CRUSTAL MOVEMENT OF THE GREAT LAKES—TIME SERIES ANALYSIS

ABSTRACT

Time series techniques were employed to determine rates of vertical crustal movement within the Great Lakes region of North America. Observations of water level elevations as recorded at gauges around the lakes, and differences in elevations between pairs of gauges were analysed for linear trends, periodicities and stochastic components. It was found that the variance of time series of elevations consisted mainly of first-order linear trends and small periodic components. Relative rates of crustal movement were computed from a linear trends analysis of elevation differences. These rates were converted to absolute rates of movement using the Nipissing zero isobase as a datum.

This study shows that, in general, the northeastern area of the Great Lakes region is rising at a rate of about 1·00 ft per 100 years relative to the southwest of the region.     

Analysis of Lake Victoria levels

ABSTRACT

Time series of levels of Lake Victoria as measured at four points are analysed for trends, periodicities and autoregressive components. Comparisons are made with results of time series analysis of Lake Superior which is a similarly sized lake. Checks are made for evidence of differential crustal movement using differences between time series on opposite sides of the lake.     

The influence of autocorrelation on the ability to detect trend in hydrological series

This study investigated using Monte Carlo simulation the interaction between a linear trend and a lag‐one autoregressive (AR(1)) process when both exist in a time series. Simulation experiments demonstrated that the existence of serial correlation alters the variance of the estimate of the Mann–Kendall (MK) statistic; and the presence of a trend alters the estimate of the magnitude of serial correlation. Furthermore, it was shown that removal of a positive serial correlation component from time series by pre‐whitening resulted in a reduction in the magnitude of the existing trend; and the removal of a trend component from a time series as a first step prior to pre‐whitening eliminates the influence of the trend on the serial correlation and does not seriously affect the estimate of the true AR(1). These results indicate that the commonly used pre‐whitening procedure for eliminating the effect of serial correlation on the MK test leads to potentially inaccurate assessments of the significance of a trend; and certain procedures will be more appropriate for eliminating the impact of serial correlation on the MK test. In essence, it was advocated that a trend first be removed in a series prior to ascertaining the magnitude of serial correlation. This alternative approach and the previously existing approaches were employed to assess the significance of a trend in serially correlated annual mean and annual minimum streamflow data of some pristine river basins in Ontario, Canada. Results indicate that, with the previously existing procedures, researchers and practitioners may have incorrectly identified the possibility of significant trends. Copyright © Environment Canada. Published by John Wiley & Sons, Ltd.     

Rainfall–runoff trends in the south‐eastern USA: 1938–2005

An extensive dataset (230 precipitation gauges and 79 stream gauges) was used to analyse rainfall–runoff relationships in 10 subregions of a 482000 km² area in the south‐eastern USA (Maryland, Virginia, North Carolina, South Carolina and Georgia). The average annual rainfall and runoff for this study area between 1938 and 2005 were 1201 and 439 mm, respectively. Average runoff/rainfall ratios during this period varied between 0·24 in the southernmost Coastal Plain subregion to 0·64 in the Blue Ridge Province. Watershed elevation and relief are the principal determinants governing the conversion of rainfall to runoff. Temporal rainfall variation throughout the south‐eastern USA ranges from ～40% above and below normal while the variation for runoff is higher, from ? 75% to + 100%. In any given year there can exist a ± 25–50% error in predicted runoff deviation using the annual rainfall–runoff regression. Fast Fourier Transform and autoregressive spectral analysis revealed dominant cyclicities for rainfall and runoff between 14 and 17 years. Secondary periodicities were typically between 6–7 and 10–12 years. The inferred cyclicity may be related to ENSO and/or Central North Pacific atmospheric phenomena. Mann–Kendall analyses indicate that there were no consistent statistically significant temporal trends with respect to south‐eastern US rainfall and runoff during the study period. The results of U‐tests similarly indicated that rainfall between 1996 and 2005 was not statistically higher or lower than during earlier in the study period. Copyright © 2008 John Wiley & Sons, Ltd.     

HYDRODYNAMIC AND EXPERIMENTAL STUDY OF UNCONFINED GROUND WATER BALANCE ELEMENTS IN REPRESENTATIVE AND EXPERIMENTAL BASINS

Abstract

A mathematical model is built for monthly river flows of the Orinoco river. The model consists of a cyclic part which explains up to 93% of the total variance of the series of monthly water levels, and a stochastic part which is shown to follow a 1st order autoregressive scheme with a primary variable or random component that behaves as “white noise” and appears to have near 60% chance of coming from a normal population. A time series of flow anomalies is obtained from long term monthly means. Statistical techniques are applied to the series of flow anomalies in order to obtain the mean number of crossings at an arbitrary level during an arbitrary time. There are also studied the mean length of an upward excursion over an arbitrary level and the mean time between successive upcrossings. The actual results for the Orinoco river are in good agreement with the statistical theory showing that this kind of analysis can be extremely useful in the design and planning of reservoir operations. (Key words: hydrology, runoff, statistical analysis.)     

Performance of detrending models of crop yield risk assessment: evaluation on real and hypothetical yield data

Detrending is a widely used technique for obtaining stationary time series data in residual analysis and risk assessment. The technique is frequently applied in crop yield risk assessment and insurance ratings. Although several trend models have been proposed in the literature, whether these models achieve consistent detrending results and successfully extract the true yield trends is rarely discussed. In the present article, crop insurance pricing is evaluated by different trend models using real and historical yield data, and hypothetical yield data generated by Monte Carlo simulations. Applied to real historical data, the linear, loglinear, autoregressive integrated moving average trend models produce different risk assessment results. The differences among the model outputs are statistically significant. The largest deviation in the county crop assessment reaches 6–8 %, substantially larger than the present countrywide gross premium rate of 5–7 %. In performance tests on simulated yield trends, popular detrending methods based on smoothing techniques proved overall superior to linear, loglinear, and integrated autoregression models. The best performances were yielded by the moving average and robust locally weighted regression models.     

Variation analysis of precipitation during past 286 years in Beijing area,China, using non‐parametric test and wavelet analysis

Non‐parametric methods including Mann–Kendall (M–K) test, continuous wavelet transform (CWT) and discrete wavelet transform analysis are applied in this paper to detect the trend and periodic trait of precipitation data series in Beijing area where the data set spans nearly 300 years from 1724 to 2009. First, the trend of precipitation variables is elaborated by the M–K test (Sequential M–K test). The results show that there is an increasing trend (the value of this trend is 1.98) at the 5%‐significance level and there are not turning points in the whole data series. Then, CWT and wavelet variance are used to check for significant periodic characteristics of data series. In the plots of wavelet transform coefficients and figure of wavelet variance, some periodic events affect the trend of the annual total precipitation series in Beijing area. 85‐year, 35‐year and 21‐year periodic events are found to be the main periodic series of long‐term precipitation data, and they are all statistically significant. Moreover, the results of non‐parametric M–K test are exhibited on seven different combinations of discrete wavelet components. D5 (32‐year periodicity) periodic component is the effective and significant component on data. It is coincident with the result (35‐year periodic event as one part of main periodicity) by using CWT analysis. Moreover, approximation mode shows potential trend of the whole data set because it is the residuals as all periodicities are removed from data series. Thus, the mode A + D5 is responsible for producing a real basic structure of the trend founded on the data. Copyright © 2012 John Wiley & Sons, Ltd.     

Regional scale rainfall–runoff modeling using VARX–MGARCH approach

Multivariate time series modeling approaches are known as useful tools for describing, simulating, and forecasting hydrologic variables as well as their changes over the time. These approaches also have temporal and cross-sectional spatial dependence in multiple measurements. Although the application of multivariate linear and nonlinear time series approaches such as vector autoregressive with eXogenous variables (VARX) and multivariate generalized autoregressive conditional heteroscedasticity (MGARCH) models are commonly used in financial and economic sciences, these approaches have not been extensively used in hydrology and water resources engineering. This study employed VARX and VARX–MGARCH approaches in modeling mean and conditional heteroscedasticity of daily rainfall and runoff records in the basin of Zarrineh Rood Dam, Iran. Bivariate diagonal VECH (DVECH) model, as a main type of MGARCH, shows how the conditional variance–covariance and conditional correlation structure vary over the time between residuals series of the fitted VARX. For this purpose, five model fits, which consider different combinations of twofold rainfall and runoff, including both upstream and downstream stations, have been investigated in the present study. The VARX model, with different orders, was applied to the daily rainfall–runoff process of the study area in each of these model fits. The Portmanteau test revealed the existence of conditional heteroscedasticity in the twofold residuals of fitted VARX models. Therefore, the VARX–DVECH model is proposed to capture the heteroscedasticity existing in the daily rainfall–runoff process. The bivariate DVECH model indicated both short-run and long-run persistency in the conditional variance–covariance matrix related to the twofold innovations of rainfall–runoff processes. Furthermore, the evaluation criteria for the VARX–DVECH model revealed the improvement of VARX model performance.     

On recent variations of the annual thunderstorm activity in the region of the Alps

Summary Observations of days with thunderstorm at twelve stations in the Alpine and Pre-Alpine region have been used to look for trends and periodicities in the thunderstorm activity in this area. The annual number of days with thunderstorms was represented by negative binomial distributions, then by means of a non-parametric trend test, parts of the series were found in which the numbers of days with thunderstorm were increasing, decreasing or uniform at the various stations. A variance spectrum analysis showed the existence of long-periodic waves and of short-periodic waves superposed on the background red noise or white noise spectrum. Partitioning the series of observations yielded that these periodicities are not stable.     

Long-term trend analysis using discrete wavelet components of annual precipitations measurements in Marmara region (Turkey)

The purpose of this study is to determine the possible trends in annual total precipitation series by using the non-parametric methods such as the wavelet analysis and Mann-Kendall test. The wavelet trend (W-T) analysis is for the first time presented in this study. Using discrete wavelet components of measurement series, we aimed to find which periodicities are mainly responsible for trend of the measurement series. We found that some periodic events clearly affect the trend of precipitation series. 16-yearly periodic component is the effective component on Bal?kesir annual precipitation data and is responsible for producing a real trend founded on the data. Also, global wavelet spectra and continuous wavelet transform were used for analysis to precipitation time series in order to clarify time-scale characteristics of the measured series. The effects of regional differences on W-T analysis are checked by using records of measurement stations located in different climatic areas. The data set spans from 1929 to 1993 and includes precipitation records from meteorological stations of Turkey. The trend analysis on DW components of the precipitation time series (W-T model) clearly explains the trend structure of data.     

Characterisation of hydroclimatological trends and variability in the Lake Naivasha basin,Kenya

Recent hydro‐climatological trends and variability characteristics were investigated for the Lake Naivasha basin with the aim of understanding the changes in water balance components and their evolution over the past 50 years. Using a Bayesian change point analysis and modified Mann–Kendall tests, time series of annual mean, maximum, minimum, and seasonal precipitation and flow, as well as annual mean lake volumes, were analysed for the period 1960–2010 to uncover possible abrupt shifts and gradual trends. Double cumulative curve analysis was used to investigate the changes in hydrological response attributable to either human influence or climatic variability. The results indicate a significant decline in lake volumes at a mean rate of 9.35 × 10⁶ m³ year^?1. Most of the river gauging stations showed no evidence of trends in the annual mean and maximum flows as well as seasonal flows. Annual minimum flows, however, showed abrupt shifts and significant (upward/downward) trends at the main outlet stations. Precipitation in the basin showed no evidence of abrupt shifts, but a few stations showed gradual decline. The observed changes in precipitation could not explain the decline in both minimum flows and lake volumes. The findings show no evidence of any impact of climate change for the Lake Naivasha basin over the past 50 years. This implies that other factors, such as changes in land cover and infrastructure development, have been responsible for the observed changes in streamflow and lake volumes. Copyright © 2015 John Wiley & Sons, Ltd.     

Detecting and quantifying sources of non-stationarity via experimental semivariogram modeling

Conventional geostatistics often relies on the assumption of second order stationarity of the random function (RF). Generally, local means and local variances of the random variables (RVs) are assumed to be constant throughout the domain. Large scale differences in the local means and local variances of the RVs are referred to as trends. Two problems of building geostatistical models in presence of mean trends are: (1) inflation of the conditional variances and (2) the spatial continuity is exaggerated. Variance trends on the other hand cause conditional variances to be over-estimated in certain regions of the domain and under-estimated in other areas. In both cases the uncertainty characterized by the geostatistical model is improperly assessed. This paper proposes a new approach to identify the presence and contribution of mean and variance trends in the domain via calculation of the experimental semivariogram. The traditional experimental semivariogram expression is decomposed into three components: (1) the mean trend, (2) the variance trend and (3) the stationary component. Under stationary conditions, both the mean and the variance trend components should be close to zero. This proposed approach is intended to be used in the early stages of data analysis when domains are being defined or to verify the impact of detrending techniques in the conditioning dataset for validating domains. This approach determines the source of a trend, thereby facilitating the choice of a suitable detrending method for effective resource modeling.     

The role of anthropogenic activities in karst spring discharge volatility

The traditional hydrological time series methods tend to focus on the mean of whichever variable is analysed but neglect its time‐varying variance (i.e. assuming the variance remains constant). The variances of hydrological time series vary with time under anthropogenic influence. There is evidence that extensive well drilling and groundwater pumping can intercept groundwater run‐off and consequently induce spring discharge volatility or variance varying with time (i.e. heteroskedasticity). To investigate the time‐varying variance or heteroskedasticity of spring discharge, this paper presents a seasonal autoregressive integrated moving average with general autoregressive conditional heteroskedasticity (SARIMA‐GARCH) model, whose the SARIMA model is used to estimate the mean of hydrological time series, and the GARCH model estimates its time‐varying variance. The SARIMA‐GARCH model was then applied to the Xin'an Springs Basin, China, where extensive groundwater development has occurred since 1978 (e.g. the average annual groundwater pumping rates were less than 0.20 m³/s in the 1970s, reached 1.20 m³/s at the end of the 1980s, surpassed 2.0 m³/s in the 1990s and exceeded 3.0 m³/s by 2007). To identify whether human activities or natural stressors caused the heteroskedasticity of Xin'an Springs discharge, we segmented the spring discharge sequence into two periods: a predevelopment stage (i.e. 1956–1977) and a developed stage (i.e. 1978–2012), and set up the SARIMA‐GARCH model for the two stages, respectively. By comparing the models, we detected the role of human activities in spring discharge volatility. The results showed that human activities caused the heteroskedasticity of the Xin'an Spring discharge. The predicted Xin'an Springs discharge by the SARIMA‐GARCH model showed that the mean monthly spring discharge is predicted to continue to decline to 0.93 m³/s in 2013, 0.67 m³/s in 2014 and 0.73 m³/s in 2015. Copyright © 2014 John Wiley & Sons, Ltd.     

Pressure effects on past regional sea level trends and variability in the German Bight

The impact on a large-scale sea level pressure field to the regional mean sea level changes of the German Bight is analysed. A multiple linear regression together with an empirical orthogonal function analysis is used to describe the relationship between the sea level pressure and the regional mean sea level considering the time period 1924–2001. Both, the part of the variability and of the long-term trend that can be associated with changes in the sea level pressure, are investigated. Considering the whole time period, this regression explains 58?% of the variance and 33?% of the long-term trend of the regional mean sea level. The index of agreement between the regression result and the observed time series is 0.82. As a proxy for large-scale mean sea level changes, the mean sea level of the North East Atlantic is subsequently introduced as an additional predictor. This further improves the results. For that case, the regression explains 74?% of the variance and 87?% of the linear trend. The index of agreement rises to 0.92. These results suggest that the sea level pressure mainly accounts for the inter-annual variability and parts of the long-term trend of regional mean sea level in the German Bight while large-scale sea level changes in the North East Atlantic account for another considerable fraction of the observed long-term trend. Sea level pressure effects and the mean sea level of the North East Atlantic provide thus significant contributions to regional sea level rise and variability. When future developments are considered, scenarios for their future long-term trends thus need to be comprised in order to provide reliable estimates of potential future long-term changes of mean sea level in the German Bight.     

SPECTRAL ANALYSIS OF GEOMORPHIC TIME SERIES: AUTO-SPECTRUM

The collection of time series data is an essential component in the investigation of earth surface processes. Spectral analysis of these time series can provide an invaluable insight into the behaviour of geophysical processes. Spectral analysis of a single time series produces an auto-spectrum which provides a representation of the amount variance of the time series as a function of frequency. Prior to spectral analysis, the time series should be plotted to identify the presence of any trends in the mean or the variance of the series, and to identify anomalies in the data which should be corrected. To satisfy the assumption of stationarity, any trend (in either the mean or variance) should be removed from the time series. Consequently, the probability density function of the time series should be plotted and compared with the Gaussian distribution. The final stage in preparing the time series for spectral analysis is to apply a taper to reduce spectral leakage and distortion of the auto-spectrum. Following the calculation of the periodogram, spectral estimates should be combined to reduce the variability associated with the estimates and thereby ensure that the autospectrum is more representative. Finally, confidence limits should be constructed around the spectral density function so that statistically significant spectral peaks (or troughs) can be identified.     

Latitude and Altitude Dependence of the Interannual Variability and Trends of Atmospheric Temperatures

—The 4-season (12-month) running means of temperatures at five atmospheric levels (surface, 850–300 mb, 300–100 mb, 100–50 mb, 100–30 mb) and seven climatic zones (60°N–90°N, 30°N–60°N, 10°N–30°N, 10°N–10°S, 10°S–30°S, 30°S–60°S, 60°S–90°S) showed QBO (Quasi-biennial Oscillation), QTO (Quasi-triennial Oscillation) and larger periodicities. For stratosphere and tropopause, the temperature variations near the equator and North Pole somewhat resembled the 50mb low latitude zonal winds, mainly due to prominent QBO. For troposphere and surface, the temperature variations, especially those near the equator, resemble those of eastern equatorial Pacific sea-surface temperatures, mainly due to prominent QTO. In general, the temperature trends in the last 35 years show stratospheric cooling and tropospheric warming. But the trends are not monotonic. For example, the surface trends were downward during 1960–70, upward during 1970–82, downward during 1982–85 and upward thereafter. Models of green-house warming should take these non-uniformities into account.     

Improved estimates of mean sea level changes in the German Bight over the last 166 years

In this paper, mean sea level changes in the German Bight, the south-eastern part of the North Sea, are analysed. Records from 13 tide gauges covering the entire German North Sea coastline and the period from 1843 to 2008 have been used to derive high quality relative mean sea level time series. Changes in mean sea level are assessed using non-linear smoothing techniques and linear trend estimations for different time spans. Time series from individual tide gauges are analysed and then ‘virtual station’ time series are constructed (by combining the individual records) which are representative of the German Bight and the southern and eastern regions of the Bight. An accelerated sea level rise is detected for a period at the end of the nineteenth century and for another one covering the last decades. The results show that there are regional differences in sea level changes along the coastline. Higher rates of relative sea level rise are detected for the eastern part of the German Bight in comparison to the southern part. This is most likely due to different rates of vertical land movement. In addition, different temporal behaviour of sea level change is found in the German Bight compared to wider regional and global changes, highlighting the urgent need to derive reliable regional sea level projections for coastal planning strategies.     

Comparative evaluation of statistical tests for time series analysis: application to hydrological time series / Evaluation comparative de tests statistiques pour l'analyse de séries temporelles: application à des séries temporelles hydrologiques

Abstract

Statistical analyses of hydrological time series play a vital role in water resources studies. Twenty-nine statistical tests for detecting time series characteristics were evaluated by applying them to analyse 46 years of annual rainfall, 47 years of 1-day maximum rainfall and consecutive 2-, 3-, 4-, 5- and 6-day maximum rainfalls at Kharagpur, West Bengal, India. The performance of all the tests was evaluated. No severe outliers were found, and both the annual and maximum rainfall series were found to be normally distributed. Based on the known physical parameters affecting the homogeneity, the cumulative deviations and the Bayesian tests were found to be superior to the classical von Neumann test. Similarly, the Tukey test proved excellent among all the multiple comparison tests. These tests indicated that all the seven rainfall series are homogeneous. Two parametric t tests and the non-parametric Mann-Whitney test indicated stationarity in all the rainfall series. Of 12 trend detection tests, nine tests indicated no trends in the rainfall series. The Kendall's Rank Correlation test and the Mann-Kendall test were found equally powerful. Moreover, the Fourier series analysis revealed no apparent periodicities in all the seven rainfall series. The annual rainfall series was found persistent with a time lag of nine years. All the rainfall series were subjected to stochastic analysis by fitting 35 autoregressive moving-average (ARMA) models of different orders. The best-fit models for the original annual rainfall and 1-, 2- and 3-day maximum rainfall series were found to be ARMA(0,4), ARMA(0,2), ARMA(0,2) and ARMA(3,0), respectively. The best-fit model for the logarithmically transformed 4-day maximum rainfall was found to be ARMA(0,2). However, for the inversely transformed 4-, 5- and 6-day maximum rainfall series, ARMA(0,1) was obtained as the best-fit model. It is concluded that proper selection of time series tests and use of several tests is indispensable for making useful and reliable decisions.     

Spatial variability of rainfall on a sub‐kilometre scale

The variability of rainfall in space and time is an essential driver of many processes in nature but little is known about its extent on the sub‐kilometre scale, despite many agricultural and environmental experiments on this scale. A network of 13 tipping‐bucket rain gauges was operated on a 1·4 km² test site in southern Germany for four years to quantify spatial trends in rainfall depth, intensity, erosivity, and predicted runoff. The random measuring error ranged from 10% to 0·1% in case of 1 mm and 100 mm rainfall, respectively. The wind effects could be well described by the mean slope of the horizon at the stations. Except for one station, which was excluded from further analysis, the relative differences due to wind were in maximum ±5%. Gradients in rainfall depth representing the 1‐km² scale derived by linear regressions were much larger and ranged from 1·0 to 15·7 mm km^?1 with a mean of 4·2 mm km^?1 (median 3·3 mm km^?1). They mainly developed during short bursts of rain and thus gradients were even larger for rain intensities and caused a variation in rain erosivity of up to 255% for an individual event. The trends did not have a single primary direction and thus level out on the long term, but for short‐time periods or for single events the assumption of spatially uniform rainfall is invalid on the sub‐kilometre scale. The strength of the spatial trend increased with rain intensity. This has important implications for any hydrological or geomorphologic process sensitive to maximum rain intensities, especially when focusing on large, rare events. These sub‐kilometre scale differences are hence highly relevant for environmental processes acting on short‐time scales like flooding or erosion. They should be considered during establishing, validating and application of any event‐based runoff or erosion model. Copyright © 2009 John Wiley & Sons, Ltd.