A model for distributed GIUH‐based flow routing on natural and anthropogenic hillslopes

Attempts to reduce the number of parameters in distributed rainfall–runoff models have not yet resulted in a model that is accurate for both natural and anthropogenic hillslopes. We take on the challenge by proposing a distributed model for overland flow and channel flow based on a combination of a linear response time distribution and the hillslope geomorphologic instantaneous unit hydrograph (GIUH), which can be calculated with only a digital elevation model and a map with field boundaries and channel network as input. The spatial domain is subdivided into representative elementary hillslopes (REHs) for each of which we define geometric and flow velocity parameters and compute the GIUH. The catchment GIUH is given by the sum of all REH responses. While most distributed models only perform well on natural hillslopes, the advantage of our approach is that it can also be applied to modified hillslopes with for example a rectangular drainage network and terrace cultivation. Tests show that the REH‐GIUH approach performs better than classical routing functions (exponential and gamma). Simulations of four virtual hillslopes suggest that peak flow at the catchment outlet is directly related to drainage density. By combining the distributed flow routing model with a lumped‐parameter infiltration model, we were also able to demonstrate that terrace cultivation delays the response time and reduces peak flow in comparison to the same hillslope, but with a natural stream network. The REH‐GIUH approach is a first step in the process of coupling distributed hydrological models to erosion and water quality models at the REH (associated with agricultural management) and at the catchment scale (associated with the evaluation of the environmental impact of human activities). It furthermore provides a basis for the development of models for large catchments and urban or peri‐urban catchments. Copyright © 2013 John Wiley & Sons, Ltd.     

INFLUENCE OF THE DRAINAGE NETWORK IDENTIFICATION METHOD ON GEOMORPHOLOGICAL PROPERTIES AND HYDROLOGICAL RESPONSE

The influence of the method of identification of the drainage network on its geomorphological characteristics and on its hydrological response is analysed. Blue lines, photo-interpreted networks and networks generated from digital elevation models (DEMs) by an automatic algorithm are compared with field observations for two small alpine catchments. The comparisons are carried out in quantitative terms by using several geomorphological indices and functions and by calculating the hydrological response of the networks as represented by their geomorphologic instantaneous unit hydrograph (GIUH). The results show that the effect of the identification method on the geomorphological indices and on the hydrological response is significant, and that the threshold area for channel initiation is not constant. Moreover, the available data show a poor correlation between local slope and threshold area. Finally, the influence of the threshold area on the shape of the GIUH is larger when the residence time on the hillslopes is of the same order as the residence time in the network. In the opposite case, the variability of the flow velocity along the network seems to play an important role. © 1997 John Wiley & Sons, Ltd.     

Stone movement on hillslopes in the Mojave Desert,California: A 16-year record

Previous research has shown that either hydraulic action or creep may be the dominant process transporting coarse debris down hillslopes in the American Southwest. This study analyses the movement over 16 years of painted stones on two hillslopes in the central Mojave Desert to ascertain which of these two processes dominate in this region. The distance moved (M) is found to be directly related to length of overland flow (X) and hillslope gradient (S), and inversely related to particle size (D). The fact that M is more highly correlated with X than with S suggests that hydraulic action rather than creep is the dominant process. It is concluded that this is probably the case over most of the Mojave and Sonoran Deserts on slopes up to at least 24°, and that it is only at higher elevations where winters are more severe that creep may become dominant.     

A similarity approach to determine response times to steady-state saturation in landscapes

Relationships describing response times for landscape saturation and subsurface flow for idealised hillslopes after a change in water balance are derived in terms of similarity parameters given by their topographic, soil and climatic attributes. The study was carried out under a set of specific assumptions.The work quantitatively describes effects of each of these attributes on travel time. For example, divergent hillslopes can exhibit travel times that are double those of convergent hillslopes, and concave slopes tend to have lower travel times than planar or convex slopes. It is shown that the hillslope travel time T is dependent on hillslope length L, hydraulic conductivity K and slope S such that T = f(L/KS, B, CR, Δq/smd) and if saturation does not occur in the lower reaches of a hillslope, this can be simplified to T = f(L/KS, B). Here, Δq is change in net external flux and smd is soil moisture deficit; CR and B are the convergence ratio and profile factor respectively. The relative importance of these attributes on response time is discussed.It is shown that subsurface flow hydrographs of simple hillslopes of different scale, derived numerically, can be collapsed into a single curve by normalising them by means of a form of response time, the time constant.Results from the analytical derivation were compared with numerical analysis with good agreement. The theory was applied to a natural catchment using observed data from a bushfire event. Comparison of analytically calculated response time with observed response time for the event showed a large discrepancy. The reason is discussed.     

Analytical procedures for weekly hydrological droughts: a case of Canadian rivers

Abstract

Two entities of importance in hydrological droughts, viz. the longest duration, L_T , and the largest magnitude, M_T (in standardized terms) over a desired time period (which could also correspond to a specific return period) T, have been analysed for weekly flow sequences of Canadian rivers. Analysis has been carried out in terms of week-by-week standardized values of flow sequences, designated as SHI (standardized hydrological index). The SHI sequence is truncated at the median level for identification and evaluation of expected values of the above random variables, E(L_T ) and E(M_T ). SHI sequences tended to be strongly autocorrelated and are modelled as autoregressive order-1, order-2 or autoregressive moving average order-1,1. The drought model built on the theorem of extremes of random numbers of random variables was found to be less satisfactory for the prediction of E(L_T ) and E(M_T ) on a weekly basis. However, the model has worked well on a monthly (weakly Markovian) and an annual (random) basis. An alternative procedure based on a second-order Markov chain model provided satisfactory prediction of E(L_T ). Parameters such as the mean, standard deviation (or coefficient of variation), and lag-1 serial correlation of the original weekly flow sequences (obeying a gamma probability distribution function) were used to estimate the simple and first-order drought probabilities through closed-form equations. Second-order probabilities have been estimated based on the original flow sequences as well as SHI sequences, utilizing a counting method. The E(M_T ) can be predicted as a product of drought intensity (which obeys the truncated normal distribution) and E(L_T ) (which is based on a mixture of first- and second-order Markov chains).

Citation Sharma, T. C. & Panu, U. S. (2010) Analytical procedures for weekly hydrological droughts: a case of Canadian rivers. Hydrol. Sci. J. 55(1), 79–92.     

Assessing plot-scale impacts of land use on overland flow generation in Central Panama

Land use in Panama has changed dramatically with ongoing deforestation and conversion to cropland and cattle pastures, potentially altering the soil properties that drive the hydrological processes of infiltration and overland flow. We compared plot-scale overland flow generation between hillslopes in forested and actively cattle-grazed watersheds in Central Panama. Soil physical and hydraulic properties, soil moisture and overland flow data were measured along hillslopes of each land-use type. Soil characteristics and rainfall data were input into a simple, 1-D representative model, HYDRUS-1D, to simulate overland flow that we used to make inferences about overland flow response at forest and pasture sites. Runoff ratios (overland flow/rainfall) were generally higher at the pasture site, although no overall trends were observed between rainfall characteristics and runoff ratios across the two land uses at the plot scale. Saturated hydraulic conductivity (K_s) and bulk density were different between the forest and pasture sites (p < 10⁻⁴). Simulating overland flow in HYDRUS-1D produced more outputs similar to the overland flow recorded at the pasture site than the forest site. Results from our study indicate that, at the plot scale, Hortonian overland flow is the main driver for overland flow generation at the pasture site during storms with high-rainfall totals. We infer that the combination of a leaf litter layer and the activation of shallow preferential flow paths resulting in shallow saturation-excess overland flow are likely the main drivers for plot scale overland flow generation at the forest site. Results from this study contribute to the broader understanding of the delivery of freshwater to streams, which will become increasingly important in the tropics considering freshwater resource scarcity and changing storm intensities.     

Effects of hillslope dynamics and network geometry on the scaling properties of the hydrologic response

This paper investigates the specific contributions of river network geomorphology, hillslope flow dynamics and channel routing to the scaling behavior of the hydrologic response as function of drainage area. Scaling relationships emerged from the observations of geomorphological and hydrological data and were reproduced in previous works through mathematical models, for both idealized self-similar networks and natural basins. Recent literature highlighted that scale invariance of hydrological quantities depends not only on the metrics of the drainage catchment but also on effective flow routing. In this study we employ a geomorphological width function scheme to test the simple scaling hypothesis adopting more realistic dynamic conditions than in previous approaches, specifically taking into account the role of hillslopes. The analysis is based on the derivation of the characteristic distributions of path lengths and travel times, inferred from DEM processing and measurements of rainfall and runoff data. The study area is located in the Tiber River region (central Italy).Results indicate that, while scaling properties clearly emerge when the hydrologic response is defined on the basis of the sole geomorphology, scale invariance is broken when less idealized flow dynamics are taken into account. Lack of scaling appears in particular as a consequence of the catchment to catchment variability of hillslope velocities.     

GEOMORPHOLOGICAL TRANSFER FUNCTION CALCULATED FROM DIGITAL ELEVATION MODELS FOR DISTRIBUTED HYDROLOGICAL MODELLING

Recently, several attempts have been made to relate the hydrological response of a catchment to its morphological and topographical features using different hypotheses to model the effect of the drainage network. Several transfer functions were developed and some of these are based on the theory of a linear model, the geomorphological unit hydrograph. The aim of this paper is to present a methodology to automatically identify the transfer function, using digital elevation models for applications in distributed hydrological modelling. The transfer function proposed herein is based on the Hayami approximation solution of the diffusive wave equation especially adapted for the routing hydrograph through a channel network. The Gardon d’Anduze basin, southern France, was retained for applications. Digital elevation models were used to extract the channel network and divide the basin into subcatchments. Each subcatchment produces, at its own outlet, an impulse response which is routed to the outlet of the whole catchment using the diffusive wave model described by two parameters: celerity and diffusivity functions of geometrical characteristics of the channel network. Firstly, a geomorphological unit hydrograph obtained by routing a homogeneous effective rainfall was compared with the unit hydrograph identified by a lumped model scheme, then the distributed model was applied to take into account the spatial variability of effective rainfall in the catchment. Results show that this new method seems to be adapted for distributed hydrological modelling; it enables identification of a transfer function response for each hydrological unit, here subcatchments, and then simulation of the contribution of each unit to the hydrograph at the outlet. © 1997 by John Wiley & Sons, Ltd.     

Connectivity of post-fire runoff and sediment from nested hillslopes and watersheds

Wildfire increases the potential connectivity of runoff and sediment throughout watersheds due to greater bare soil, runoff and erosion as compared to pre-fire conditions. This research examines the connectivity of post-fire runoff and sediment from hillslopes (< 1.5 ha; n = 31) and catchments (< 1000 ha; n = 10) within two watersheds (< 1500 ha) burned by the 2012 High Park Fire in northcentral Colorado, USA. Our objectives were to: (1) identify sources and quantify magnitudes of post-fire runoff and erosion at nested hillslopes and watersheds for two rain storms with varied duration, intensity and antecedent precipitation; and (2) assess the factors affecting the magnitude and connectivity of runoff and sediment across spatial scales for these two rain storms. The two summer storms that are the focus of this research occurred during the third summer after burning. The first storm had low intensity rainfall over 11 hours (return interval <1–2 years), whereas the second event had high intensity rainfall over 1 hour (return interval <1–10 years). The lower intensity storm was preceded by high antecedent rainfall and led to low hillslope sediment yields and channel incision at most locations, whereas the high intensity storm led to infiltration-excess overland flow, high sediment yields, in-stream sediment deposition and channel substrate fining. For both storms, hillslope-to-stream sediment delivery ratios and area-normalised cross-sectional channel change increased with the percent of catchment that burned at high severity. For the high intensity storm, hillslope-to-stream sediment delivery ratios decreased with unconfined channel length (%). The findings quantify post-fire connectivity and sediment delivery from hillslopes and streams, and highlight how different types of storms can cause varying magnitues and spatial patterns of sediment transport and deposition from hillslopes through stream channel networks.     

Predictors of the peak width for networks with exponential links

We investigate optimal predictors of the peak (S) and distance to peak (T) of the width function of drainage networks under the assumption that the networks are topologically random with independent and exponentially distributed link lengths. Analytical results are derived using the fact that, under these assumptions, the width function is a homogeneous Markov birth-death process. In particular, exact expressions are derived for the asymptotic conditional expectations ofS andT given network magnitudeN and given mainstream lengthH. In addition, a simulation study is performed to examine various predictors ofS andT, includingN, H, and basin morphometric properties; non-asymptotic conditional expectations and variances are estimated. The best single predictor ofS isN, ofT isH, and of the scaled peak (S divided by the area under the width function) isH. Finally, expressions tested on a set of drainage basins from the state of Wyoming perform reasonably well in predictingS andT despite probable violations of the original assumptions.     

Variation in curve numbers derived from plot runoff data for New South Wales (Australia)

The curve number method is a simple one parameter (the curve number) rainfall runoff model. While its theoretical underpinning has been questioned it remains a powerful hydrological tool in the absence of detailed data and is therefore used extensively in hydrological models. This study aims to characterize the variation in maximum retention values (S), which underlie curve numbers, for a range of agricultural treatments across a large spatial area in New South Wales (NSW), Australia. The data used for the analysis spans several decades of rainfall runoff observations. A range of different derivation methods result in variation in mean and variance of S. In particular, methods that emphasize the larger storms result in greater S and thus lower runoff. For larger spatial scales, emphasis on larger storms gives more reliable estimates of S. Systematic variation in S arises from variations in treatment, pre‐runoff soil moisture, rainfall depth, and variations in cover. On the basis of the analysis, a table of curve number values for different land uses found in NSW is presented. The resulting distributions of S and curve numbers provide guidance for rainfall runoff modelling studies in the agricultural important areas of NSW. Copyright © 2011 John Wiley & Sons, Ltd.     

A semi-empirical method for predicting hydrological drought magnitudes in the Canadian prairies

Abstract

A hydrological drought magnitude (M _T ) expressed in standardized terms is predicted on annual, monthly and weekly time scales for a sampling period of T years in streamflow data from the Canadian prairies. The drought episodes are considered to follow the Poisson law of probability and, when coupled with the gamma probability distribution function (pdf) of drought magnitude (M) in the extreme number theorem, culminate in a relationship capable of evaluating the expected value, E(M _T ). The parameters of the underlying pdf of M are determined based on the assumption that the drought intensity follows a truncated normal pdf. The E(M _T ) can be evaluated using only standard deviation (σ), lag-1 autocorrelation (ρ) of the standardized hydrological index (SHI) sequence, and a weighting parameter Φ (ranging from 0 to 1) to account for the extreme drought duration (L _T ), as well as the mean drought duration (L_m ), in a characteristic drought length (L_c ). The SHI is treated as standard normal variate, equivalent to the commonly-used standardized precipitation index. A closed-form relationship can be used for the estimation of first-order conditional probabilities, which can also be estimated from historical streamflow records. For all rivers, at the annual time scale, the value of Φ was found equal to 0.5, but it tends to vary (in the range 0 to 1) from river to river at monthly and weekly time scales. However, for a particular river, the Φ value was nearly constant at monthly and weekly time scales. The proposed method estimates E(M _T ) satisfactorily comparable to the observed counterpart. At the annual time scale, the assumption of a normal pdf for drought magnitude tends to yield results in close proximity to that of a gamma pdf. The M _T , when transformed into deficit-volume, can form a basis for designing water storage facilities and for planning water management strategies during drought periods.

Editor D. Koutsoyiannis; Associate editor C. Onof

Citation Sharma, T.C. and Panu, U.S., 2013. A semi-empirical method for predicting hydrological drought magnitudes in the Canadian prairies. Hydrological Sciences Journal, 58 (3), 549–569.     

The influence of elevation error on the morphometrics of channel networks extracted from DEMs and the implications for hydrological modelling

Stream network morphometrics have been used frequently in environmental applications and are embedded in several hydrological models. This is because channel network geometry partly controls the runoff response of a basin. Network indices are often measured from channels that are mapped from digital elevation models (DEMs) using automated procedures. Simulations were used in this paper to study the influence of elevation error on the reliability of estimates of several common morphometrics, including stream order, the bifurcation, length, area and slope ratios, stream magnitude, network diameter, the flood magnitude and timing parameters of the geomorphological instantaneous unit hydrograph (GIUH) and the network width function. DEMs of three UK basins, ranging from high to low relief, were used for the analyses. The findings showed that moderate elevation error (RMSE of 1·8 m) can result in significant uncertainty in DEM‐mapped network morphometrics and that this uncertainty can be expressed in complex ways. For example, estimates of the bifurcation, length and area ratios and the flood magnitude and timing parameters of the GIUH each displayed multimodal frequency distributions, i.e. two or more estimated values were highly likely. Furthermore, these preferential estimates were wide ranging relative to the ranges typically observed for these indices. The wide‐ranging estimates of the two GIUH parameters represented significant uncertainty in the shape of the unit hydrograph. Stream magnitude, network diameter and the network width function were found to be highly sensitive to elevation error because of the difficulty in mapping low‐magnitude links. Uncertainties in the width function were found to increase with distance from outlet, implying that hydrological models that use network width contain greater uncertainty in the shape of the falling limb of the hydrograph. In light of these findings, care should be exercised when interpreting the results of analyses based on DEM‐mapped stream networks. Copyright © 2007 John Wiley & Sons, Ltd.     

Inverse modelling using PS‐InSAR data for improved land subsidence simulation in Las Vegas Valley,Nevada

Las Vegas Valley has had a long history of groundwater development and subsequent surface deformation. InSAR interferograms have revealed detailed and complex spatial patterns of subsidence in the Las Vegas Valley area that do not coincide with major pumping regions. This research represents the first effort to use high spatial and temporal resolution subsidence observations from InSAR and hydraulic head data to inversely calibrate transmissivities (T), elastic and inelastic skeletal storage coefficients (S_ke and S_kv) of the developed‐zone aquifer and conductance (CR) of the basin‐fill faults for the entire Las Vegas basin. The results indicate that the subsidence observations from InSAR are extremely beneficial for accurately quantifying hydraulic parameters, and the model calibration results are far more accurate than when using only groundwater levels as observations, and just a limited number of subsidence observations. The discrepancy between distributions of pumping and greatest levels of subsidence is found to be attributed to spatial variations in clay thickness. The Eglington fault separates thicker interbeds to the northwest from thinner interbeds to the southeast and the fault may act as a groundwater‐flow barrier and/or subsidence boundary, although the influence of the groundwater barrier to this area is found to be insignificant. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of the nonlinear storage–discharge relation for hillslopes through 2D numerical modelling

Storage–discharge curves are widely used in several hydrological applications concerning flow and solute transport in small catchments. This article analyzes the relation Q(S) (where Q is the discharge and S is the saturated storage in the hillslope), as a function of some simple structural parameters. The relation Q(S) is evaluated through two‐dimensional numerical simulations and makes use of dimensionless quantities. The method lies in between simple analytical approaches, like those based on the Boussinesq formulation, and more complex distributed models. After the numerical solution of the dimensionless Richards equation, simple analytical relations for Q(S) are determined in dimensionless form, as a function of a few relevant physical parameters. It was found that the storage–discharge curve can be well approximated by a power law function Q/(LK_s) = a(S/(L²(? ? θ_r)))^b, where L is the length of the hillslope, K_s the saturated conductivity, ? ? θ_r the effective porosity, and a, b two coefficients which mainly depend on the slope. The results confirm the validity of the widely used power law assumption for Q(S). Similar relations can be obtained by performing a standard recession curve analysis. Although simplified, the results obtained in the present work may serve as a preliminary tool for assessing the storage–discharge relation in hillslopes. Copyright © 2012 John Wiley & Sons, Ltd.     

Hydropedology of a mildly‐arid loess covered area,southern Israel

Extensive loess covered areas characterize the mildly arid areas of western Israel, where average annual rainfall is 280 mm. Hydrological data point to a peculiar hydrological behavior of the ephemeral streams. The frequency of sporadic flash floods is very high. However, even in extreme rain events peak discharges are extremely low. Hydrographs are usually characterized by very steep rising and falling limbs, representative of saturated areas, extending over a limited part of the watershed. Following this observation we advanced the hypothesis that storm channel runoff originated in the channel itself, with negligible contribution from the adjoining hillslopes. The study was based on two complementary approaches. The hydrological approach was based on the detailed analysis of rainfall–runoff relationships in a small watershed (11 km²) and on the analysis of the hydrological characteristics of the drainage network. The second approach was based on the toposequence concept. Several boreholes were dug along a hillslope 400 m long. Chemical data obtained show no significant difference in the downslope direction. Similar results were also obtained for the particle size distribution and soil moisture content. Data obtained perfectly fit the concept of ‘Partial Area Contribution’ as it presents an extreme case of hydrological discontinuity at the hillslope–channel interface. The lack of pedological trends in the downslope direction is an additional indication of the limited connectivity between the hillslopes and the adjoining channel. The limited connectivity is attributed to the prevalence of low rain intensities in the study area. The present study is also relevant to our understanding of pedological processes in dryland areas. The high frequency of intermittent low intensity rainstorms limits runoff generation and flow distances, and casts doubt on the general application of the toposequence approach. Copyright © 2016 John Wiley & Sons, Ltd.     

The information content theory for the estimation of the topographic index distribution used in TOPMODEL

This paper analyses the significance of the entropy concept in the topography parameterization within the model TOPMODEL proposed by Beven and Kirkby (1979), by means of the hydrological behaviour of an experimental basin in southern Italy. For a significant number of flood events recorded at the basin outlet, the performance of TOPMODEL for different spatial distributions of the topographic index, ln(a/tan β), has been observed. Performance is related to the information content estimated as an entropy measure, corresponding to each of the spatial distributions of the topographic index, with the aim of identifying the procedures most suitable to represent the hydrological process of rainfall–runoff. The results obtained have shown that for flood events corresponding to brief, heavy precipitation, some procedures provide better performances than others. Moreover, these improvements are justified by greater information content in the corresponding spatial distributions of the topographic index. Finally, TOPMODEL performances for some procedures have been analysed, varying the resolution scale of the topographic index. For analogous hydrological performances, scale change produced variations in some of the subsurface hydraulic parameters. These variations were proportional to a spatial variability measure of the topographic index distribution, derived from the corresponding information content. © 1997 John Wiley & Sons, Ltd.     

River flow mass exponents with fractal channel networks and rainfall

An important problem in hydrologic science is understanding how river flow is influenced by rainfall properties and drainage basin characteristics. In this paper we consider one approach, the use of mass exponents, in examining the relation of river flow to rainfall and the channel network, which provides the primary conduit for transport of water to the outlet in a large basin. Mass exponents, which characterize the power-law behavior of moments as a function of scale, are ideally suited for defining scaling behavior of processes that exhibit a high degree of variability or intermittency. The main result in this paper is an expression relating the mass exponent of flow resulting from an instantaneous burst of rainfall to the mass exponents of spatial rainfall and that of the network width function. Spatial rainfall is modeled as a random multiplicative cascade and the channel network as a recursive replacement tree; these fractal models reproduce certain types of self-similar behavior seen in actual rainfall and networks. It is shown that under these modeling assumptions the scaling behavior of flow mirrors that of rainfall if rainfall is highly variable in space, and on the other hand flow mirrors the structure of the network if rainfall is not so highly variable.     

Drought analysis of monthly hydrological sequences: a case study of Canadian rivers

Abstract

Two parameters of importance in hydrological droughts viz. the longest duration, L_T and the largest severity, S_T (in standardized form) over a desired return period, T years, have been analysed for monthly flow sequences of Canadian rivers. An important point in the analysis is that monthly sequences are non-stationary (periodic-stochastic) as against annual flows, which fulfil the conditions of stochastic stationarity. The parameters mean, μ, standard deviation, σ (or coefficient of variation), lag1 serial correlation, ρ, and skewness, γ (which is helpful in identifying the probability distribution function) of annual flow sequences, when used in the analytical relationships, are able to predict expected values of the longest duration, E(L_T ) in years and the largest standardized severity, E(S_T ). For monthly flow sequences, there are 12 sets of these parameters and thus the issue is how to involve these parameters to derive the estimates of E(L_T ) and E(S_T ). Moreover, the truncation level (i.e. the monthly mean value) varies from month to month. The analysis in this paper demonstrates that the drought analysis on an annual basis can be extended to monthly droughts simply by standardizing the flows for each month. Thus, the variable truncation levels corresponding to the mean monthly flows were transformed into one unified truncation level equal to zero. The runs of deficits in the standardized sequences are treated as drought episodes and thus the theory of runs forms an essential tool for analysis. Estimates of the above parameters (denoted as μ_av, σ_av, ρ_av, and γ_av) for use in the analytical relationships were obtained by averaging 12 monthly values for each parameter. The product- and L-moment ratio analyses indicated that the monthly flows in the Canadian rivers fit the gamma probability distribution reasonably well, which resulted in the satisfactory prediction of E(L_T ). However, the prediction of E(S_T ) tended to be more satisfactory with the assumption of a Markovian normal model and the relationship E(S_T ) ≈ E(L_T ) was observed to perform better.