is the accumulated reference rainfall of the corresponding hour. The conditional error, EINST-ACCU, of RREF-INST with respect to RREF-ACCU is defined as: EINST-ACCU = RREF-INST/RREF-ACCU
(5.2)
Due to the multiplicative nature of the error the statistical distribution of EINST-ACCU is considered log-normal, with mean and variance components defined as MuINST-ACCU and SINST-ACCU, respectively. The above error-modeling framework is used to statistically generate instantaneous reference rain rates (RREF-INST) from the hourly-accumulated area-
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averaged reference rainfall values (RREF-ACCU) determined from gauges—this is discussed later in the section. An instantaneous satellite rain retrieval, on the other hand, may exhibit one of the following possible outcomes: (1) when it actually rains the satellite retrieval can be nonzero (successful rain detection) or zero (false no-rain detection), while (2) when it does not rain the satellite retrieval can be also zero (successful no-rain detection) or non-zero (false rain detection). The successful rain detection probability, P1, is defined as a function of the RREF-INST. The functional form is identified through calibration with actual data as discussed in the following section. The false no-rain detection is derived from P1 as (1-P1). The successful no-rain detection, P0, is the unitary probability that satellite retrieval is zero when RREF-INST is zero, which is also determined from actual data. The false rain detection probability is then derived from P0 as (1-P0). A probability density function (Dfalse) is introduced to characterize the probability distribution of the satellite retrieval in false rain detection. This function is also identified through calibration on the basis of actual sensor data. The non-zero instantaneous satellite rain retrieval, RSAT, is statistically related to corresponding non-zero instantaneous reference rainrate, RREF-INST, as,
RSAT = RREF − INST .ε S
(5.3)
where the multiplicative satellite error parameter, εs, is assumed log-normally distributed. The log-normality of the distribution is justified by the non-negative property of εs. A log transformation of the log(RSAT)-log(RREF-INST) statistical relationship transforms the
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error εs to a Gaussian deviate ε with N(µ,σ) statistics, where µ and σ are the mean and standard deviation, respectively. To determine the multiplicative mean (MuINST) and standard deviation (SINST) of εs the following conversion is used in terms of µ and σ,
Mu INST = exp ( µ + 0.5σ 2 )
(5.4)
S INST = [exp(σ 2 ) - 1] exp(2µ + σ 2 )
(5.5)
2
The error parameter ε (hereafter also referred to as ‘log-error’) can be spatially and temporally auto-correlated. Only temporal auto-correlation is considered in this study because the watershed scale is represented by a single PM satellite retrieval pixel (~100 km2, see Figure 4.1, left panel). A lag-one autocorrelation function was used to model the correlated error sequence, which for a Gaussian random variable leads to the following equations for the propagation of µ and σ2,
µ i = µ + ( ρ ) (ε i −1 − µ )
(
σ 2 i = σ 2 1 - (ρ 2 )
(5.6)
)
where time index i represents discrete hourly time step while ρ2 is the lag-one autocorrelation of ε. The work by HAD04 had shown evidence of the runoff error being insensitive to the lag-one auto-correlated rain retrieval error for 3 and 6 hourly sampling scenarios; hence ρ2 was constrained to a fixed value of 0.4 in this study.
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The RSREM operation is summarized in the flow chart of Figure 5.2. When at a certain hour, i, the hourly-accumulated area-averaged reference rainfall is non-zero (RREFACCU,i>0)
we use Equations 5.1 and 5.2 to generate realization of an instantaneous
reference rainrate (RREF-INST,i). Namely, a Bernoulli trial is conducted to model the PODINST by generating a uniform U[0,1] random number, rn. If rn is less than PODINST (which is determined as a function of RREF-ACCU,i) an instantaneous reference rainrate, RREF-INST,i value is calculated on the basis of Equation 5.2 by randomly generating a lognormally distributed deviate, LN [MuINST-ACCU, SINST-ACCU], representing EINST-ACCU. Next, when the instantaneous reference rainrate is non-zero (RREF-INST,i>0) the satellite error model (RSREM) decides as to whether the satellite rainfall is zero or not through another set of Bernoulli trials. If the uniform U[0,1] random number, rn, generated is less than P1 (which is determined as function of RREF-INST,i) then the instantaneous satellite retrieval RSAT,i is non-zero and modeled through Equation 5.3. Otherwise, RSAT,i is assigned a zero value. Similarly, during a non-rainy hour (RREF-INST,i=0) Bernouilli trial is used again to decide as to whether the satellite rainfall will be zero or non-zero. If the uniformly random deviate rn is less than P0, then RSAT,i is assigned a zero value. Otherwise, the nonzero satellite rainfall value is determined through random sampling on the basis of the false alarm probability density (Dfalse) function.
b.
RSREM Calibration
Having mathematically formulated the algorithm for RSREM, the next step is to calibrate the model parameters based on actual data at the sensor resolution. Calibrated values and/or functional forms for the following need be identified: (1) PODINST, MuINST-
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ACCU
and SINST-ACCU parameters of the instantaneous rainfall representativeness error; and (2)
the P1, P0, Dfalse, MuINST and SINST parameters of RSREM. To evaluate the PODINST, MuINST-ACCU and SINST-ACCU parameters six months of high-resolution (1 minute) rainfall data from a gauge network in the area were used; the storm intensities and durations were of similar magnitudes with the storms studied herein. A time series of 10-minute rain accumulations at 10X10 and 25X25 km2 area-averages were first created, by averaging the gauge rainfall reported within each domain size to make it representative of the PM and IR retrieval scales. Within the PM (10X10 km2) domain there were 4 gauges, while the IR domain had a total of 8 (inclusive of all the gauges in the PM domain) gauges. A study reported by Habib and Krajewski (2002) has shown that a 5-10 min accumulation can be considered representative of an instantaneous remote sensing measurement at 4X4 km2 grid—in this study this space-time scale similarity may be representative of even larger time integrations given the coarser spatial resolution of both IR and PM retrievals. Figure 5.3 shows the dependency of PODINST on RREF-ACCU. It is observed that at hourly-accumulated rainfall values beyond 1 mm/hr the 10-minute sample will have probability greater than 95% to report rainfall. A sigmoid function of the form shown in Equation 5.7 (below) appeared a statistically good fit to model PODINST.
POD INST (R REF − ACCU ) =
A ACCU
1 + exp (-B ACCU R REF − ACCU )
(5.7)
The other error parameters, MuINST-ACCU and SINST-ACCU, controlling the conditional error are reported in Table 5.2. To generate instantaneous reference rainrates for the PM
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retrieval, hourly-accumulated reference rainfall values were computed from an average of 7 gauges located within a 10X10 km2 grid surrounding the basin, and subsequently corrupted by the representative error model comprising Equations 5.1 and 5.2.
To
generate corresponding instantaneous reference rainrates for the IR retrieval, the hourlyaccumulated reference rainfall values were computed from the entire network of 11 gauges that are within the 25X25 km2 grid area (see Figure 4.1, left panel, for a schematic on the gauge network configuration). As common reference (i.e., the ‘truth’) for PM and IR retrievals coincident rain profile, estimates from TRMM Precipitation Radar (PR) (Kummerow et al., 2000) were used. The PR estimates are based on a radar profiling retrieval that is superior to any overland passive microwave technique (Iguchi et al., 2000). The primary aspects of the retrieval are the precipitation classification, which is facilitated by the high vertical resolution (250 meters) reflectivity profile measurements, and an inversion algorithm that is controlled by a surface reference technique for path integrated attenuation and a reflectivity-to-rainfall relationship with parameters differentiated for convective and stratiform rain regimes (Iguchi et al., 2000; Meneghini et al., 2000). Ground validation studies on PR have shown high correlation (>0.9) and low (3 hour), hydrologic simulations were performed at 3 hourly time steps using the relevant calibrated TOPMODEL parameters. For the assessment of the combined PM-IR retrievals, which are available at hourly scale, due to the frequent IR data, hydrologic simulations were performed at the hourly time step using the hourly-calibrated TOPMODEL parameters. For each storm event, an ensemble of 20,000 realizations of synthetic satellite precipitation retrievals from RSREM were propagated through the hydrologic model to derive an equal number of simulated hydrographs for computation of error statistics in runoff simulation. The following hydrologic error parameters were assessed in this study: (1) Normalized Mean Absolute Error in peak runoff (PR); (2) Normalized Mean Absolute Error in time to peak (TP) and, (2) Normalized Mean Absolute Error in runoff volume (RV), defined as:
Error in PR =
Error in TP =
1 N sim
1 Nsim
Error in RV =
N sim
∑ i =1
Nsim
∑ i =1
1 N sim
Peak Runoff i − Peak Runoff REF +NAEREF_PR Peak Runoff REF
Timeto Peaki − Timeto PeakREF + NAEREF_TP Timeto PeakREF
N sim
∑ i =1
(5.9)
Runoff Volumei - Runoff VolumeREF + NAEREF_RV Runoff VolumeREF
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where Nsim is the total number of simulation runs (20,000); subscript i indicates the simulation index; and subscript “REF” implies the hydrologic parameter that was derived from reference runoff. The second term in Equation 5.9, NAEREF_X, represents the hydrologic modeling error for each hydrologic parameter X (PR, TP, and RV) due to reference runoff.
a.
Assessment of PM retrievals
First the PM retrievals are assessed alone. In this case the instantaneous rain rate fields are assumed constant between successive PM overpasses, which is a reasonable scenario for real-time operation. As the TOPMODEL was run using 3-hourly time steps, simulated runoff was interpolated at hourly intervals by a polynomial function. Because the first overpass by a PM sensor may occur at anytime during a storm event all possible sampling start times (rounded off to the closest hour) have been considered for the examined storms. For this purpose, the start of the PM sampling was shifted up to 11 hours, which was the maximum revisit time during an event (see Table 5.1, column 10). The same procedure was followed to determine the error statistics in terms of the above runoff parameters for the planned GPM-based 3-hourly canonical PM sampling. Although GPM revisit time intervals would vary between one and six hours (with an average revisit time be 3 hours) an idealized scenario of 3-hourly canonical sampling has been assumed. The 3-hourly error statistics were used to normalize the error statistics determined for the current PM-based prediction as:
Error Statistic Ratio (ER) =
Runoff Error Statistic (current PM sampling) Runoff Error Statistic (3 - hourly sampling)
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(5.10)
These Error Ratios (ER), determined for the PR, TP, and RV parameters, are used to quantify the factor by which the current PM sampling scenario is more uncertain than the planned GPM 3-hourly sampling in terms of flood prediction. It is argued that ERs are more informative than using absolute errors in understanding the implications of the retrieval and sampling error in runoff. The upper panels of Figure 5.6 show the PM-based hydrographs along with the 90% confidence limits for one of the storms in the database (Storm 9). The runoff prediction uncertainty of the current PM retrievals (right panel) appears to be higher (approximately 50%) than those associated with the 3-hourly PM sampling (left panel). The uncertainty in peak runoff was nearly 350 m3/s (compared to the 300 m3/s for the 3hourly sampling), while the time to peak tended to be overestimated by about 20 hours. This can be explained by the high revisit times (~11 hours) between successive PM overpasses, which can completely misinterpret the peak rainfall rates of a storm (see Figure 5.1, lowermost panel). Table 5.4 shows the ER values of the runoff parameters for all 15 storms. The mean ER values of peak runoff, time to peak runoff and runoff volume are 1.52, 2.17 and 1.50, respectively. Overall, the flood prediction uncertainty based on the current PM retrieval and sampling is found to be higher than what would be achieved by the planned 3-hourly GPM sampling in the ranges of 50 – 100%. An interpretation of this is that the GPM can be expected to be more reliable for flood prediction than current PM sensors but significant uncertainty would still persist as indicated in Figure 5.6.
Figure 5.7a shows the response of storm morphology on flood prediction uncertainty associated with current PM retrieval and sampling. In the upper left panel,
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the impact of storm duration to error in time to peak is evident. As storm durations become longer (from 3 to 7 days) the Time to Peak Runoff is predicted with less error (a reduction from 220% to 30% is observed). This is explained by the fact that the PM satellite sampling effect becomes less important as storm duration increases. It is also observed that as the Rain Fraction (%) increases the error in runoff volume reduces moderately (Figure 5.7a, upper right panel). It is noted that due to the high PM rain detection probability, as the hours during a storm event associated with rain increase the error for a PM sensor to retrieve the storm’s total rainfall accumulation decreases. There is insignificant sensitivity observed in the mean conditional rain rate (lower left panel, Figure 5.7a) and variability of rainrate (lower right panel, Figure 5.7a) on the error in runoff volume. When the response of the storm morphology on flood prediction uncertainty for the current PM sampling is compared to the planned GPM 3-hourly sampling (shown in Figure 5.7b), the following two notable effects are observed. First, the hydrologic error statistics (mainly error in Time to Peak and Runoff Volume) are lowered by an average of 10-30%. Second, the overall sensitivity of the four storm morphological parameters on the flood prediction uncertainty appears to diminish, as there are no obvious gradients and widespread scatter observed in Figure 5.7b. Consequently, the increased sampling in GPM should allow more consistent flood prediction than current PM sampling for storm events whose morphological properties are within the domain analyzed herein. b. Assessment of Combined PM-IR retrievals In this section the utility of merging IR retrievals with PM is assessed. The simulated (through RSREM) hourly IR rainfall retrievals were used to fill up the gaps
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between successive simulated PM retrievals and repeated the runoff simulation experiment on the basis of the framework described in section 5.4a. There are two major distinctions to be highlighted in the herein application: (1) the time step used in the hydrologic model simulation was now hourly and, (2) the hourly-accumulated reference rainfall for the IR retrieval pertained to averaged-rainfall over a 25X25 km2 grid area surrounding Posina basin as discussed before in section 5.3b (see also Figure 4.1, left panel). Results of this simulation experiment are used to determine an error parameter named the Merging Error Ratio (Merging ER) and defined as follows:
Merging ER =
1 nstorm Runoff Error Statistics (merged PM - IR retrievals) i ∑ Runoff Error Statistics (PM retrievals) i n i =1
(5.11)
Here n is equal to 15 (the total number of storm events) with i being the storm index. Merging ER values are determined for all three runoff parameters. A Merging ER value of less than one indicates that the merging of IR with PM retrievals reduces the overall uncertainty in the associated runoff parameter compared to the stand-alone use of PM rain retrieval input. The example shown in Figure 5.6 (lower panels) indicates that the use of IR retrievals with current performance statistics can, for a catastrophic flood event like Storm 9, worsen the flood prediction uncertainty for both the GPM and current PM sampling scenarios. Even though the 90% confidence limits are narrower when filling sampling gaps with IR, the reference runoff is enveloped only sparsely by these bounds (for example the first peak is completely missed for Storm 9). This indicates that the hydrologic model had lost its predictive accuracy through the use of the IR retrieval. Our 122
findings in this study indicate that the worsening in flood prediction for Storm 9 can be attributable to the relatively lower rain detection accuracy of IR retrievals at finer scale, which results in a net underestimation of total rain volume compared to that by the four PM retrievals (upper right panel, Figure 5.6). Although a portion of this underestimation may have been compensated by the higher IR false alarm rates, the inaccurate temporal characterization of rain fluxes renders the corresponding estimation of runoff in time similarly erroneous. The next section presents a more detailed analysis of the response of PM-IR retrieval in runoff, considering all 15-storm events jointly. Figure 5.8a presents the impact of combined PM-IR retrieval for various hypothetical levels of IR retrieval accuracy and current PM sampling.
As IR
measurement accuracy can be sensitive to scale and the retrieval technique, a range of possible IR uncertainty levels are expected in future algorithm improvements. Keeping the PM retrieval error characteristics, the IR’s Dfalse and probability of no-rain detection P0 fixed at the calibrated values (shown in Table 5.2), the IR’s maximum rain detection probability was co-varied with the IR multiplicative conditional retrieval error standard deviation (SIR). The variation of maximum rain detection probability was performed from 0.2 (very low rain detection probability) to 1.0 (an optimistic scenario where IR equals the maximum detection probability of PM) by varying parameter AINST in Equation 5.8 (0.75 forms the IR rain detection level for the 3B41RT). The error standard deviation SIR was co-varied from 0.9 (a fairly optimistic scenario comparable to PM, see Table 5.2) to 1.90 (a pessimistic scenario) (1.51 forms the current level of SIR). The co-variation of these two IR retrieval parameters was assessed in terms of their reduction in flood
123
prediction uncertainty by evaluating the Merging ER parameter (Equation 5.11). The temporal correlation, ρ2 of IR error, was fixed to 0.4. Several distinct features are worth noting from Figure 5.8a. While Merging ER in both peak runoff and runoff volume decreases below one in the optimistic regions of IR retrieval performance (Figure 5.8a, leftmost and rightmost panels), it was always higher than one for Time to Peak (Figure 5.9a, middle panel). It seems that IR retrievals’ high false alarm rate (nearly twice as much of the PM) coupled with its relatively lower probability of no-rain detection (P0) obscured the positive effect of improving rain detection and retrieval error variance on the prediction of Time to Peak Runoff. On the other hand, IR retrievals are found to reduce uncertainty moderately by 15-20% in Peak Runoff prediction when they are associated with high rain detection levels (P1 IR > 0.90) and with comparably lower conditional error standard deviation (SIR 0.70) seems necessary. The stronger gradients along the axis of rain detection probability (abscissa of rightmost panel) indicate that the runoff volume error is more responsive to improvements in IR rain detection than the other two hydrologic parameters (PR and TP). Figure 5.9b presents the impact of combined PM-IR retrievals for varying levels of IR retrieval accuracy associated with the 3-hourly GPM sampling scenario. While a pattern of error interaction qualitatively similar to that presented for the current PM sampling is observed, there is a distinctive aspect emerging from the increased sampling frequency in GPM. The Merging ER values are about 20% to 30% higher than those of current PM sampling for both the Peak Runoff and Runoff Volume parameters (see
124
leftmost and rightmost panels of Figure 5.9b). This indicates that with an improved PM sampling, the benefit of combining IR with PM retrievals to reduce flood prediction uncertainty is expected to reduce by 20-30%. This agrees with a previous study by HAD04 where it was found that runoff error reduction for a canonical 3-hourly PM sampling was 30-50% lower than that of a 6-hourly PM sampling. For Time to Peak parameter the combined PM-IR retrievals increase the prediction uncertainty compared to using solely PM-based predictions.
5.5 Conclusions
Hydrologic assessment of satellite rainfall retrievals for flood prediction demands the recognition that as the space and time scales become smaller, the sensor’s precipitation detection and retrieval accuracy become increasingly more complex. This study revealed that current PM retrieval and sampling scenario can be 50-100% more uncertain than the planned GPM era 3-hourly sampling scenario in terms of flood prediction for medium-sized watersheds. The merging of IR with PM retrievals on the basis of current retrieval error characteristics showed that it would worsen flood prediction uncertainty, especially in terms of the time to peak prediction, and for catastrophic storm events. Considering that the accuracy of IR retrievals varies by scale and retrieval technique, and that improvements are expected in new algorithms, various levels of its measurement uncertainty were assessed here in terms of the reduction in flood prediction uncertainty when merged with PM retrievals. It was found that for certain levels (some of them very optimistic for an IR retrieval scheme at a fine
125
resolution) of rain detection efficiency and conditional retrieval error variability of IR rain estimates can lead to significant reduction of prediction uncertainty in terms of runoff volume and peak runoff parameters. No error reduction in time to peak was achieved however. Probably, to reduce the error in time to peak, further improvement such as reduction in IR retrieval’s false alarm rates coupled with an even higher rain detection ability may be necessary. Results from this study are limited to major floods resulting from long-lasting storms (>2days) and saturation-excess runoff generation mechanisms from medium sized mountainous watersheds (50-500 km2). Mountainous basins at these scales are prone to high flood risks, while satellite observations for many of those regions is probably the only data source for measuring rainfall. Nevertheless, the results of this study cannot be generalized to other scales and runoff generation mechanisms. For example, the work needs to be expanded to larger basins and other land surface environments (e.g., vegetated versus dry regions, and/or basins dominated by infiltration excess runoff) to better understand the interactions of precipitation error with hydrology. The herein study highlighted the need for improving satellite retrieval error characteristics to achieve improved hydrologic forecasts. Recent techniques have sought such improvements in IR rain estimation accuracy through pattern recognition of cloud features (Xu et al., 1999), assimilation of lightning information (Morales and Anagnostou, 2003; Chronis et al., 2004), and assimilation of Microwave information (Todd et al., 2001), to name a few. Similarly, PM retrievals continue to evolve regularly in terms of new overland techniques (McCollum and Ferraro, 2003) and improved sampling and resolutions as there is a transition to the GPM era. It is worthwhile, therefore, to understand the effect of scale on
126
the precipitation measurement error from a host of single and multi-sensor retrievals, and how this propagates in hydrologic prediction. Furthermore, short duration storms and larger size watersheds need to be studied to understand the temporal sampling problem and the aspect of spatial variability of satellite measurement error in runoff prediction. These are some of the many aspects that need to be addressed to achieve more meaningful applications of satellite rainfall observations on flood hydrology.
127
Table 5.1 Statistical summary of storm events with the simulated PM sampling overpasses of the assumed complement of four PM sensors.
No
Date
Rainfall Statistics
Discharge
PM Sampling (2 SSM/I+AMSR-E+TMI)
(month
Duration
year)
Rain
Mean
Std.
CV
Fract.
Cond.
Dev
(%)
Max.
Sampling
Max
Mean
Discharge
Coverage
Revisit
Revisit
Time
Time
(hrs)
(hrs)
(hrs) (%)
mm/hr
(m3/s)
mm/hr
(hrs)
1.
08 1987
72
34.0
3.90
5.8
148
54.40
14
11
5.1
2.
10 1987
96
40.8
2.80
2.7
96
75.72
21
11
4.3
3.
07 1989
96
29.0
3.20
2.9
91
31.49
21
11
4.3
4.
11 1990
96
35.8
2.80
2.9
103
64.37
21
11
4.3
5.
12 1990
108
40.1
3.61
3.9
108
76.41
25
11
4.5
6.
03 1991
72
32.3
2.49
2.9
115
32.59
14
11
5.1
7.
10 1991
84
44.4
3.85
4.5
170
117.40
17
11
4.6
8.
04 1992
120
58.3
1.35
2.3
107
56.89
26
11
4.4
9.
10 1992
120
86.7
4.24
4.5
107
192.50
26
11
4.4
10.
12 1992
144
36.3
1.94
1.9
99
41.60
32
11
3.9
11.
09 1993
132
61.5
2.98
3.5
116
49.40
30
11
4.2
12.
11 1994
72
55.0
3.65
4.1
112
106.90
14
11
5.1
13.
10 1996
96
85.4
3.65
3.2
87
156.50
21
11
4.3
14.
11 1996
120
81.7
1.84
2.3
124
70.80
26
11
4.4
15
12 1997
84
76.0
1.78
2.3
128
70.26
17
11
4.6
128
Table 5.2 Mean error model parameters calibrated for PM (2A12) and IR (3B41RT) sensor retrievals on the basis of coincident TRMM Precipitation Radar rainfall fields.
2A12
3B41RT
(10 X 10 km)
(25 X 25 km)
AACCU (AINST)
1.05 (1.0)
1.01(1.35)
BACCU (BINST)
4.5 (3.5)
5.5 (1.0)
λ
0.9
0.5
MuINST-ACCU (MuINST)
1.0 (1.27)
1.0 (1.52)
SINST-ACCU (SINST)
0.24 (0.94)
0.19 (1.51)
Lag-one correlation (ρ2)
0.40
0.40
No Rain Det Probability (P0)
0.93
0.88
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Table 5.3 Hydrologic Modeling error of TOPMODEL based on observed runoff versus reference runoff differences.
Storm No.
Normalized Absolute Error - NAEREF Runoff Volume Peak Runoff Time to Peak Runoff
1
0.038
0.310
0.080
2
0.027
0.083
0.000
3
0.008
0.043
0.000
4
0.022
0.048
0.000
5
0.035
0.016
0.075
6
0.210
0.540
0.056
7
0.098
0.110
0.038
8
0.099
0.062
0.031
9
0.083
0.232
0.000
10
0.309
0.019
0.000
11
0.069
0.129
0.017
12
0.007
0.071
0.000
13
0.128
0.072
0.011
14
0.119
0.025
0.016
15
0.115
0.089
0.051
Mean
0.091
0.123
0.025
Note: Normalized Absolute Error (NAEREF) = | Xobs-Xref |/Xobs
130
Table 5.4 Runoff prediction Error Ratio (ER) values of PM retrievals for the 15 storm events evaluated for the current PM sampling scenario.
Error Ratio (ER) Storm No.
Peak Runoff
Time
to
Peak
Runoff Volume
Runoff 1
1.09
1.01
1.13
2
1.31
1.77
1.46
3
6.29
1.67
3.47
4
1.01
2.72
1.23
5
1.23
1.16
1.17
6
1.07
1.04
1.19
7
1.31
3.09
1.38
8
1.11
3.56
1.30
9
1.12
1.09
1.13
10
1.22
1.00
1.17
11
1.04
1.27
1.69
12
1.14
7.04
1.59
13
1.15
1.08
1.19
14
1.16
4.09
1.24
15
2.70
1.01
2.12
Mean
1.52
2.17
1.50
131
Figure 5.1 An example of the assumed complement of four PM sensor overpasses (sampling) for Storm 9 (October, 1992). The assumed constellation comprises SSM/IF14, SSM/I-F15, TMI and AMSR-E.
132
i = time step index RREF-ACCU, i
i=1
i=i+1 Is RREF-ACCU,i >0.0 mm/hr ? YES
NO RREF-INST,i=RREF-ACCU,i
Compute PODINST from Eqn. 5.7 Generate uniform rn (0-1)
Is rn ≤ PODINST? YES
NO
RREF-INST,i
RREF-INST,i
From Eqn.5.2
=0.0 mm/hr
Is RREF-,INST,i>0.0 mm/hr ? YES
NO
Compute P1 from Eqn. 5.8
Obtain P0 from Table 5.2
Generate uniform rn (0 –1)
Generate uniform rn (0-1)
Is rn ≤ P1? YES
RSAT-INST,i from Eqn.5.3
Is rn ≤ P0 ? NO
YES
RSAT-INST,i
NO
RSAT-INST,i
= 0.0
=0.0
RSAT-INST,i from Dfalse (Table 5.2)
Figure 5.2 Flow-chart for RSREM with the pertinent modifications.
133
Figure 5.3 Probability of Detection of instantaneous reference rainrate PODINST, as a function of accumulated reference rainrate : Prob( RREF-INST >0.0 | RREF-ACCU >0.0). The dashed line are the best fits to the data, which determine the error model parameters of RSREM.
134
Figure 5.4 An example of matched TMI (2A12), IR (3B41RT) and TRMM-PR (2A25) retrieved rain maps over Central Africa. Left panel is TMI (Orbit No. 24225); Middle Panel is PR (Orbit No. 24225); Right panel is IR (1300 hrs UTC), on February 13, 2002. PR and IR are shown at the 25 km X 25 km2 resolution.
135
Figure 5.5 Probabilities of rain detection (right panels) and probability distributions of rain rates in false alarms (left panels) for PM (upper panels) and IR (lower panels) retrievals. These experimental results are determined on the basis of coincident TRMM PR overland rainfall data around the Globe. The solid lines are best fits to the data, which determine the error model parameters presented in the paper.
136
Figure 5.6 Flood prediction uncertainty (5% and 95% upper/lower quantiles) of Storm 9 (October, 1992) for the GPM based 3 hourly sampling (left), and current PM sampling (right). Upper panels are based on PM retrievals, and lower panels on the merged IR-PM rain products.
137
Figure 5.7a Effect of storm morphology on the uncertainty of flood prediction driven by current PM retrieval sampling scenarios. Upper left panel: Error in Time to Peak vs. storm duration; Upper right panel: Error in Runoff Volume vs. rain fraction; Lower left panel: Error in Runoff Volume vs. mean conditional rain rate; Lower right panel: Error in Runoff Volume vs. standard deviation of conditional rain rate.
138
Figure 5.7b Same as in Figure 5.7a, but for flood predictions driven by the GPM based 3-hourly PM sampling scenario.
139
Figure 5.8a Contours of Merging ER values representing the relative runoff prediction error of the combined PM-IR rain products to PM retrieval alone (current PM sampling scenarios), for various levels of IR rain estimation accuracy (SIR vs. maximum IR rain detection probability). Left panel: Merging ERs in Peak Runoff; Middle Panel: Merging ERs in Time to Peak Runoff; Right Panel: Merging ERs in Runoff Volume. The solid circle represents the performance level of current IR algorithm.
140
Figure 5.8b Same as in Figure 5.8a, but for the GPM based 3-hourly sampling scenario.
141
6 ON THE DEPENDENCY OF SOIL MOISTURE PREDICTION ACCURACY TO SATELLITE RAINFALL ESTIMATION AND LAND SURFACE MODELING UNCERTAINTIES Abstract This study aims at evaluating the uncertainty in the prediction of soil moisture from offline land surface models (LSM) forced by hydro-meteorological and radiation data. The focus is on two major sources of uncertainty: (1) in the precipitation input, which in global applications is typically based on satellite remote sensing; and, (2) in the modeling due to non-uniqueness of soil hydraulic parameters. The study is facilitated by data (precipitation, radiation, soil moisture) driven simulation experiments comprising a LSM and stochastic models for error characterization. The LSM used in the study is NOAHLSM.
The modeling uncertainty is represented by the Generalized Likelihood
Uncertainty Estimation (GLUE) technique. Satellite rainfall estimates pertain to halfhourly Infra-red (IR) retrievals from geo-synchronous platforms.
The rain retrieval
uncertainty is characterized on the basis of a Remote Sensing Rainfall Error Model (RSREM).
The combined uncertainty—i.e., rainfall estimation and modeling
(RSREM+GLUE)—is compared with the partial assessment of uncertainty, i.e., accounting for either the modeling (GLUE) or rain retrieval uncertainty (RSREM). Comparisons are made as a function of site characteristics on the basis of two geographically distant locations: in Champaign (Illinois) associated with farmland-type vegetation and Perkins (Oklahoma) associated with dense vegetation. Soil moisture prediction uncertainty is found to be 50%-100% higher for the dense vegetated site. It is
142
shown that precipitation error may explain moderate to low proportion of the overall soil moisture simulation uncertainty, which depends on the level of modeling accuracy— 50%-60% for high model accuracy, and 20%-30% for low model accuracy. This study exemplifies the need for improved understanding of rainfall-modeling error interaction aimed at identifying optimal satellite rain retrieval error characteristics for use in land data assimilation systems.
6.1
Introduction
Soil moisture, defined as the water content in the upper layer (i.e., about 2 meters) of soil, is the hydrologic variable that controls the interactions (and feedbacks) between land surface and atmospheric processes. It influences—(1) the partition of incoming radiation into sensible heat and latent heat fluxes, (2) the separation of surface rainfall into infiltration and surface runoff, and (3) the rate of evapo-transpiration.
The
importance of accurate characterization of spatio-temporal variability of soil moisture has been stressed numerous times in recent literature (Schaake et al., 2004; Albertson and Montaldo, 2003; Pan et al., 2003; Hornberger et al 2001; Houser et al., 1998; Entekhabi and Rodriguez-Iturbe, 1994; among others). Soil moisture estimates based on Land Data Assimilation Systems (LDAS) are comprised of a physically based land surface model (LSM) forced by hydro-meteorological data and schemes for assimilating remotely sensed land surface variables. Of particular importance is the characterization of uncertainty in the prediction of soil moisture by LSMs. This would facilitate the statistical (ensemble) pre-storm initialisation of distributed hydrologic models used in the
143
prediction of floods, and the development of ensemble land surface boundary conditions for regional atmospheric models issuing short to medium-range quantitative precipitation forecasts. Furthermore, knowledge of uncertainty is key to advancing the efficiency of land data assimilation techniques (McLaughin, 2002). An example is the LDAS that is currently in operation over North America employing four different LSMs (Robock, et al., 2003; Luo et al., 2003). The two main sources of uncertainty in LSM predictions are: (1) errors in the forcing variables and in particular the precipitation; and, (2) the LSM formulation error (manifesting as non-uniqueness in soil hydraulic and vegetation parameters). Figure 6.1 provides a schematic of this LSM error concept. On one hand, understanding the error in rainfall measurement has implications for the Global Precipitation Measurement (GPM), which is a mission to be launched by the international community by 2009 (Smith, 2001; Bidwell et al., 2002). GPM is expected to provide rainfall measurements from space at scales finer than what are globally available today. Hence, satellite rainfall estimates may gradually become the dominant component of the atmospheric forcing data for LDAS. Recent satellite hydrologic application studies, in anticipation of the GPM, have focused mainly on runoff simulations (Hossain and Anagnostou, 2004a; Hossain et al., 2004b; Nijssen and Lettenmaier, 2004). On the other hand, assessment of the modeling formulation of LSMs (i.e., validation) is equally important. This is because despite the physical complexity, these models suffer from parameter non-uniqueness where a wide range of parameter sets exhibits equally acceptable simulations against observations (this shall referred to as ‘parameter equifinality’) (Schulz and Beven, 2003). LSMs may also have error due to its structural formulation. This is however not the topic of study herein.
144
Current LDAS formulations have provisions for using multiple state-of-the-art LSMs in the assimilation technique (Robock et al., 2003), which can optimally be achieved on the basis of error characterization. The literature review on pertinent research efforts to characterize soil moisture predictive variability by LSMs (e.g., Maurer et al., 2002; Maurer et al., 2001; Nijssen et al., 2001) shows that studies, addressing the relative impacts of both (precipitation and modeling) uncertainty sources, are lacking. This is also supported by the recent statement from Schaake (2003) where he has asked how a hydrologic model would respond to an ensemble of equally likely traces of remote sensing retrievals that would be repeated for an ensemble of equally likely model parameter sets (i.e., different hydrologic model realizations). The combined assessment of uncertainty would allow comparisons with partial assessments (i.e., rainfall or modeling error alone), which can potentially identify the implications of each error source (and hence a strategy for reducing the prediction uncertainty).
In hydrologic remote sensing, this recognition of both sources of
uncertainty implies that a partial assessment may not be adequate to describe the full range of variability in soil moisture prediction. Studying the statistical characterization of soil moisture prediction error associated with both uncertainty sources is the key to advancing the use of rainfall remote sensing in LDAS. A similar paradigm of scientific inquiry for catchment-scale hydrology has proved useful in enhancing the usage of radar rainfall estimation in flood simulations over mountainous basins (Hossain et al., 2004a). Even though the importance of uncertainty assessment is well known, accurate assessment through exhaustive Monte Carlo (MC) simulations has always been recognized as computationally prohibitive for physically-based LSMs (see Hossain et al.,
145
2004c,d; Hossain and Anagnostou, 2004b). This perhaps explains the current absence of full-scale uncertainty assessment of LSMs driven by remotely sensed rainfall datasets. Yet, the MC sampling technique, due to its lack of restrictive assumptions and completeness in sampling the input error structure, is generally considered the preferred (and easiest) method for uncertainty assessment (Isukapalli and Georgopoulos, 1999; Beck, 1987; Kremer, 1983). These two conflicting issues impose significant limitations on full-scale uncertainty assessment. An MC assessment combining both sources of uncertainty will naturally be computationally prohibitive. The study conducted herein is motivated by the need to explore the full range of uncertainty of soil moisture prediction driven by satellite rain retrievals. In that respect, it attempts to address the relative impacts and interactions of uncertainty in satellite rainfall retrieval and LSM simulations. The model parameter uncertainty is represented by the Generalized Likelihood Uncertainty Estimation (GLUE, Beven and Binley, 1992) technique. Uncertainty in satellite rain retrieval is modeled on the basis of a Remote Sensing Rainfall Error Model (RSREM) developed by Hossain and Anagnostou (2004a). Satellite rainfall estimates pertain to retrievals from half-hourly geo-stationary satellite Infra-red (IR) observations. The combined assessment of uncertainty—due to rainfall and modeling (RSREM+GLUE)—is compared with partial assessments that accounted for either the modeling (GLUE only) or rain retrieval uncertainty (RSREM only). Comparisons are also made as a function of vegetation regime by conducting the study on two sites: (1) a sparse farmland vegetation site (in Champaign, Illinois); and (2) a densely vegetated site (in Perkins, Oklahoma).
146
The paper is organized as follows. In section 6.2 the study region and data are described followed by a description of the LSM used in section 6.3. In section 6.4, the stochastic error models used to characterize the uncertainty in the two error sources (satellite rainfall and LSM) are presented. In section 6.5, the simulation framework and results are presented.
Section 6.6 presents the major conclusions and suggested
extensions of this study.
6.2
Study Region and Data
Two regions were chosen for the study: (1) Champaign in Illinois; and, (2) Perkins in Oklahoma (hereafter the two regions will be abbreviated as “CHAMP_ILL” and “PERK_OK”, respectively). CHAMP_ILL is a farmland located 40.01o North and 88.37o West. The site characteristics are typical of those found throughout Midwestern US with most of the land in agricultural production. The soil is loamy with a bulk density of 1.5 gm/cm3. The study period is one year (1998) when soybeans were planted on the farm. Atmospheric and radiation forcing data from a flux measuring system installed in the farm were recorded every 30 minutes for that year. The major atmospheric data comprised rainfall, temperature, humidity, surface pressure and wind. The radiation forcing data pertained to downward solar (short-wave) and downward long-wave radiation flux measurements. Soil moisture measurements at only the 5 cm depth are considered here, because at deeper depths the measurements were found unreliable (Hossain et al., 2004c; see also User’s Guide described next). This data is public domain and available as part of standardized testing protocols for simulation codes of the NOAH-
147
LSM (also discussed next). For more information on the study region and data measurement protocols the reader is referred to the User’s Guide, Public Release Version 2.5 available at ftp://ftp.emc.ncep.noaa.gov/mmb/gcp/ldas/noahlsm/ver_2.5. PERK_OK is located at 35.99o North and 97.05o West in Payne County, Oklahoma. It has perennial ground cover, intermittent farmland with broad-leaf deciduous trees. All requisite hydro-meteorological data except the downward longwave radiation for NOAH-LSM operation are available at half-hourly intervals from the Oklahoma
Mesonet
network
(for
more
details
on
Mesonet
see
http://www.mesonet.ou.edu). The average composition of soil in a 75 cm column at PERK_OK is about 45% sand, 35% silt and 20% clay and is classified as sandy-clay loam. Soil moisture measurements are available at three depths: 5 cm, 25 cm and 60 cm from surface. The soil moisture evolution at these three depths can be considered representative of the daily, weekly and seasonal response, respectively. The year under study for PERK_OK is 2001.
Long-wave radiation was calculated from in-situ
measurements of air temperature (Tair, oC) and relative humidity (RLH, %) as follows. First, the saturation vapor pressure esat (mb) was calculated from air temperature (Bras, 2001):
[
]
esat = 33.8639 (0.00738Tair + 0.8072) 8 − 0.0000191.8Tair + 48 + 0.001316 (6.1)
Using knowledge of relative humidity, the ambient vapor pressure e (mb) was then computed as:
148
e = RLH × e sat
(6.2)
The long-wave emissivity Ea was then computed as (Idso, 1981),
E a = 0.70 + 5.95(10 −5 )e ⋅ exp(
1500 ) (Tair + 273.0)
(6.3)
Finally, we used the Stefan-Boltzmann law (Equation 6.4) to predict the long-wave radiation LW (W/m2) as,
LW = E aσ (Tair + 273.0) 4
(6.4)
where, σ = 5.67 X 10-8 J m-2 s-1 K-1
The above formulation for computing longwave radiation does not take into account information on cloud cover (which was unavailable in this study). The consideration of CHAMP_ILL and PERK_OK sites allowed the exploration of the role of vegetation cover in the propagation of precipitation and modeling error to soil moisture. CHAMP_ILL represents a relatively dryer regime with low to moderate evapotranspiration (ET) due to the prevalence of seasonal cropland. The ET usually peaks around the months of July and August (the growing season for soybeans), during which the mean monthly rainfall is low (see Hossain et al., 2004c for more details). PERK_OK, on the other hand, is a relatively greener region with a larger proportion of deciduous
149
trees. For a more detailed comparison of the vegetation regimes, the reader is referred to the information page on land surface vegetation parameters of the US at http://ldas.gsfc.nasa.gov/LDAS8th/green/LDASgreen.shtml. A point to note is that the comparison across vegetation regimes is limited to soil moisture predictions at the 5cm depth due to limited soil moisture data availability.
6.3
The Land Surface Model
(a) Model Description The LSM used in this study is the NOAH-LSM (also known as The Community NOAH-LSM) (Chen et al., 1996; Pan and Mahrt, 1987). NOAH-LSM was chosen as it is a popular operational model and more importantly, it is one of the four LDAS LSMs currently being evaluated over the United States (Robock et al., 2003). NOAH-LSM is a stand-alone, uncoupled (offline), column (1-D) version used to execute single-site land surface simulations at 30 minutes intervals. In this traditional 1-D uncoupled mode, near surface atmospheric and radiation data are required as input forcing. NOAH-LSM simulates soil moisture (both liquid and frozen), soil temperature, snow pack, depth, snow pack water equivalent, canopy water content and the energy and water flux terms in terms of the surface energy balance and surface water balance.
A four-layer soil
configuration (comprising a total depth of 2 meters) is adopted in the NOAH-LSM for capturing daily, weekly and seasonal evolution of soil moisture and mitigating possible truncation error in discretization (Sridhar et al., 2002). The lower 1-meter acts as gravity drainage at the bottom, and the upper 1-meter of soil serves as root zone depth. For more
150
details on the physical description of the model, one may refer to Sridhar et al. (2002) and Chen et al. (1996). In line with the minimum requirements for spin-up (Cosgrove et al., 2003) and to reduce the impact of snow cover, which was not the focus of this study, we truncated our effective study period for CHAMP_ILL and PERK_OK to the May 1– October 30, 1998 and 2001 periods, respectively.
(b) Model Fine-Tuning Preliminary investigation with NOAH-LSM found it necessary to adjust some of the NOAH-LSM vegetation parameters to make the model more representative of the point-scale soil moisture flux simulations at the two study regions. The adjustment procedure studied by Hossain et al. (2004c) is adopted here (see Chapter 7B, page 217218). Figures 6.2a and 6.2b show the effect of fine-tuning during the study period for CHAMP_ILL and PERK_OK, respectively. It is seen that NOAH-LSM is able to simulate soil moisture variability at the 5cm depth (for CHAMP_ILL) and at 5, 25 and 60 cm depths for PERK_OK. The overall correlation of model predicted to measured soil moisture was calculated 0.8 (0.9) for PERK_OK (CHAMP_IL).
(c) Model Parameter Uncertainty NOAH-LSM parameter (model) uncertainty was accounted for the following five soil hydraulic parameters that were considered most sensitive to soil moisture simulation: (1) maximum volumetric soil moisture content (porosity) (SMCMAX, m3/m3); (2) saturated matric potential (PSISAT, m) (3) saturated hydraulic conductivity K (SATDK, m s-1); (4) parameter ‘B’ of soil-water retention model of Clapp and Hornberger (1978)
151
(BB); and (5) soil moisture wilting point at which ET ceases (SMCWLT, m3/m3). The range (upper/lower) and optimal values for those parameters are shown in Table 6.1. These values were selected based on an empirical study by Clapp and Hornberger (1978), in-situ land surface information, and considering the sampling requirements of GLUE (Beven and Binley, 1992). It was assumed that the parameter uncertainty domain represented by the 5-D hyperspace characterizes adequately the parameter equifinality, which is responsible for most of the modeling uncertainty in soil moisture simulation. GLUE was chosen as the framework to characterize the modeling uncertainty in the NOAH-LSM formulation for the simulation of soil moisture. While GLUE is not the only modeling uncertainty assessment tool currently available (Misirli et al., 2003; Thiemann et al., 2001; Tyagi and Haan, 2001; Krzysztofowicz, 2000; Young and Beven, 1994), the simplicity of the theory behind this framework is what makes it convenient and very easy to implement (Beven and Freer, 2001). GLUE has therefore found extensive application in the assessment of modeling uncertainty of many hydrologic variables like streamflow, flood inundation, ground water flow, land surface fluxes, etc. (Schulz and Beven, 2003; Christaens and Feyen, 2002; Beven and Freer, 2001; Schulz et al., 2001; Romanowicz and Beven, 1998; Franks et al., 1998; Franks and Beven, 1997; Freer et al., 1996; among many others). Recently, the GLUE technique has also proved to be a powerful tool in understanding the implications of radar rainfall error adjustment on flood prediction uncertainty (Hossain et al., 2004a). GLUE is based on MC simulation: a large number of model runs are performed, each with random parameter values each sampled from uniform probability distribution (e.g. Table 6.1). The acceptability of each run is assessed through comparison of the
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predicted versus observed hydrologic variables on the basis of a selected likelihood measure. Simulations with likelihood values below a certain threshold are rejected as non-behavioral. The likelihoods of these non-behavioral parameters are set to zero and are thereby removed from subsequent analysis.
Following the rejection of non-
behavioral runs, the likelihood weights of the retained (i.e., behavioral) runs are rescaled so that their cumulative total is one (Freer et al., 1996). In this study the GLUE method was applied to uncertainty estimation of soil moisture simulation by NOAH-LSM at the 5 cm depth (for CHAMP_ILL) and at 5 cm, 25 cm and 60 cm depth (for PERK_OK). Thus at each time step (at 30 minute intervals), the predicted soil moisture from the behavioral runs are likelihood weighted and ranked to form a cumulative distribution of soil moisture simulation from which quantiles can be used to represent modeling uncertainty. While GLUE is based on a Bayesian conditioning approach, the likelihood measure is achieved through a goodness of fit criterion as a substitute for a more traditional likelihood function. The classical index of efficiency, ENS (Nash and Sutcliffe, 1970), was considered as the measure of likelihood,
⎡ σ 2e ⎤ E NS = ⎢1 - 2 ⎥ ⎣ σ obs ⎦
(6.5)
where σe is the variance of errors and σobs the variance of soil moisture observations. This likelihood measure is consistent with the requirements of the GLUE method, as it increases monotonically with the similarity of behavior. A point to note is that for PERK_OK, the index of efficiency was computed as a depth-weighted average (weighted by the thickness of each soil layer). 153
To implement the GLUE methodology, each parameter of NOAH-LSM was specified the range of possible values shown in Table 6.1. Constant (calibrated) values for all other NOAH-LSM parameters were used. Model predictions of soil moisture were carried out, and the model likelihood measure was calculated using the efficiency index of Equation 6.5. From the specified parameter ranges, MC simulations were conducted that allowed the selection of a large number of behavioral parameter sets characterized by a simulation efficiency index value greater than an assigned minimum threshold value. For further details on GLUE implementation the reader is referred to Beven and Binley (1992), Freer et al. (1996) and Beven and Freer (2001). The GLUE method has a drawback that limits its application for computationally complex models. It requires analysis of multiple simulation scenarios based on uniform random sampling of the model parameter hyperspace. This requirement can be prohibitive for models that are slow-running (Bates and Campbell, 2001; Beven and Binley, 1992). Hossain and Anagnostou (2004b) and Hossain et al. (2003c) provide an extensive review about this limitation, and propose an efficient sampling technique as an addendum to GLUE. In this technique, the uncertainty in soil moisture simulation (model output) is approximated through a Hermite polynomial chaos expansion of normal random variables that represent the model’s parameter (model input) uncertainty (see Chapter 7B for more details). The unknown coefficients of the polynomial are calculated using limited number of model simulation runs. The calibrated polynomial is then used as a fast-running proxy to the slower-running LSM to predict the degree of representativeness of a randomly sampled model parameter set. The study herein has employed this efficient sampling scheme formulated by Hossain and Anagnostou (2004b)
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and Hossain et al. (2004c) to substantially reduce the computational burden of the analyses.
6.4
Remote Sensing Rainfall Error Model
The one-dimensional (1-D) Remote Sensing Rainfall Error Model (hereafter referred to as RSREM) developed by Hossain and Anagnostou (2004a) was used to characterize the satellite rainfall retrieval error. RSREM has already been discussed in section 5.3 and hence only the most pertinent aspects of the uncertainty framework are presented herein. The rain retrieval considered in this study was from satellite IR, as at global scale, these observations offer the finest temporal sampling characteristics (1/2hourly) necessary to resolve the dynamic variability of soil moisture in the root zone. Herein the hourly-averaged IR rainfall fields produced by NASA’s Multi-satellite Precipitation Analysis (MPA) algorithm (Huffman et al., 2003) were considered as representative of the current level of IR rainfall estimation characteristics. This community release product is known as 3B41RT. Hossain and Anagnostou (2004a) had calibrated RSREM parameters for 3B41RT over the US on the basis of coincident rain profile estimates from TRMM Precipitation Radar (Kummerow et al., 2000). Figure 6.3a shows the cumulative hyetographs of actual IR (3B41RT) rainfall products and the corresponding Mesonet rainfall data over PERK_OK for the year 2002 (January 1 to October 30). The 3B41RT rainfall is compared against the quantile envelop associated with 5%-95% percentiles, predicted by RSREM using as input the Mesonet rain rates (Figure 6.3a).
The 3B41RT rainfall hyetographs, which is considered an observed
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realization, is enveloped by the RSREM quantiles. In Figure 6.3b, a similar quantile envelop of SREM-1D simulations is shown for CHAMP_ILL. There are no 3B41RT products available for the retrospective period of 1998 over CHAMP_ILL.
6.5
Simulation Framework and Results
The focus of this study was on the relative impact and non-linear interaction of uncertainties in modeling and satellite rainfall estimation.
It was assumed that the
reference rainfall (from surface measurements) and the optimal parameters for NOAHLSM represent deterministic processes. Hence, they yield accurate predictions of soil moisture with low uncertainty (see Figures 6.2a and 6.2b). If it is further assumed that RSREM and GLUE sample adequately the error structure in satellite rainfall and NOAHLSM parameters, respectively, then, based on these two assumptions the following logical inferences can be constructed: (1) Propagation of multiple realizations of RSREM rainfall processes via NOAH-LSM at optimal parameters will reflect the partial uncertainty in soil moisture prediction due to satellite rainfall estimation error (uppermost panel-Figure 6.4A); (2) Propagation of reference rainfall to NOAH-LSM via multiple GLUE model parameter realizations will reflect the partial uncertainty in soil moisture prediction due to modeling uncertainty (middle panel-Figure 6.4B); (3) Combining RSREM and GLUE on NOAH-LSM will reflect the total uncertainty in soil moisture prediction due to both sources of uncertainty (lowermost panel-Figure 6.4C). As a demonstration of the relative impacts and interactions of uncertainties, multiple (500) realizations were conducted from RSREM (for demonstrating inference 1, Figure 6.4A) and using GLUE (for inference 2, Figure 6.4B). For GLUE, the 500 best
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behavioral parameter sets were sampled from ranges in Table 6.1 with an ENS greater than 0.4 (using gauge rainfall as input). For the combined uncertainty assessment (inference 3, Figure 6.4C), the full-blown MC uncertainty assessment comprising 250,000 (500 RSREM rainfall realizations times 500 GLUE parameter sets) NOAH-LSM runs was executed to identify the full range of predictive variability. The wideness of prediction quantiles in soil moisture simulation is considered a measure of prediction uncertainty. This wideness, defined as Uncertainty Ratio (UR), is the time integrated uncertainty in soil moisture volume bounded by the quantile width (between upper and lower percentiles) normalized by the time-integrated observed soil moisture volume. The UR at n% quantile width (ranging from 10% to 90%), URn, is defined as follows:
∑ (SM NT
URn =
sim j , 50 + n 2
− SM sim j , 50 − n 2
)
j =1
(6.6)
NT
∑ SM
obs j
j =1
where, j is the time-step index of simulation, NT the total number of time-steps in the simulation period. Superscripts sim and obs refer to simulated and observed soil moisture, respectively. UR represents the bulk variability in prediction expressed relatively to the magnitude of the observed variable. In Figures 6.5 and 6.6 (a, b and c) we show each of the three inferences for CHAMP_ILL and PERK_OK, respectively. The uncertainty limits of simulation are shown at the 90% quantile width. It appears that there is no significant dependency of uncertainty as a function of depth (for PERK_OK, Figures 6.6). It should be noted that the partial uncertainty due to modelling and the combined (total) uncertainty are conditioned upon the subjective threshold used to select the behavioral parameter sets
157
(here it was fixed at ENS>0.4). It is observed that the partial uncertainties due to rainfall estimation, and modelling, are considerably higher in PERK_OK (compare the uppermost and middle panels of Figures 6.5 and 6.6). This indicates that the vegetation and hydraulic properties can have a notable effect on error interaction. Sites with denser vegetation sites are associated with stronger non-linear error propagation, which causes the notable difference observed between the two sites. The increasing sensitivity to precipitation error as the regime transforms from CHAMP_ILL to PERK_OK vegetation/soil type environment is shown in Figure 6.7a. The simulation experiment resulted to this figure used meso-forcing meteorological data of CHAMP_ILL, while the soil-hydraulic parameters were varied from the optimal value for CHAMP_ILL to the optimal value for PERK_OK (shown in Table 6.1). In the figure, the optimal values are shown rescaled from 0 to 1 where 0 and 1 represents the CHAMP_ILL and PERK_OKlike regime, respectively. The vertical axis of this figure shows the relative increase (in %) of UR at the 90th quantile width (UR90, Equation 6.7). The most sensitivity is observed for parameter DKSAT. This is expected as this parameter represents the soil’s bulk drainage property introducing most of the sensitivity in the model. PERK_OK regime has a representative DKSAT value about an order higher than that of CHAMP_ILL (Table 6.1). This may imply that faster soil water movement in the root zone due to denser vegetation enhances the non-linear interaction of precipitation error in soil moisture prediction. Similarly, Figure 6.7b shows the relative level of error propagation from rainfall input to soil moisture as function of quantile width assuming optimal model performance (run at optimal parameter sets shown in Table 6.1). The URn where n is varied from 10%-90% is used here to characterize the level of uncertainty in precipitation
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and the predicted soil moisture. What is evident is that soil moisture uncertainty is significantly dampened in the rainfall-soil moisture transformation process in a highly non-linear fashion. It is shown that the satellite IR hourly rain input uncertainty increases exponentially with quantile width to over than twice the magnitude of the estimated rainfall (which is commonly expected for IR retrievals at high resolution).
The
corresponding IR rain estimation error propagation to soil moisture prediction is, though, associated with a significant non-linear smoothing: i.e., the UR converges to values well below 0.4 (i.e., > 85% error reduction). This dampening is notably more significant for CHAMP_ILL (90% error reduction) than PERK_OK (85% error reduction) where the error propagation is enhanced due to the vegetated environment. The greater sensitivity of modelling uncertainty to soil moisture prediction uncertainty (and hence larger uncertainty estimation) for the PERK_OK regime can also be explained by a sensitivity analysis called the Generalized Sensitivity Analysis (GSA, Spear and Hornberger, 1980) of the NOAH_LSM model parameters (Figures 6.8a and 6.8b). GSA represents the sensitivity of the behavioral parameter values’ cumulative distribution function to varying likelihood thresholds. Figures 6.8a and 6.8b show the Cumulative Distribution Functions (CDF) of the five NOAH-LSM parameters for three different likelihood thresholds (i.e., ENS > 0.4, 0.5, and 0.7), for CHAMP_ILL and PERK_OK, respectively. It is observed that differences in the shapes of the CDF functions as a function of increasing threshold are much larger for the PERK_OK regime than the CHAMP_ILL regime, thereby indicating greater model parameter sensitivity. This explains the wider uncertainty limits in soil moisture prediction due to modelling error observed for the PERK_OK regime (in the 5 cm depth).
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Next, the relative significance of the two uncertainty sources (precipitation versus modeling), is studied. This warrants a more detailed characterization of the role of the behavioral threshold used in GLUE. Since this threshold is essentially subjective, it is important to recognize that its value may increase or decrease (from ENS=0.4) to represent various levels of parametric uncertainty (or model accuracy) at the operational scenario. Consequently, the behavioral parameter sets (all having ENS>0.4) were grouped into three model performance categories – (1) HIGH (high modeling accuracy: 0.750.7).
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Figure 6.9a The categorization of model parameter sets in HIGH (0.750.5, lower panel). This observation justifies the wisdom of using a more efficient parameter sampling scheme for GLUE based on interpolation of the parameter response surface. The interpolator is able to demonstrate sampling efficiency in predicting correctly the nature of a sampled set (behavioral or non-behavioral?) even at high degrees of acceptance criterion. For NashSutcliffe efficiency likelihood measure, the SR value for interpolator is always found to be above 0.90 and about 0.10 higher than that of NN method (Figure 7B.4, upper panel). The SR value of the interpolator for Exponential efficiency likelihood measure appears to decrease moderately to 0.80 at the high acceptance criterion of EEXP> 0.60 (lower panel, Figure 6B.4), and become less than that of the NN method. However, for this case, the interpolator versus NN method difference is found to be small (less than 15%). Overall,
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when compared with the uniform random sampling, the interpolator is noted to be able to reduce the wastage of computational time due to non-behavioral runs in the ranges of 10% -70%. Table 7B.2 summarizes the mean values (of the 100 sub-divisions of the 500,000 sets) for BS values for the interpolator and NN method using the Nash-Sutcliffe efficiency as the likelihood measure. Similar statistics were observed for the Exponential efficiency likelihood measure, and is therefore not reported herein. The interpolator is moderately conservative (BS < 1.0) compared to the NN method in accepting a sampled parameter set as behavioral. This is not necessarily considered a drawback of the interpolator as it can be executed as many times as needed to generate the desired sample size of behavioral parameter sets. The more qualifying aspect is whether the interpolator exhibits regions of local attractions in the response surface that are inconsistent with the uniform random sampling (discussed next). In Figures 7B.5a and 7B.5b, certain calibration aspects of the interpolator and the NN method are explored for Nash-Sutcliffe and Exponential efficiency likelihood measures, respectively. The upper panels show the effect of the choice of calibration sample points for interpolator for three different criteria (selection of points based on a minimum Efficiency value of 0.3, 0.5 and 0.7). The lower panels show the effect of the ‘n’ –the number of nearest neighbors – in interpolating the likelihood value by the NN method for 6, 12 and 24 nearest neighbors. The choice of calibration points can have an impact on the efficiency (SR value) of the interpolator with the best performance achieved when the choice of points are highly behavioral (i.e., Emin>0.7). For NN method, the choice of n appears to have a negligible impact, although for both schemes, it
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is observed that the variability in prediction increases as the acceptance criterion increases. Furthermore, the sampling efficiency (in terms of SR) of the NN method appears to decrease in the moderate likelihood measure ranges (0.20.4. The plots represent an ensemble of 5000 parameter sets.
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Figure 7B.6b Same as in Figure 7B.6a, but for the interpolator.
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Figure 7B.6c Same as Figures 7B.6a and 7B.6b, but for the fifth NOAH-LSM parameter-
SMCWLT.
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Figure 7B.7a The GLUE uncertainty estimation of soil moisture simulation at 90%
confidence limits for uniform random sampling (upper panel) and interpolator (lower panel). Nash-Sutcliffe efficiency likelihood measure >0.4 was used as the acceptance criterion for behavioral parameter sets. Uncertainty estimation for each panel was conducted from the 5000 sampled sets shown in Figures 7B.6a, 7B.6b and 7B.6c.
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Figure 7B.7b Same as in Figure 7B.7a, but for Exponential efficiency likelihood
measure >0.4 as the acceptance criterion for behavioral parameter sets.
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Figure 7B.8 Response of Exceedance Probability to width of predicted confidence
limits. Left panel – Nash-Sutcliffe efficiency likelihood measure; Right panel – Exponential efficiency likelihood measure.
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8 CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER STUDY
8.1 Major Conclusions The major conclusions of the study are summarized as follows.
1. Use of radar scans closest to the ground for rainfall measurement offered runoff prediction uncertainty similar to that obtained from a dense gauge network. The principal sources of errors in radar rainfall estimation, such as mean field bias and VPR, magnified in the non-linear rainfall-runoff transformation as runoff prediction uncertainty. For mitigation of these error sources, combined adjustment procedures for MFB and VPR effect, such as those developed by Dinku et al. (2002), were found effective for reducing uncertainty of runoff predictions.
2. A probabilistic discharge prediction scheme formulated within the parameter uncertainty framework of GLUE yielded 50% less variability in predicting time to peak error than the conventional optimum parameter set approach. However, in terms of most probable hydrograph, the scheme yielded prediction accuracy similar to the deterministic scheme.
3. The Passive Microwave (PM) sensor’s 3-hourly rainfall sampling gave comparable flood prediction uncertainties with respect to the hourly sampling, typically used in runoff modeling. The runoff prediction error, though, was
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magnified up to a factor of 3 when rainfall estimates were derived from 6-hourly PM sampling intervals.
4. Current PM sampling was found to be associated with flood prediction uncertainty approximately 50%-100% higher than that of a canonical 3-hourly sampling planned for the Global Precipitation Measurement (GPM). It was found that a reduced standard error (0.90) can make IR retrievals useful in reducing uncertainty in the prediction of peak runoff. To reduce the error in time to peak, further improvement such as the reduction in the IR retrieval’s false alarm rates coupled with an even higher POD may be necessary. In terms of overall runoff volume, combined moderate improvements in POD and error variance of current IR retrieval algorithms were found sufficient for the reduction of prediction uncertainty.
5. Soil moisture prediction uncertainty was found to be larger for a more vegetated regime (compared to a less vegetated regime). IR rain retrievals at current performance level underestimated the total uncertainty. However, this underestimation was found to be a function of the level of modeling uncertainty.
6. The use of Latin Hypercube Sampling (LHS) technique for accelerating sampling of rainfall error structure revealed that the 80% and higher confidence limits in runoff simulation error could be predicted with the same degree of reliability as
247
the Monte Carlo method, but with almost two orders of magnitude fewer simulations.
7. A stochastic response surface method (SRSM) for accelerating sampling of GLUE model parameters was able to reduce computational burden of uniform MC sampling in the ranges of 10%-70%.
8.2 Major Limitations
Major limitations of the study are as follows:
1. Results from this study were limited to major floods resulting from lasting storms (>2 days) and saturation-excess runoff generation mechanisms for medium sized (~100 km2) watersheds. Hence, results from this study cannot be generalized to other scales and runoff generation mechanisms. In general, use of the same watershed (Posina) and hydrologic model (TOPMODEL) (in Chapters 2, 3, 4, 5 and 7A) limited the applicability of the conclusions drawn in this study to conditions significantly different from those found in the Alpine watershed.
2. The assessment of satellite (PM and IR) rainfall retrievals for runoff simulation was conditioned upon the assumption that TOPMODEL was deterministic in modeling the rainfall-runoff transformation at the watershed outlet. For such a semi-distributed and conceptual model this assumption should be relaxed for a
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more detailed physical understanding of the error propagation in flood prediction based on remotely-sensed rainfall.
8.3 Recommendations for Further Study
Some of the recommended extensions of the study conducted herein include: (i) the consideration of ground radar error structures and adjustment algorithms different from those used in Chapter 2; (ii) the implementation of the analysis framework for continuous hydrological simulation instead of event-based flood modeling described in Chapter 2; and (iii) examining the effects of different likelihood functions and parameter sampling ranges within the GLUE methodology.
Recent techniques have sought
improvements in IR rain estimation accuracy through pattern recognition of cloud features (Xu et al., 1999b), assimilation of lightning (Morales and Anagnostou, 2003; Chronis et al., 2004) and Passive Microwave information (Todd et al., 2001), to name a few. Similarly, PM retrievals continue to evolve regularly in terms of new overland techniques (Dinku and Anagnostou 2004; McCollum and Ferraro 2003) and improved sampling and resolutions as we transition to the GPM era. It is worthwhile, therefore, to understand the effect of scale on the precipitation measurement error from a host of single and multi-sensor retrievals, and study the nature of propagation in hydrologic prediction. Furthermore, short duration storms and larger size watersheds need to be studied to understand the temporal sampling problem and the aspect of spatial variability of satellite measurement error in runoff prediction (for Chapters 4 and 5).
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Although the work presented in Chapter 6 revealed the nature of combined uncertainty due to precipitation input/modeling error sources, there are certain aspects that justify further study. In order to reduce the uncertainty in IR retrievals, optimal merging with the less-frequent, but more accurate Passive Microwave (PM) rainfall estimates should be explored. Since PM estimates have more accurate detection of rain, it is expected that the use of PM or PM-IR merged estimate would allow the model to better predict the wetting up and drying down of soil. There are currently several LSM currently in use by Land Data Assimilation Systems. These LSMs need to be explored of their relative impacts and interactions of uncertainties with precipitation. For stochastic interpolation of the parameter response surface (Chapter 7B), notable extensions include: (i) application of the interpolator to other physically-based models and hydrologic variables within the GLUE framework; (ii) investigating the conditions or assumptions that give rise to a chaotic and non-chaotic behavior in the hydrologic system and thereby attempt to connect the relationship of the hydrologic variable to the order of polynomial chaos expansions; and (iii) investigating the effect of the dimensional size of the parameter hyperspace on the sampling efficiency of the interpolator. It has also been suggested that the gradient information of the parameters with respect to model output, when assimilated in the polynomial chaos expansion, an increase in the prediction accuracy of the interpolator can be expected (Isukapalli and Georgopoulos, 1999). Another potential use of the interpolator sampling scheme would be in applications to large-scale land surface simulations where model parameters are distributed as a matrix (2-D spatial domain) over large scales (>10,000 km2) (note: in this study the parameters were a vector). For such applications, research is needed to explore
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convenient ways to mathematically reformulate the interpolator to handle such distributed parameters in spatial format. Lastly, the most notable aspect worthy of further study is the expansion of the 1-D simulation of rainfall retrieval error (RSREM) to two-dimensional space. The current application was limited to only point simulation of the surface hydrologic processes (e.g. surface runoff and soil moisture). However, the most useful analyses for uncertainty and data assimilation techniques can be expected when the two-dimensional spatial aspects of rainfall retrieval error and the corresponding propagation are considered. RSREM therefore needs to be expanded to incorporate the full spatio-temporal variability of satellite rainfall retrieval error so that the error propagation can be studied as a 2-D random process.
8.4 Closing Statement
This study has shown that the use of the explicitly defined uncertainty tools for input and hydrologic prediction can offer an objective framework to improve upon the problem definition for questions such as, How confident is the flood prediction based on remote sensing rainfall estimates? Combined with the numerical sampling techniques that strive to achieve computational efficiency, the joint use of the uncertainty tools has allowed a more detailed statistical characterization of prediction uncertainty associated with the input and modeling error interaction that is the key to investigating potential improvements in remote sensing rain retrieval algorithms. In past studies, feedback has been negligible between the efforts at addressing hydrologic prediction uncertainty and
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the efforts to characterize uncertainty of remote sensing of rainfall. Within the limitations highlighted afore, this study attempted to provide this feedback. It is now hoped that results from this study will promote formulation of more reliable rainfall remote sensing algorithms for surface hydrologic applications (such as flood prediction). The statistical approach demonstrated herein is independent of hydrologic model and/or rain retrieval algorithm. Thus, it could be easily applied to study the prediction uncertainty of other hydrologic variables or remote sensing techniques.
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