Flooding normally occurs during periods of excessive precipitation or thawing in the winter period (ice jam). Flooding is typically accompanied by an increase in river discharge. This paper presents a statistical model for the prediction and explanation of the water discharge time series using an example from the Schoharie Creek, New York (one of the principal tributaries of the Mohawk River). It is developed with a view to wider application in similar water basins. In this study a statistical methodology for the decomposition of the time series is used. The Kolmogorov-Zurbenko filter is used for the decomposition of the hydrological and climatic time series into the seasonal and the long and the short term component. We analyze the time series of the water discharge by using a summer and a winter model. The explanation of the water discharge has been improved up to 81%. The results show that as water discharge increases in the long term then the water table replenishes, and in the seasonal term it depletes. In the short term, the groundwater drops during the winter period, and it rises during the summer period. This methodology can be applied for the prediction of the water discharge at multiple sites.
It has been shown that there is a connection between increased river flooding and climate change [
Over the last decade, Schoharie Creek has experienced a significant increase in water discharge [
Water discharge depends on the watershed area and the tributaries that drain into the stream. Schoharie Creek (hydrologic unit: 02020005 with a drainage basin of approximately 886 mi2 (2300 km2) in New York, USA, flows north 93 miles (150 km) from the foot of Indian Head Mountain in the Catskill Mountains through the Schoharie Valley to the Mohawk River. It is one of the two principal tributaries of the Mohawk River, with the other being West Canada Creek. Eight further tributaries drain into the Schoharie Creek. It is critical to model flooding in order to help local governments and communities to take precautionary steps to minimize or to prevent damage on property from flooding.
Several studies on flooding in Schoharie or in different locations (e.g., in Germany, Indus River in India, or in Elbe River in Czech [
In previous studies regarding water discharge, the time series of the water discharge variable has rarely been decomposed into different components [
The main purpose of this paper is to present a novel methodology to explain and improve the predictive capability of river water discharge time series by using climatic and hydrological variables. We introduce a new model which incorporates separate treatments of frequency scales. Those scales have been determined based on spectral and cospectral analysis, and they provide a physical explanation of the hydrology of this area. In particular, we separate the data into seasonal and long and short term components by using the Kolmogorov-Zurbenko (KZ) filter [
One model is designed for the winter period and another model for the summer period. This is because flooding in Schoharie Creek may be caused by heavy rainfall during the summer or rapid snowmelt or “ice jam” during the prolonged winter period [
Daily time series of water discharge, groundwater level, tides, and climatic variables have been obtained through the online public database of National Oceanic and Atmospheric Administration (NOAA;
In several studies, it has been shown that the separation of a time series into different components is essential in order to avoid contributions from different covariance structures between the components of the time series [
The KZ filter, which separates long term variations from short term variations in a time series [
Specifically, the KZ filter is a low pass filter, defined by
In the following sections, we describe the method for prediction of the seasonal and long and short term components of the water discharge time series using Schoharie Creek in New York as an example of application area.
Here, we apply the previous methodology for the decomposition of the time series to explain and predict the water discharge time series of Schoharie Creek. For the explanation and prediction of the water discharge, we use the climatic variables and the groundwater level. The logarithm of the water discharge from the study area was measured from January 2005 to February 2013, and it has been presented in Figure
Time series of logarithm of water discharge data (ft3/sec) during the period January 2005–February 2013.
Table
Correlation matrix of the raw data between the water discharge, the climatic variables, and the groundwater level.
Variables | Water discharge | Groundwater level | Temperature | Tide | Wind speed | Precipitation |
---|---|---|---|---|---|---|
Water discharge | 1 | −0.65 | −0.481 | 0.254 | 0.203 | 0.109 |
Groundwater level | −0.65 | 1 | 0.178 | −0.277 | −0.094 | 0.033 |
Temperature | −0.481 | 0.178 | 1 | 0.041 | −0.193 | −0.007 |
Tide | 0.254 | −0.277 | 0.041 | 1 | 0.105 | −0.008 |
Wind speed | 0.203 | −0.094 | −0.193 | 0.105 | 1 | 0.139 |
Precipitation | 0.109 | 0.033 | −0.007 | −0.008 | 0.139 | 1 |
The relationship of the water discharge with the climatic variables can be strengthened by separating the seasonal and long and short term variations in all the time series. The separation of scales in the time series can also be verified through examination of the coherence between the variables. Table
Periods derived by the graph of periodograms of the water discharge variable and the climatic variables.
Variable | Period (days) | ||
---|---|---|---|
First peak of period | Second peak of period | Third peak of period | |
Water discharge | 365 | 594 | |
Groundwater level | 365 | 991 | 198 |
Tide | 365 | 185 | 14 |
Precipitation | 27 | 7 | 4 |
Wind speed | 365 | ||
Temperature | 365 |
Periodogram of the temperature variable. The
Periodogram of the precipitation variable. The
For the decomposition of the time series, we use the
The KZ filter is applied to the logarithm of daily water discharge and produces a time series devoid of short term variations and consisting only of the long term variations of the time series (
To explain the long term component of water discharge and its relationship with groundwater level and climatic variables, we examine the filtered daily temperature, tide, wind speed, and precipitation of the sum of four continuous days. For the analysis, we denote the long term components of the water discharge, temperature, tide, wind speed, precipitation, and groundwater level time series with
The correlation between the long term component of the water discharge and the precipitation is weak (Table
Correlation matrix between the long term components of the water discharge, the climatic variables, and the groundwater level.
Variable | Water discharge | Groundwater level | Temperature | Tide | Wind speed | Precipitation (4 days) |
---|---|---|---|---|---|---|
Water discharge | 1 | −0.744 | −0.641 | 0.39 | 0.596 | 0.175 |
Groundwater level | −0.744 | 1 | 0.26 | −0.412 | −0.43 | −0.224 |
Temperature | −0.641 | 0.26 | 1 | 0.033 | −0.691 | 0.103 |
Tide | 0.39 | −0.412 | 0.033 | 1 | 0.462 | −0.114 |
Wind speed | 0.596 | −0.43 | −0.691 | 0.462 | 1 | −0.131 |
Precipitation (4 days) | 0.175 | −0.224 | 0.103 | −0.114 | −0.131 | 1 |
For the prediction of the long term component of water discharge, a linear regression is performed using the filtered logarithm of the water discharge, the filtered climatic variables, and the groundwater level time series with an
From expression (
Because expression (
The seasonal component of a time series represents the year-to-year fluctuations of the corresponding variable. In order to predict the seasonal component of the water discharge time series, we use seasonal components of the climatic variables and the groundwater level. In particular, the seasonal component of the water discharge,
To investigate the relationship between the seasonal component of the water discharge and the climatic variables, we estimate their correlation matrix (Table
Correlation matrix between the seasonal components of the water discharge, the climatic variables, and the groundwater level.
Variable | Water discharge | Groundwater level | Temperature | Tide | Wind speed | Precipitation (4 days) |
---|---|---|---|---|---|---|
Water discharge | 1 | 0.63 | 0.579 | 0.893 | 0.418 | 0.27 |
Groundwater level | 0.63 | 1 | 0.492 | 0.831 | 0.406 | 0.012 |
Temperature | 0.579 | 0.492 | 1 | 0.627 | 0.3 | −0.046 |
Tide | 0.893 | 0.831 | 0.627 | 1 | 0.428 | 0.089 |
Wind speed | 0.418 | 0.406 | 0.3 | 0.428 | 1 | 0.044 |
Precipitation (4 days) | 0.27 | 0.012 | −0.046 | 0.089 | 0.044 | 1 |
To predict the seasonal component of the water discharge, we perform linear regression using the seasonal components of the climatic variables and the groundwater level. The coefficient of determination,
For the prediction of the short term component of the water discharge, we consider the short term components of the climatic variables and groundwater level. The short term component of the water discharge time series,
For the short term component of the water discharge, we consider two different models. One describes the prediction of the short term component during the summer period (May through September) and the other during winter (December through March). We consider two different models because flooding in the rivers in New York State is caused by extensive precipitation (e.g., extensive rainfall or tropical storms during the summer period) or by rapid snowmelt (e.g., ice jams during the prolonged winter period). The overall explanation of the winter model is maximum during the prolonged winter period (December to March), while the summer model shows maximum explanation during the prolonged summer period (May to September).
To predict the short term component of the water discharge time series, we perform a linear regression by using the short term components of daily temperature, precipitation, the sum of four continuous days of daily precipitation, and daily groundwater level. The short term components of the above variables are denoted by
Correlation matrix between the short term components of the water discharge, the climatic variables, and the groundwater level for the summer period.
Water discharge | Groundwater level | Temperature average (4 days) | Precipitation | Precipitation (4 days) | |
---|---|---|---|---|---|
Water discharge | 1 | −0.379 | −0.247 | 0.124 | 0.502 |
Groundwater level | −0.379 | 1 | 0.099 | 0.194 | −0.069 |
Temperature average (4 days) | −0.247 | 0.099 | 1 | 0.004 | −0.249 |
Precipitation | 0.124 | 0.194 | 0.004 | 1 | −0.004 |
Precipitation (4 days) | 0.502 | −0.069 | −0.249 | −0.004 | 1 |
For the prediction of the short term component of the water discharge for the summer period, a linear regression is performed by using the short term components of the above climatic variables and the groundwater level with coefficient of determination,
Flooding in the rivers during the winter period occurs through rapid snowmelt due to the increase of air temperature. For this reason, the average temperature is used for the prediction model in the linear regression. The maximum correlation between water discharge and average temperature occurs for the average temperature over four days. Moreover, for the prediction model, we consider the variables: tide, wind speed, groundwater level, and the sum of four days of precipitation. The short term components of the above variables are denoted by
Correlation matrix between the short term components of the water discharge, the climatic variables, and the groundwater level for the winter period.
Water discharge | Groundwater level | Temperature average (4 days) | Tide | Wind speed | Precipitation (4 days) | |
---|---|---|---|---|---|---|
Water discharge | 1 | 0.398 | 0.57 | 0.732 | 0.25 | 0.229 |
Groundwater level | 0.398 | 1 | 0.408 | 0.597 | 0.12 | −0.005 |
Temperature average (4 days) | 0.57 | 0.408 | 1 | 0.444 | 0.136 | 0.038 |
Tide | 0.732 | 0.597 | 0.444 | 1 | 0.181 | 0.117 |
Wind speed | 0.25 | 0.12 | 0.136 | 0.181 | 1 | 0.155 |
Precipitation (4 days) | 0.229 | −0.005 | 0.038 | 0.117 | 0.155 | 1 |
To predict the short term component of the water discharge time series, we perform a linear regression by using the short term components of the above climatic variables with resultant coefficient of determination,
A river can act as a gaining stream, receiving water from the groundwater system, or as a losing stream, losing water to the groundwater system. The water table’s height reflects a balance between the rate of replenishment, through precipitation, and removal through discharge and withdrawal. Any imbalance either raises or lowers the water table, acting with the opposite effect to the groundwater level value (because groundwater level value is measured as the distance of the water level depth below surface). In the summer period, the water river discharge of the Schoharie Creek increases as it flows downstream as tributaries and groundwater contribute additional water (recalling that eight tributaries contribute to Schoharie Creek before it reaches Mohawk River). During summer, groundwater resources appear to increase to ample levels (recharge). Natural replenishment will decrease the measured value of the groundwater level and enhance the river water discharge. Hence, it produces a negative association.
In regions where there is a prolonged winter period, the rainfall to replenish the water table is often scarce, and the rate of water table recharge will be less than the river’s water discharge. The groundwater will drop and may result in very low water level (depletion). Groundwater level value will increase (water level drops), and water discharge in the stream increases (primarily through snow melt), which produces a positive association.
Due to the separation of scales in the time series, the long term component of water discharge shows a negative correlation with the long term component of the groundwater level and temperature (Table
The negative correlation between the long-term components of the water discharge and the groundwater level is due to the increased rainfall during the last decade in Schoharie Creek area. This phenomenon has also been observed for the short term components during the summer period when precipitation is intense. A positive correlation between the water discharge and the groundwater level takes place during the prolonged winter period.
The correlation between the short term component of the water discharge and precipitation during the winter is lower than the short term components of those variables during the summer. This is because in areas that experience prolonged winter seasons, rainfall does not contribute to water table in the same rate as it would replenish the water table during the summer period.
To estimate the total explanation of the model for the water discharge of the summer period, we combine expressions (
Summary of variances and the coefficient of determinations for the seasonal and long and short term component for the summer and winter period.
Long term | Seasonal term | Short term | |
---|---|---|---|
Summer period | |||
Variance | 0.588 | 0.178 | 0.232 |
|
0.833 | 0.912 | 0.447 |
|
|||
Winter period | |||
Variance | 0.531 | 0.162 | 0.307 |
|
0.833 | 0.912 | 0.719 |
Raw water discharge data (blue line) along with the prediction model (purple line) for the summer period (May through September 2006).
By combining expressions (
Raw water discharge data (blue line) along with the prediction model (purple line) for the winter period (December of 2009 through March of 2010).
As a consequence of summer and winter model results, we prove that the decomposition of the time series improves our ability to describe and predict the time series variations of water discharge by approximately two times. In particular, the unexplained variance derived by the raw data described in expression (
This study focuses on predicting the daily water discharge time series using available climatic variables and the groundwater level. We prove that the decomposition of the time series is essential due to the presence of short term variations in the time series. We use the KZ filter to decompose the time series into the seasonal and long and short term components to provide a physical based explanation for the time series of the water discharge. The long term component is associated with long term changes, the seasonal component with year-to-year fluctuations, and the short term component with short term variations.
In this study, we prove that the seasonal and long and short term components of water discharge can be explained through consideration of climatic variables and groundwater level. The results show that as water discharge increases in the long term the water table replenishes, while in the seasonal term it depletes. In the short term, the groundwater drops during the winter period and rises during the summer period. The short term component of the water discharge is related to synoptic weather fluctuations and short term effects of rain, storms, and cyclones during the summer period and the rapid increase in temperature as well as storms during the winter period. As it is described in expressions (
After the application of the KZ filter and the decomposition of all time series, we apply different multivariate models to each component of the water discharge time series. Different scales of time series are associated with different correlation structures (Tables
In our paper, we show that the accuracy of the water discharge time series prediction can be increased up to 81.1% for the winter period and 75.6% for the summer period by incorporating the KZ filter in a separation of the scales of the time series [
The prediction of the daily water discharge time series using the climatic variables and groundwater level can be substantially improved through the decomposition of the time series. The decomposition of the different components (scales), which are the seasonal and long and short term components, avoids erroneous results and approximately doubles the prediction accuracy of the water discharge time series relative to raw data. The resulting isolation of the short term variations by the decomposition of the time series shows a summer period with a water table replenishment and a winter period with a water table depletion. The design of multivariate models (winter and summer) can improve the prediction of flooding caused by storms, rapid snowmelt, and ice jams.
The authors declare that there is no conflict of interests regarding the publication of this paper.