Hydrological Predictability, Scales, and Uncertainty Issues
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The survival and well-being of human civilization depends on water. Human civilization is especially vulnerable to large variations in the water cycle such as flood and drought that disrupts food supplies and can cause havoc to day-to-day operations. Many of these extreme events have occurred in recent years including large droughts and extreme floods in many parts of the world. The looming threat of climate change has the additional potential to make the impacts of extreme water cycle events an even greater threat to society. The ability to have foreknowledge of these extremes in the water cycle can provide time for preparations to reduce the negative impacts of these extremes on society. Predictions of these extreme events require models of the hydrometeorological system, including all its associated uncertainties, and appropriate observations systems to provide input data to these models. Ensemble forecasts using statistical and physically based models that also account for forecast uncertainties have great potential to make the needed predictions of future hydrometeorological events.
This chapter discusses the basis for predictability, predictive scales, and uncertainty associated with hydrometeorological prediction. Although much uncertainty may be associated with some hydrometeorological predictions, ensemble forecasting techniques offer a way to quantify this uncertainty making it possible to have more useful predictions for decision makers and for the ultimate benefit to society.
KeywordsPredictability Uncertainty GCM ESP Spatial scales Temporal scales
The dependence on water for the survival and well-being of human populations has made an inseparable link between human society and the water cycle. In particular, human society is especially vulnerable to large fluctuations in the amount of available water that come in the form of flood and drought and disrupts food supplies and can cause havoc to day-to-day operations. Many such extreme events have occurred in recent years including large droughts and extreme floods in many parts of the world (Marchi et al. 2010; Karl et al. 2012; Villarini et al. 2013; Smith et al. 2013). Furthermore, the looming threat of climate change presents a very real possibility that the fluctuations of available water could intensify (Huntington 2006; Sheffield and Wood 2008) and have a larger impact on society. Although extreme variability in available water is a clear and present threat to society, it is not a modern peril. Since the very beginning of civilization, human beings have had to deal with the elements of nature for survival, especially water, with dynasties literally thriving or collapsing in response to when, where, and how much water came, stayed, and went into the societal living environment.
Skillful foresight of water flow would help not only mitigate the disastrous effects of damaging floods and droughts, but also provide a tool for managing water as a valuable resource. This is particularly relevant for managing residential use, irrigation, power generation, navigation, and environmental and ecosystem protection. Hydrometeorological forecasts predict the amount of water at specific locations and times. It is often associated with prediction of natural hazards such as severe storms, floods, droughts, landslides, and coastal inundation by storm surges.
The term hydrometeorological is derived from a combination of the words hydrology and meteorology, both key components to the earth’s water cycle. Hydrology pertains to the components of the water cycle that take place over land. This includes movement of water after it exits the atmosphere as precipitation and flows through the ground, lakes, rivers, and streams until it returns to the atmosphere through evaporation or flows into the ocean. Meteorology, on the other hand, deals with the transport of water within in the atmosphere. This includes evaporation from land and sea and movement of evaporated water in the atmosphere until it ultimately falls as precipitation. It is simple to see that the transport of water through the hydrologic system begins where the meteorological system ends and vice versa. This coupling between the hydrology and meteorology is what makes up the water cycle of the earth. The ability to predict the dynamics of this system is essential for any form of water prediction. Therefore, the term hydrometeorological forecasting includes prediction of both the hydrological and metrological components of the water cycle.
Hydrometeorological forecasting is usually done with the aid of process-based or statistical hydrometeorological models. These can be further categorized into meteorological models and hydrological models. Meteorological models are designed to mimic water and energy cycles in the atmosphere and over land. Hydrological models are designed to emulate water and energy cycles that occur over and within the land surface. Hydrometeorological models are generally run in two modes: (i) simulation mode, where hydrometeorological processes of the past are emulated; and (ii) forecast mode, where future hydrometeorological processes are predicted. The commonly simulated/forecasted variables include precipitation, air temperature, river stage or streamflow discharge, snow aerial coverage, snow water content, evaporation, soil moisture, groundwater storage and discharge, river sediment, and other variables with practical application to water systems including upstream anthropogenic activities that affect downstream flows.
Depending on the length of the forecast time horizon, known as lead time, hydrometeorological forecasting may be classified into short-, medium-, and long-range forecasting. Short-range forecasts have lead times of a few hours to a few days into the future. Medium-range forecasts have lead times of a few weeks. Long-range forecasts have lead times ranging from a few weeks to 1 year or more. From a meteorological perspective, short-range predictions are associated with weather and long range is associated with climate. Short- and medium-range forecasts predict how actual hydrometeorological states change in time and space, while the skill in long-range forecasts is in estimates of average hydrometeorological states expected over monthly or seasonal time periods over large areas.
Historically, hydrometeorological forecasts, especially short-range forecasts, have been made as deterministic, single-value estimates of the magnitude, location, and timing of likely hydrometeorological events. Two examples of deterministic hydrometeorological forecasts are: “the temperature will be 35 °C in Hong Kong tomorrow,” and “the streamflow discharge of Mississippi River at Vicksburg, Mississippi, will be 516,000cfs on Sunday.” Deterministic forecasts have been used in operational hydrometeorological forecasting centers around the world since before computer-based hydrometeorological models came to existence many decades ago. Even today, they are still widely used for short-range forecasting throughout the world. Deterministic forecasts can often satisfy many public needs.
Recently, forecast users are becoming increasingly aware of the limitations of deterministic, single-value hydrometeorological forecasts, especially under severe or extreme hydrometeorological conditions. As pointed out by Ramon Krzysztofowicz (2001), “A deterministic forecast may create the illusion of certainty in a user’s mind, which can easily lead the user to suboptimal action.”
One example involving forecaster lack of confidence in a single-value prediction that led to issuance of an incorrect warning that resulted in devastating consequences occurred during a torrential storm in Beijing, China. On July 21, 2012, the numerical weather prediction (NWP) model run by Beijing Meteorological Bureau predicted a major storm would occur over Beijing that day. The Beijing Meteorological Bureau only issued a “blue code” warning for Beijing before the storm that lasted from 9:00 am until 3:00 am in the next morning. This warning implied that over 50 mm of rainfall would occur over the city in the ensuing 12-h period. But the actual rainfall amount was over 170 mm for all areas of Beijing City, with downtown receiving 220 mm and over 460 mm in the Southwestern Fanshan District over the storm period. Because of this incorrect, underpredicted warning, inadequate emergency measures were taken by the city government, leading to a virtual shutdown of city traffic soon after the storm began, due to almost all highway underpasses flooded and all public transportation interrupted. A professional football game and a large music concert with tens of thousands of people in attendance went on as scheduled in the evening, compounding the traffic paralysis. The final casualty tally included 79 deaths and property damages in excess of ¥10B in Chinese Yuans.
The irony is that the forecasted rainfall from the numerical weather prediction (NWP) model run by Beijing Meteorological Bureau was closer to the actual rainfall amount than that indicated by the warning level. This severely underestimated warning was issued because the forecaster lacked confidence in the deterministic forecast of this extreme storm that later was determined to be a 1-in-60-year event. This is just one example of the failure of a deterministic hydrometeorological forecast. Reality is that they often fail, especially during extreme events, because a single model prediction lacks information about the certainty or uncertainty of what will actually occur.
In contrast to single-value deterministic forecasts, ensemble forecasts are made using Monte Carlo simulation in which a single model or multiple models are run numerous times to generate multiple samples of future states of the dynamical hydrometeorological system. Different model outcomes are generated by perturbing uncertain factors such as model forcing, initial and boundary conditions, and/or model physics. Ensemble techniques are attractive because they not only offer an estimate of the range of possible future states of the hydrometeorological system, but also offer a way to quantify the risks of a catastrophic hydrometeorological event occurring.
The goal of this chapter is to provide an overview of hydrometeorological ensemble forecasting in terms of understanding the history and rationale of ensemble hydrometeorological forecasting and uncertainty estimation. This chapter discusses major aspects of predicting hydrometeorological systems and provides a broad overview of the topics covered in this book. This chapter is a mile wide and only an inch deep. But references to other parts of this book will be given to facilitate more in-depth understanding of the concepts presented in this section.
This chapter has five remaining sections. Section 2 provides a brief history and application of hydrometeorological forecasts and is followed by an introduction to the nature of prediction in its many forms and subtleties in Sect. 3. Aspects of prediction discussed in Sect. 3 are then applied to hydrometeorological predictability through discussion of the rationale for hydrometeorological prediction in Sect. 4. Section 5 discusses the components of an ensemble hydrometeorological forecast system emphasizing the uncertainties and skill in hydrometeorological prediction. This is then followed by a brief summary in Sect. 6.
2 History and Application of Ensemble Hydrometeorological Forecasting
Ensemble approaches to scientific understanding can be traced to the emergence of Monte Carlo simulation as a practical way to solve complicated computational problems. The term “Monte Carlo simulation” was coined by scientists working on the Manhattan Project in Los Alamos, New Mexico, in the 1940s, who used random numerical techniques to obtain solutions for highly complex fluid dynamical computational problems. Monte Carlo simulation is, in a way, similar to many games popularly played in the gambling city Monte Carlo in Europe. Today, Monte Carlo simulation usually refers to the use of random number generation together with numerical models to obtain approximate solutions to complicated computational problems.
Edward Lorenz’s discovery of the chaotic nature of the atmospheric system in 1965 provided theoretical justification that ensemble hydrometeorological forecasting is the only practical way to deal with the uncertainty in the atmospheric initial conditions and the inherent nonlinearity of the atmospheric system. Edward Epstein recognized in 1969 that the atmosphere could not be completely described with a single forecast run due to inherent uncertainty, and proposed a stochastic dynamic model that produced means and variances for the state of the atmosphere (Epstein 1969). Although these Monte Carlo simulations showed skill, in 1974 Cecil Leith revealed that they produced adequate forecasts only when the ensemble probability distribution was a representative sample of the probability distribution in the atmosphere (Leith 1974).
Epstein presented a theoretical Stochastic-Dynamic approach, which adds stochastic terms to dynamical equations to account for various uncertainties (Epstein 1969). However, his approach was not practical in operational applications as the number of model equations is too many to be implemented numerically given the limited computation resources at the time. Later, Leith proposed a Monte Carlo approach as a practical implementation of Epstein’s Stochastic-Dynamic approach, which represented the first attempt at ensemble forecasting (Leith 1972). In this approach, a hydrometeorological model was run in an ensemble mode. In the 1970s, only the uncertainty in initial condition was represented with only a few ensemble members due to the computational constraint. As computational power has grown exponentially over the years, uncertainty in initial condition, model dynamics, model structure, and even model parameters are often considered simultaneously in ensemble forecasting today. For example, ensemble Kalman filter (ETKF) has been used to provide probabilistic estimate of both the state variables as well as model parameters in many meteorological applications (Cornick et al. 2009; Evensen 2009).
It was not until 1992 that ensemble forecasts began being prepared by the European Centre for Medium-Range Weather Forecasts and the National Centers for Environmental Prediction (Molteni et al. 1996; Toth et al. 1997). The ECMWF Ensemble Prediction System uses singular vectors to simulate the initial probability density, while the NCEP Global Ensemble Forecasting System uses a technique known as vector breeding. Now ensemble forecasting is an essential and routine component of the suite of meteorological forecast products for many countries.
Realizing the importance of ensemble forecasting, the World Meteorological Organization (WMO) sponsored a 10-year international program in 2003 – The Observing System Research and Predictability Experiment (THORPEX) to accelerate improvement in ensemble meteorological forecasting for lead times ranging from 1 day to 2 weeks. As part of the THORPEX, ten numerical weather prediction centers around the world have provided their medium-range ensemble meteorological forecasts to THORPEX Interactive Grand Global Ensemble (TIGGE) database since the end of 2006.
From a hydrological perspective, ensemble forecasting strategy has been used since the 1970s. The U.S. National Weather Service (NWS) produced Extended Streamflow Prediction (ESP) by using historical temperature precipitation data from different years to represent future precipitation and temperature forcing (Day 1985). Climatological ESP has been used extensively in reservoir and water supply operations applications. US nation-wide ensemble hydrological forecasting began in the late 1990s with the advent of the Advanced Hydrologic Prediction Service (AHPS) that uses climatological ESP. At approximately the same time, ECMWF began to produce hydrological ensemble forecasts by making use of meteorological ensemble forecasts from NWP models.
In the 2000s, the NWS California Nevada River Forecast Center (CNRFC) began issuing short-range hydrologic ensemble forecasts that used preprocessed single-value precipitation and temperature forecasts from atmospheric models and from operational hydrometeorological forecasters. In 2010, NWS began implementing a national Hydrological Ensemble Forecast System (HEFS) that included atmospheric forecast preprocessing techniques to provide reliable forcing, at both short range and long range, for hydrologic forecast models at the space and time scales required by these models.
A major driver for global adoption of ensemble hydrological forecasting is the founding of the Hydrological Ensemble Prediction Experiment (HEPEX) in 2004. The main objective of HEPEX is to bring the international hydrological community together with the meteorological community to demonstrate how to produce reliable “engineering quality” hydrological ensemble forecasts that can be used with confidence to assist the emergency management and water resources sectors to make decisions that have important consequences for the economy and for public health and safety.
3 Understanding Prediction and Models
To fully understand ensemble hydrometeorological prediction, it is first necessary to grasp the basics of predictions and models. The Webster online dictionary defines prediction as “a statement about what will happen or might happen in the future” (Merriam-Webster). Predictions can take many forms and be founded on any reasoning. But predictions are only useful if they are made from a sound understanding that provides a strong base for the prediction. Often, the basis of the prediction is called the premise of predictability. The premise is the rationale for the prediction and ultimately limits the potential success of the prediction. Regardless of the premise, prediction usually takes one of two forms: deterministic or probabilistic.
Deterministic prediction gives a single outcome of the thing being predicted. In many societal applications, a deterministic prediction is wanted in order to make a decision that is related to the event being predicted. If the prediction were perfect, then a deterministic statement would provide all the information needed to make an informed decision. But there are very few perfect predictions. Therefore, it becomes important to account for the uncertainty in the prediction. A probabilistic prediction does not give a single outcome, but provides a range of possible outcomes and an estimate of the likelihood of their occurrence. Although probabilistic statements do not provide single outcomes, they enable informed decisions based on the additional information about the predictive uncertainty.
The importance of predictability and the difference between deterministic and probabilistic predictions can be illustrated through the simple example of flipping a coin. Consider, for example, a coin toss where a competitor predicts before the toss which side of the coin (heads or tails) will come out upright. If that happens, they win; otherwise, they lose. In such situations, it is in the person’s best interest to be able to correctly predict the outcome from flipping a coin to ensure they receive the maximum benefit or avoid the penalty.
In this example, the competitor’s choice of heads or tails is an example of a deterministic, single-value prediction. Alternatively, the statement that there is a 50% chance of heads and a 50% chance of tails would be an example of a probabilistic prediction for the same event. The probabilistic prediction does not give a direct outcome, but gives an assessment of all outcomes. The competitor must then make their own deterministic prediction, but the competitor’s choice can be informed by the probabilistic prediction.
The basis or premise of predictability of the probabilistic prediction is key for the ultimate utility of the prediction. For example, the 50-50 chance of the coin flip outcome could be dictated by the event space and the assumption that each side of the coin is equally likely. Although this assumption of a fair sided coin is reasonable for this simple example, a prediction could be refined by observations and incorporating these observations into a simple statistical model. An example of developing such a model would be to flip the coin multiple times and recording the outcome, with the outcomes being used to calculate the relative frequency of each outcome. For example, if after four coin flips there was one head and three tails, the probabilistic prediction of all future coin flips would be 25% heads and 75% tails. For this prediction, the premise of predictability would be based on four observed experiments. Although this is an assessment of all possible outcomes, it would not be considered statistically robust given the few samples used to derive the prediction. On the other hand, if the same prediction of 25% heads and 75% tails were based on 1000 samples, this would be considered a statistically robust prediction (but unlikely one) and would be a clear indicator of a biased coin. Such foreknowledge about the biased coin would provide considerable advantage to the competitor in this example.
Statistical models based on observations are not the only way to estimate the likelihoods of the eventual out comes of an event. Physically based models based on the underlying physics are also very useful. An example of a physically based model for the coin flip outcomes would be a mathematical model that numerically simulates the rotation of the coin and its interaction with the surrounding environment throughout the trajectory of the coin: based on properties of the coin itself, like its size and weight as well as the initial height of the coin and the magnitude and location of the force exerted on it. The premise of predictability for such a model would be that there is enough information about the properties of the coin and physical representation of its environment that could be represented mathematically to understand the outcome of flipping a coin. Using this model for a single prediction would be an example of a deterministic prediction. But the only way this model would consistently give the correct outcome of flipping a coin would be if the physics and inputs in the model where exact. Because the mathematical formulas and measurements for a coin toss simulation would be imperfect, there is little chance a single run of the model would provide more utility than a simple guess. In this case, the inability of the model to provide useful information about the coin flip would come from the uncertainty in model input and model physics that would likely overwhelm any predictive power of knowing the properties, initial height, or force exerted on the coin.
Additional model runs provide a more robust assessment of the possible outcomes than a single realization. Different model inputs based on uncertainties could be used to generate multiple model outcomes that could be combined to give a probabilistic description of the event. Multiple runs of a model form an ensemble and are necessary when making predictions using models that have underlining uncertainties.
4 The Rationale for Hydrometeorological Forecasting
Predicting the hydrometeorological system is similar to predicting the result of flipping a coin in that statistically or physically based numerical models can be used to make deterministic or probabilistic forecasts, but unlike flipping a coin, which only has two outcomes, the hydrometeorological system has many outcomes. Mathematical and statistical models used to predict the hydrometeorological system are extremely complex. Although they incorporate numerical representations of the physics that drive the water cycle, there are still many uncertainties in the inputs, parameters, and models themselves. Therefore, it is essential that these models use ensemble techniques to provide the maximum amount of information about the hydrometeorological system. But even with ensemble techniques, any prediction is only as good as the basis for prediction. This section discusses the premise of predictability in hydrometeorological models and outlines the rationale for ensemble hydrometeorological forecasting.
In this figure, short-range predictions are for the period ranging from zero out to about 15 days. Hydrometeorological predictability in the short range is impacted by the initial hydrologic conditions, i.e., the current state of the hydrologic system. However, short-range hydrometeorological predictability also greatly depends on the predictability of the atmosphere. Short-range predictability of the atmosphere is primarily driven by atmospheric initial conditions whose effect is generally considered to be limited to the first 15 days (Lorenz 1969, 1982). Predictability of the atmosphere is strongest in the first few days when predictions can be very precise and are often applied to small basins and at subdaily time-steps (de Roo et al. 2003). Because of the high level of predictability at short lead times, short-range predictions are particularly applicable to flood events that can develop quickly at specific locations. Short-range predictions of low flows can also be useful for other purposes such as controlling reservoir operations to ensure ecological habitat.
Climate hydrometeorological predictions are for time scales ranging from about 2 weeks to many months in the future or longer. At these time scales, the main basis for atmospheric predictability is the influence of boundary conditions (land and sea) on the atmosphere. At climate scales, slowly varying atmospheric boundary conditions influence the transport of water in the coupled ocean-atmosphere-land system. Land and sea surface boundary conditions both contribute to predictability at the climate scale, although the sea surface plays a large role at longer lead times (Palmer and Anderson 1994; Goddard et al. 2001).
Hydrologic initial conditions also play a critical role in hydrometeorological predictions at climate scales, but their role diminishes rapidly with forecast lead time (Li et al. 2009). One major exception is the influence of snow. Snow water initial conditions can be a basis for long-range hydrologic predictability in spring and summer for areas with substantial snow accumulation.
The precision (i.e., level of detail) of climate scale predictions is usually much coarser than short-range predictions. This is because atmospheric predictions are influenced by atmospheric noise and require temporal and spatial averaging in order to detect the predictive signal. Therefore, hydrometeorological predictions at climate time scales are most applicable for large basins on seasonal or longer timescales. The coarser level of detail of climate scale predictions make them most applicable to drought that typically forms over long time periods and large areas. Although droughts are the main use of climate scale hydrometeorological predictions, the ability to predict anomalously wet time periods at climate scales is also beneficial.
Hydrometeorological predictions for both short- and long-time scales are typically made as separate atmospheric and hydrological predictions. Because forecast uncertainty is important, these predictions, both atmospheric and hydrologic, are often made as ensemble predictions with ensemble member time series having daily, or possibly, subdaily time steps.
The remainder of this section discusses some examples of the premise of hydrometeorological prediction. First, the premise of predictability of the hydrologic system due to initial conditions is discussed and is followed by a discussion of the predictability of the atmosphere and the combined impact of hydrologic and atmospheric predictability. Examples of hydrometeorological prediction given in this section are for the climate scale. However, similar methods and applications could easily be demonstrated for short-range forecasts. For a showcase of other applications of ensemble hydrometeorological predictions, see Part 10 of this book .
4.1 Hydrologic Predictability
Hydrologic predictability depends on initial hydrologic conditions such as soil moisture, groundwater, snow and current streamflow, as well as on atmospheric predictability because hydrologic processes depend on precipitation and temperature forcing as well as initial conditions.
To illustrate how initial hydrologic conditions affect predictability, consider the example of a large rain event over a basin. If the previous month before the rain event were abnormally dry, then soils would be dry and baseflow would be low. Under these conditions, a heavy rain event would result in high infiltration rates and a smaller amount of direct runoff. The small amount of direct runoff would combine with low but increasing baseflow to produce the total discharge that would not likely cause extreme flooding. In contrast, if conditions prior to the heavy rainfall event were abnormally wet, then the basin would have wet soils at or near saturation and a larger initial baseflow. In this scenario, the same storm event would produce low infiltration and a large amount of direct runoff. Combining the large amount of direct runoff with the initially high baseflow could result in a very large discharge that might induce extreme flooding. This heavy rain event could lead to even more extreme conditions if snow were present. This could produce even more streamflow and would be called a “rain on snow event.”
This example illustrates why knowing initial hydrologic conditions is a crucial part of hydrometeorological prediction. This requires measuring or estimating soil moisture, snowpack, and streamflow throughout the basin. These variables are routinely measured in some locations. But it is not feasible to make such measurements everywhere hydrologic predictions are needed. For this reason, hydrologic models have been developed to provide temporally and spatially continuous estimates of initial hydrologic conditions as well as hydrologic forecasts.
Hydrologic models can use a variety of methods to transform atmospheric forcing variables (precipitation, temperature, humidity, radiation) into estimates of soil moisture, snowpack, runoff, and streamflow. When observations or estimates of atmospheric forcing are used to drive hydrologic models, this is often referred to as offline modeling, because the atmospheric model is not coupled to the hydrologic model. Traditionally, hydrologic forecast models have been used in an off-line mode for hydrologic forecasting.
Hydrologic forecast models can be applied in several different modes. One is the forecast mode where a hydrologic forecast model is used to make hydrologic forecasts for some lead time. Another is a simulation mode where a hydrologic model is run, possibly for many years, using historical observations. Short-term simulations run in parallel with forecast operations and provide initial hydrologic conditions for forecast runs. Long-term simulations give information about how well a hydrologic model can predict streamflow when input forcing is based on observations. These simulations can be used to make estimates of hydrologic model error. Finally, hydrologic models can be used in a reforecast mode to generate hydrologic reforecasts from atmospheric reforecasts. Such reforecasts can be used to develop and test water management and flood control models. For a full treatment of hydrologic models and their use in forecasting, see Part 4 of this book.
Most regions have large seasonal variability in the hydrologic cycle that can further inform hydrologic forecasts. A forecast technique that uses atmospheric climatology together with estimates of initial hydrologic conditions is known as Extended Streamflow Prediction or ESP (Day 1985). This technique has been used by the U.S. National Weather Service since 1974 (shortly after the first continuous hydrologic simulation models became available and began to be used operationally by NWS).
ESP was first implemented in the western US to make seasonal snowmelt runoff forecasts as an alternative to existing regression-based statistical forecasts of seasonal snowmelt runoff volumes. It is the earliest known example of ensemble prediction being used for operational streamflow forecasting. The original, climatological ESP uses initial hydrologic conditions estimated by running the forecast model forward each day using the previous day’s initial conditions together with recent observations of atmospheric forcing to generate continuing day-to-day updates of initial hydrologic conditions. It uses future atmospheric forcing selected from the climatological record. Different ensemble members are generated from precipitation and temperature time series taken from different past years of the climatological record beginning on the current forecast day of the year.
ESP has more utility than either a simple persistence model or a single-value hydrologic forecast model. If a single-value forecast were wanted, that could be derived as the ensemble mean. The dark blue line in Fig. 3 is the ensemble mean for the ESP forecast. The ensemble mean includes variability that is associated with the seasonal variability of the atmospheric forcing. Climatological ESP forecasts tend to the climatology of the streamflow as the influence of the initial condition dissipates. This limits the predictability of the climatological ESP method at longer forecast lead times. Despite this limitation, the climatological ESP method is still widely used in operational hydrologic forecasting .
4.2 Atmospheric Predictability
Atmospheric predictability depends on two factors: the internal dynamics of the atmosphere and the variability of atmospheric boundary conditions (with land and sea surfaces). It also may depend on knowledge of past climatology. A lower limit of atmospheric predictability is a prediction that the future will be like the past climatology.
How internal atmospheric dynamics limits atmospheric predictability was studied by Edward Lorenz using a toy model of the atmospheric system. Lorenz made a seminal discovery that equations describing the atmospheric system exhibit chaotic behavior, in which a minute perturbation in the initial condition would result in dramatically different future states (Lorenz 1965). This effect, known as the “Lorenz attractor” or “butterfly effect,” implies that deterministic forecasts of atmospheric events beyond a few days are not possible because the prescribed initial condition used to initialize a forecast cannot be certain, regardless how accurate the observations may be.
This does not mean that atmospheric forecasts beyond a few days are totally impossible. By accounting for uncertainty in initial conditions, it is possible to provide a probabilistic estimate of future atmospheric states for lead times up to 2 weeks (Froude et al. 2013). An ensemble approach is well suited to represent uncertainty in initial conditions. This can be achieved by running models in an ensemble mode with different initial conditions sampled from a probability distribution function (PDF) of initial conditions. Because of the predictability limit of the atmospheric system due to chaos (Straus and Shukla 2005), deterministic details of hydrometeorological events cannot be forecast beyond 2 weeks. Accordingly, long-range hydrometeorological forecasts are generally produced as probabilistic forecasts of events or as ensemble time-series forecasts (Hoskins 2013).
Predictability of the atmosphere beyond 2 weeks comes from models called coupled General Circulation Models (GCMs) that include the dynamics of ocean and land boundary conditions as well as the physics of the atmosphere and uncertainty in initial conditions. Accordingly, the premise for predictability of the atmosphere has two parts, weather time-scales and climate time-scales. As shown in Fig. 1, weather scale predictions of the atmosphere is generally considered to be the first 15 days and the predictability primarily lies in the initial conditions of coupled ocean-atmosphere-land system. As the influence of initial atmospheric conditions deteriorates, boundary conditions become increasingly important for maintaining predictability of the atmosphere. The influence of the land boundary conditions on the predictability of the atmosphere can be important at all time-scales but is strongest for longer weather scales and shorter climate scales. For very long climate scales, the predictability of the atmosphere is primarily driven by sea surface boundary conditions and by ocean dynamics that affect the sea surface. Atmospheric predictions increasingly rely on ensemble techniques using multiple realizations generated by numerical models.
As atmospheric models were improved and began to be run in ensemble mode, statistical techniques were developed to use atmospheric precipitation and temperature forecasts to specify future precipitation and temperature forcing for hydrologic forecast models. Atmospheric model forecasts were not used directly as input to hydrologic models because: (i) atmospheric forecasts are for much larger spatial scales than hydrologic models; and (ii) the climatology of atmospheric forecasts is different than the climatology of the atmospheric forcing needed by the hydrologic forecast models. Despite these differences, atmospheric forecasts still have useful information about future hydrologic forcing. Therefore, statistical techniques were developed to account for uncertainty in the relationship between the atmospheric forecasts and the temperature and precipitation events that actually occurred over the local hydrologic forecast areas.
The first operational statistical processing techniques to generate ensemble forcing for hydrologic ensemble forecast models from the atmospheric forecasts were developed by the U.S. National Weather Service (Schaake et al. 2007). This required a record of past atmospheric reforecasts using the current atmospheric forecast model (Hamill et al. 2007). The first atmospheric reforecasts became available in the early 2000s from an ensemble forecast version of the NWS Global Forecast System (GFS). Only GFS ensemble mean forecasts were used in the statistical processing techniques (MEFP) that are part of the National Weather Service Hydrologic Ensemble Forecast System (HEFS).
5 Ensemble Hydrometeorological Forecasting and Uncertainty
Ensemble hydrometeorological forecasting systems are complex and consist of many components that are imbedded with various levels of uncertainty. Different systems will account for different levels of uncertainties, which have direct impacts on the usefulness of the predictions. For example, there is a large amount of uncertainty in the persistence model prediction (Fig. 2). But it is not quantified, which limits the usefulness of the prediction. One of the major benefits of using ensemble predictions is that it provides a means to quantify uncertainty. The climatological ESP method (Fig. 3) quantifies the uncertainty of the evolution of initial hydrologic conditions by accounting for some aspects of the uncertainty of the atmospheric forcing. Similarly, the GCM-based hydrometeorological ensemble prediction (Fig. 4) also accounts for uncertainty in future atmospheric forcing of the hydrologic forecast model and provides a prediction with smaller uncertainty bounds than the climatological ESP for the first few months. Since there likely will always be uncertainty in hydrometeorological predictions, it is necessary to make sure all aspects of uncertainty are accounted for. If all uncertainty is not accounted for, then there is a risk of the forecast being over confident. This appears to have happened for the climatological ESP and GCM predictions in Figs. 3 and 4 for the forecasts of months 5 and 6. This illustrates that although both the ESP and GCM-based forecasts account for some aspects of uncertainty in the prediction, other aspects of uncertainty are not completely represented.
5.1 Input Uncertainty
5.1.1 Temporal Skill in GCM Short-Range Precipitation Forecasts
Forecast skill in short range GFS precipitation forecasts is temporally scale dependent. This is illustrated in Fig. 6 for winter season precipitation events in the North Fork of the American River in California. Figure 6 is an example of how forecast skill for events of increasing duration can exceed forecast skill of shorter duration events.
The measure of forecast skill used in Fig. 6 is the Pearson coefficient of correlation between GFS ensemble mean forecasts and observed values, including all zero values, for all forecasts made during a 30-day window on either side of January 15 between 1979 and 2002. The lowest curve in this figure is the correlation between 6 h forecasts and observed values as a function of the valid time (in future 6 h periods) of the 6 h forecast. The correlation values tend to decrease with lead time and a small daily diurnal cycle persists throughout the 14-day total forecast period. The highest curve in this figure is the correlation between the average forecast and observed precipitation values where the duration of the average begins at forecast creation time and ends at the indicated validation time. This curve shows that the GFS-based skill in predicting the total precipitation to be expected for the next 14 days is the same as the skill in predicting the amount of precipitation that will occur in the first 6 h. Clearly most of the information about future precipitation in this example is in the average or accumulated amount of precipitation over various future time periods. GFS forecast skill behaves this way in this location because most of the winter precipitation along the west coast is caused by large-scale storms that may last for several days. It is very difficult to predict the magnitude, timing, and exact location of every burst of precipitation during these events. A major source of error is temporal phase error: precipitation may occur before or after the time it was predicted. But averages over time, smooth the effect of these phase errors .
5.1.2 Analysis of Short-Range GFS Precipitation Forecast Skill in the US
This section provides a general overview of the skill of GFS-ensemble mean precipitation forecasts for different forecast lead times at different times of the year and at different locations throughout the US. The measure of forecast skill selected for this study is the Pearson coefficient of correlation between observations and GFS ensemble mean forecasts. Data presented here are for 24 basins distributed across the US. The forecast variable chosen for this study is the cumulative precipitation amount for different numbers of days in the future. This variable was chosen because the forecast skill in future cumulative precipitation amounts often tends to increase with time from the forecast creation date before it tends to decrease.
5.1.3 Analysis of Seasonal CFS Precipitation Forecast Skill in the US
This section provides a general overview of the skill of CFS seasonal precipitation forecasts for different forecast lead times at different times of the year and at different locations throughout the US. The measure of forecast skill used is the Pearson coefficient of correlation between observations and GFS ensemble mean forecasts. Data presented here are for 24 basins distributed across the US.
5.1.4 Analysis of Uncertainty in GCM Seasonal Forecasts
At location (a), the correlation is the strongest in January followed by October and the correlation is weakest during April and July. Location (b) has the highest skill in July followed by April and January. The correlation is weak in October. This demonstrates large differences in GCM predictability of precipitation with only relatively small changes in forecast spatial location. Such seasonal differences in GCM skill will cause seasonal variability of skill in hydrometeorological forecasts that utilize the GCM.
In addition to seasonal differences in predictability, the two locations also exhibit differences in skill at different temporal and spatial scales. For example, at location (a), for January there is a clear decrease in skill from 15 days to the 1 month forecast. Furthermore, the highest correlations are at large temporal scales 0–6 months and at large spatial scales (17°). In comparison, location (b) has the highest correlations in July of which the strongest correlation occurs at the finer spatial and temporal scales. This behavior is flipped in April when the correlation is low at finer temporal and spatial scales and is higher at coarser spatial and temporal scales.
The variation of correlation with spatial scale and season is likely connected with the attribution or the premise of predictability for location and time of year. For example, the predictability for location (a) in January is likely linked to slowly varying processes that impacts larger areas. This would be consistent with the predictive signal being related to spatial and temporal averaging. The obvious connection would be the slowly varying sea-surface boundary conditions in the GCM. In contrast, the acute predictability in July for location (b) would indicate that there is a more local aspect to its attribution. This could be due to the land boundary conditions in the GCM and the predictability they provide. This illustrates the importance of understanding how location, season, and scale influence GCM forecasts that subsequently influence hydrometeorological forecasts. Furthermore, aspects of location, season, and scale have implications for the application of these predictions. For example, the precipitation forecast in Fig. 11 for location (a) would be more suited for long-term drought prediction as opposed to short-range flood forecasts. In contrast, in July for location (b), there is greater potential for application to short-term flood forecasting. It is also evident there is little benefit for using the GCM for short-term forecasts at location (b) for January and April and seasonal prediction at location (a) for April and July. It is important to recognize that these results are specific to these locations and to this GCM. Other forecast models and locations could have very different predictive characteristics .
5.2 Hydrologic Model Uncertainty
Another major source of uncertainty in hydrometeorological predictions is associated with the hydrologic forecast models. An example of hydrologic model uncertainty can clearly be seen in Figs. 2, 3, and 4 by comparing the hydrologic model simulations (that use observed precipitation forcing) and the gauge observations. In this example, the hydrologic model is well correlated with the gauge observations, but there is a clear bias that varies with season. Although not shown here, there are other seasons and locations when correlations between the model simulations and the gauge observations are very low. Since these hydrologic model simulations incorporate observations of atmospheric forcing, these results are not limited by predictability of the atmosphere. Some of the uncertainties in the hydrologic model simulation shown in Figs. 2, 3, and 4 are associated with the atmospheric observations used to drive the hydrologic model as well as uncertainties in the hydrologic model itself.
Given these difference between hydrologic model simulations and observations, there are several ways to improve the predictions. One way is to calibrate the hydrologic model against observations. Hydrologic model calibration involves changing key model parameters to better match the observations. This can be done manually, but is often most effective when using an optimization algorithm that searches parameter space to find optimal parameter sets. Due to model complexity and structural model errors, usually no single parameter set gives the optimal solution. Instead, several parameter sets provide similar levels of consistency between the model and observations. These parameter sets can be used to run an ensemble of hydrologic model predictions using the same initial conditions and atmospheric forcing in order to quantify the uncertainty of the prediction associated with the estimation of the model parameters. Calibrating a model can be challenging due to the large amount of computations to effectively search the parameter space and requires a good observational data set. Part 5 of this book discusses model calibration and addresses many of the challenges with estimating model parameters and provides case studies of model calibration.
Even after calibration, there can still be local biases and errors between the hydrologic model simulation and observations. These errors can often be reduced by postprocessing the hydrologic prediction. Postprocessing requires a record of observations as well as a record of hydrologic simulations. Instead of changing model parameters, the observations and simulations are used to create a statistical relationship between the model and the observations. This relationship can remove bias in the predictions and can be used to generate ensemble traces of possible values of the observations to account for uncertainty in the predictions. Postprocessing of hydrometeorological predictions is discussed in Part 7 in this book.
The other important input to the hydrologic model is initial hydrologic conditions. This is very important for prediction but much uncertainty is associated with it. Key elements of initial hydrologic conditions include soil moisture, snow, and streamflow. There are techniques for measuring or estimating all of these variables through both in situ and remotely sensed methods. Although there is a perception that observations are perfect, there is uncertainty associated with them due to measurement methods and the fact that these measurements are not actual measurements of initial values of model state variables. Therefore, the best estimates of hydrologic initial conditions combine all observations with model estimates and information about the model errors and observational uncertain using data assimilation techniques. This can provide temporally and spatially continuous estimates of initial hydrologic conditions that incorporate uncertainty in the estimates. Part 6 of this book discusses data assimilation in hydrometeorological ensemble forecasting .
Although there are different models and methods for making ensemble hydrometeorological predictions, not all are equal in terms of their ability to provide reliable and skillful predictions. So, it becomes important to identify the best models and methods for each application (Roundy et al. 2015). Forecast verification is an important part of hydrometeorological prediction. There are many ways to verify forecasts. They range from simple deterministic measures to complex metrics that incorporate uncertainties. Many studies have compared ensemble hydrometeorological forecasts both from a climatological ESP approach and GCM-based atmospheric predictions (Mo et al. 2012; Yuan et al. 2013). These studies show that the GCM-based predictions provide some benefit over climatological ESP, but the skill of GCM-based forecasts varies in space (with location) and time of year. Part 8 of this book discusses forecast verification techniques and their communication.
5.4 Forecast Products and Services
Preparedness and response actions of emergency management authorities and the general public are highly dependent on the availability and dissemination of timely, skillful, and reliable hydrometeorological forecasting information. The usefulness of forecasts depends on how much confidence the forecast user has in them. Deterministic single-value hydrometeorological forecasts lack the uncertainty information that is needed for formulating proper actions. On the other hand, ensemble hydrometeorological forecasts offer numerous potential benefits (Krzysztofowicz 2001): (1) they are scientifically more “honest” than deterministic forecasts as they contain uncertainty information, which allows the forecast user to take risk information into account and make rational decisions; (2) they enable risk-based criteria for issuing disaster watches and warnings and for formulating emergency responses that are based on explicitly stated detection probabilities; and (3) they bring potential economic benefits of forecasts to society as a whole, which are achieved by initiating the necessary preventive measures or avoiding unnecessary overreactions to potential disasters. As the skill in hydrometeorological forecasting increases, society will continue to reap rich benefits.
One specific example of application of hydrometeorological ensemble forecasts is through the regulation and control of reservoirs. Reservoirs serve multiple purposes, including flood protection, electricity generation, water supply, recreation, and environmental and ecological protection. Skillful and reliable hydrometeorological forecasts are becoming increasingly important for reservoir operations to improve their socioeconomic, environmental, and ecological values. Deterministic single-value hydrometeorological forecasts are not suitable for developing reservoir operations rules, which have generally been developed in an ensemble (i.e., probabilistic) framework using ensemble inputs to drive the reservoir operation models. Traditionally, a reservoir operation rule was developed by treating historical hydrometeorological data from different years as ensemble inputs. With the availability of real-time ensemble forecasts based on NWP and climate models, reservoir operation is an area well positioned to take advantage of the predictive uncertainty information in ensemble hydrometeorological forecasts. Krzysztofowicz showed that the economic gain from a probabilistic temperature increases with the error in deterministic forecast (see Fig. 1.1 in Krzysztofowicz 1983). Stalling performed an evaluation which showed the economic benefit of hydrological forecasts in reservoir operation exceeds $1B per annum (Stallings 1997). A challenge today is to develop improved operating rules that use ensemble hydrologic forecasts in ways that also satisfy constituent expectations for system operation .
Continuous hydrometeorological prediction of the water cycle is driven by estimates of initial hydrologic conditions and atmospheric forecasts of forcing variables such as precipitation and temperature. The premise of predictability for hydrometeorological predictions relies on the initial state of the atmosphere and the prediction of the land and sea boundary conditions as well as the initial state of the hydrologic system. Hydrometeorological models based on physical processes can be used to make hydrometeorological forecasts. But these predictions have strong variability in their skill depending on scale, season, and location as well as other factors associated with model parameterization, model structural errors, and input uncertainty.
There are many sources of uncertainty in hydrometeorological forecasting including uncertainties associated with forcing, initial and boundary condition, model structure and parameters, and observational datasets. A schematic illustrating an ensemble hydrometeorological forecasts system and its major uncertainties is presented Fig. 5. These uncertainties are introduced at different stages in the forecasting process and propagate through the model, and eventually are manifested as uncertainty in final forecast products. It would be ideal to be able to account for these uncertainties in the model equations, solve those equations numerically, and render a probabilistic forecast.
Forecast uncertainty can be accounted for by using an ensemble of hydrometeorological models with multiple parameter sets to account for parameter estimation and multiple hydrometeorological models to account for model structure and errors, incorporating in situ and remotely sensed observations with the off-line model simulations to account for the initial hydrologic state and utilizing observed climatology or predictions from GCMs to account for the uncertainty associated with the atmospheric forcing. It can also include ensemble postprocessing techniques to improve the reliability of ensemble forecasts. Combining all these methods to estimate the uncertainty associated with the hydrologic prediction could result in a large number of ensemble members that would be computationally intensive and could result in very large uncertainty bounds.
The latest trend includes grand ensemble forecasting, i.e., the ensemble of ensemble forecasts generated by different weather or climate models. Many ensemble strategies are emerging. These range from “poor-men’s ensemble,” a simple combination of all deterministic forecasts from multiple models, to ensemble forecasts from multiple models, to “super-ensemble,” in which ensemble members are taken from ensemble forecasts from different models using a regression approach to favor members with higher correlation to observations. More recently, other grand ensemble strategies include a Bayesian model-averaging approach, which creates weighted probabilistic forecasts from the probabilistic forecasts generated by individual models. The weighting is assigned based on the likelihood of a model being correct in representing the real world. In the end, the most efficient way of dealing with uncertainty in hydrometeorological forecasting is through ensemble techniques. Finally, another strategy to improve hydrologic ensemble predictions is to postprocess individual hydrologic ensemble members. That approach was used to produce reliable ensemble inflow forecasts for the New York City water supply “Operations Support Tool.” This is discussed in Part 9 of this book.
Although uncertainty will likely never completely disappear, reduction of uncertainty through model improvements, better estimation of initial hydrologic conditions, and improvements in GCM forecast and their use in hydrometeorological models will reduce uncertainty and improve the skill and reliability of hydrometeorological predictions. As uncertainty decreases, forecast usefulness will increase and provide decision makers with information needed to prepare for hydrometeorological extremes and ensure continued value for the life and the well-being of society.
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