Uncertainty assessment of a process-based integrated catchment model of phosphorus

Despite the many models developed for phosphorus concentration prediction at differing spatial and temporal scales, there has been little effort to quantify uncertainty in their predictions. Model prediction uncertainty quantification is desirable, for informed decision-making in river-systems management. An uncertainty analysis of the process-based model, integrated catchment model of phosphorus (INCA-P), within the generalised likelihood uncertainty estimation (GLUE) framework is presented. The framework is applied to the Lugg catchment (1,077 km²), a River Wye tributary, on the England–Wales border. Daily discharge and monthly phosphorus (total reactive and total), for a limited number of reaches, are used to initially assess uncertainty and sensitivity of 44 model parameters, identified as being most important for discharge and phosphorus predictions. This study demonstrates that parameter homogeneity assumptions (spatial heterogeneity is treated as land use type fractional areas) can achieve higher model fits, than a previous expertly calibrated parameter set. The model is capable of reproducing the hydrology, but a threshold Nash-Sutcliffe co-efficient of determination (E or R ²) of 0.3 is not achieved when simulating observed total phosphorus (TP) data in the upland reaches or total reactive phosphorus (TRP) in any reach. Despite this, the model reproduces the general dynamics of TP and TRP, in point source dominated lower reaches. This paper discusses why this application of INCA-P fails to find any parameter sets, which simultaneously describe all observed data acceptably. The discussion focuses on uncertainty of readily available input data, and whether such process-based models should be used when there isn’t sufficient data to support the many parameters.     

LEOPOLD A. HEINDL,US GEOLOGICAL SURVEY

Abstract

The distributed parameter model ANSWERS was used to predict runoff and soil loss from three agricultural watersheds in the arid zone of India. Model input parameters such as landform, drainage, soil and land use/land cover were derived from Landsat Thematic Mapper false colour composites and limited ground truth. The model predicted hydrographs and sediment graphs within acceptable limits. ANSWERS underpredicted the total soil loss by factors of 2.6 to 3.6. Reasons for these results are discussed.     

THE INFLUENCE OF LATERAL INHOMOGENEITIES ON THE DIPOLE METHODS*

The theory of electrical dipole soundings proved that this method can produce resistivity measurements, which are comparable with those obtained by electrical soundings of the Wenner or Schlumberger type. Their main advantage is the use of short cable lengths, which is important if the depth of penetration should be large. A considerable disadvantage of the dipole method is the great sensitivity to lateral discontinuities. Though these have an influence on the Schlumberger arrangement as well, they can disturb a dipole sounding to such an extent than an interpretation based on a horizontal layer case is no more possible. There are six different dipole arrays, which differ from each other with respect to the angle enclosed by the two dipole orientations-the current dipole AB and the measuring dipole MN. The theoretical comparison of the dipole arrays with the Schlumberger array concerning their sensitivity to lateral discontinuities is a useful basis for the choice of the most suitable configuration. Considering geological subsurface conditions the right choice of a dipole array can give an optimal result, i.e. a dipole sounding for which the sensitivity to lateral discontinuities is as small as possible under the given circumstances.     

Modelling suspended sediment concentration using artificial neural networks for Gangotri glacier

The dynamics of suspended sediment involves inherent non‐linearity and complexity because of existence of both spatial variability of the basin characteristics and temporal climatic patterns. This complexity, therefore, leads to inaccurate prediction by the conventional sediment rating curve (SRC) and other empirical methods. Over past few decades, artificial neural networks (ANNs) have emerged as one of the advanced modelling techniques capable of addressing inherent non‐linearity in the hydrological processes. In the present study, feed‐forward back propagation (FFBP) algorithm of ANNs is used to model stage–discharge–suspended sediment relationship for ablation season (May–September) for melt runoff released from Gangotri glacier, one of the largest glaciers in Himalaya. The simulations have been carried out on primary data of suspended sediment concentration (SSC) discharge and stage for ablation season of 11‐year period (1999–2009). Combinations of different input vectors (viz. stage, discharge and SSC) for present and previous days are considered for development of the ANN models and examining the effects of input vectors. Further, based on model performance indices for training and testing phase, a suitable modelling approach with appropriate model input structure is suggested. The conventional SRC method is also used for modelling discharge–sediment relationship and performance of developed models is evaluated by statistical indices, namely; root mean square error (RMSE), mean absolute error (MAE) and coefficient of determination (R²). Statistically, the performance of ANN‐based models is found to be superior as compared to SRC method in terms of the selected performance indices in simulating the daily SSC. The study reveals suitability of ANN approach for simulation and estimation of daily SSC in glacier melt runoff and, therefore, opens new avenues of research for application of hybrid soft computing models in glacier hydrology. Copyright © 2015 John Wiley & Sons, Ltd.     

THE EFFECTS OF INACCURATE INPUT PARAMETERS ON DEPOSITION OF SUSPENDED SEDIMENT

I.~OntjcnONEngineeringPredictionsofsedimenttransPOrtinopen-Channelsystems,especiallyifunsteadyflowconditionmustbeconsidered,isfraughtwithambigUityduetospatialandtemporalvariability,measurementerrors,lumtedsamplingoftheParameters,bounceandscallconditionS,andsink/sourcetenns.Duringrecentyearsmanysophisticatedmathematicalmodelshavebecomeavailabletocalculateunsteadyflowinone,tWoorthreedimensionS.ThesemodelswillbecoupledwithtranSportequationSforbedloadorsuSpendedloadtocalculateforexamplelongt…     

Impacts of model initialization on an integrated surface water–groundwater model

Integrated hydrologic models characterize catchment responses by coupling the subsurface flow with land surface processes. One of the major areas of uncertainty in such models is the specification of the initial condition and its influence on subsequent simulations. A key challenge in model initialization is that it requires spatially distributed information on model states, groundwater levels and soil moisture, even when such data are not routinely available. Here, the impact of uncertainty in initial condition was explored across a 208 km² catchment in Denmark using the ParFlow.CLM model. The initialization impact was assessed under two meteorological conditions (wet vs dry) using five depth to water table and soil moisture distributions obtained from various equilibrium states (thermal, root zone, discharge, saturated and unsaturated zone equilibrium) during the model spin‐up. Each of these equilibrium states correspond to varying computation times to achieve stability in a particular aspect of the system state. Results identified particular sensitivity in modelled recharge and stream flow to the different initializations, but reduced sensitivity in modelled energy fluxes. Analysis also suggests that to simulate a year that is wetter than the spin‐up period, an initialization based on discharge equilibrium is adequate to capture the direction and magnitude of surface water–groundwater exchanges. For a drier or hydrologically similar year to the spin‐up period, an initialization based on groundwater equilibrium is required. Variability of monthly subsurface storage changes and discharge bias at the scale of a hydrological event show that the initialization impacts do not diminish as the simulations progress, highlighting the importance of robust and accurate initialization in capturing surface water–groundwater dynamics. Copyright © 2015 John Wiley & Sons, Ltd.     

THE CLASSIFICATION OF PHYSICO-GEOLOGICAL MODELS OF MINERAL DEPOSITS*

We describe the concept of physico-geological models (PGMs) in geophysical exploration. They represent a “general model”, a spatial combination of a set of particular models (disturbing bodies). The modeling is called complete, incomplete or approximate, depending on the degree of characterization of the PGM by parameters such as dimension, shape and petrophysical property. Each of the three modeling types can be realized as a conceptual, and analytical, or a material PGM. Both deterministic and stochastic PGMs exist; deterministic models are mainly used to investigate the possibilities of a geophysical method, while stochastic models serve to substantiate complex geophysical interpretations. Depending on the geological problem, PGMs are subdivided into multi-alternative models (geological mapping, prediction, general prospecting) and double alternative models (specialized prospecting). An exploration-oriented classification of the PGMs of mineral deposits is discussed. According to this classification the variety of known genetic deposit types is reduced to a limited number of generalized PGM types. The development of typical PGMs is illustrated with examples of magnetitic deposits of Siberia.     

The effect of differences between rainfall measurement techniques on groundwater and discharge simulations in a lowland catchment

Several rainfall measurement techniques are available for hydrological applications, each with its own spatial and temporal resolution and errors. When using these rainfall datasets as input for hydrological models, their errors and uncertainties propagate through the hydrological system. The aim of this study is to investigate the effect of differences between rainfall measurement techniques on groundwater and discharge simulations in a lowland catchment, the 6.5‐km² Hupsel Brook experimental catchment. We used five distinct rainfall data sources: two automatic raingauges (one in the catchment and another one 30 km away), operational (real‐time and unadjusted) and gauge‐adjusted ground‐based C‐band weather radar datasets and finally a novel source of rainfall information for hydrological purposes, namely, microwave link data from a cellular telecommunication network. We used these data as input for the, a recently developed rainfall‐runoff model for lowland catchments, and intercompared the five simulated discharges time series and groundwater time series for a heavy rainfall event and a full year. Three types of rainfall errors were found to play an important role in the hydrological simulations, namely: (1) Biases, found in the unadjusted radar dataset, are amplified when propagated through the hydrological system; (2) Timing errors, found in the nearest automatic raingauge outside the catchment, are attenuated when propagated through the hydrological system; (3) Seasonally varying errors, found in the microwave link data, affect the dynamics of the simulated catchment water balance. We conclude that the hydrological potential of novel rainfall observation techniques should be assessed over a long period, preferably a full year or longer, rather than on an event basis, as is often done. Copyright © 2016 The Authors. Hydrological Processes. Published by John Wiley & Sons Ltd.     

An Assessment of Two-Dimensional Past Sea Level Reconstructions Over 1950–2009 Based on Tide-Gauge Data and Different Input Sea Level Grids

We compare different past sea level reconstructions over the 1950–2009 time span using the Empirical Orthogonal Function (EOF) approach. The reconstructions are based on 91 long (up to 60?years) but sparsely distributed tide-gauge records and gridded sea level data from two numerical ocean models over 1958–2007 (the DRAKKAR/NEMO model without data assimilation and the simple ocean data assimilation ocean reanalysis-SODA-) and satellite altimetry data over 1993–2009. We find that the reconstructed global mean sea level computed over the?~60-year-long time span little depends on the input spatial grids. This is unlike the regional variability maps that appear very sensitive to the considered input spatial grids. Using the DRAKKAR/NEMO model, we test the influence of the period covered by the input spatial grids and the number of EOFs modes used to reconstruct sea level. Comparing with tide-gauge records not used in the reconstruction, we determine optimal values for these two parameters. As suggested by previous studies, the longer the time span covered by the spatial grids, the better the fit with unused tide gauges. Comparison of the reconstructed regional trends over 1950–2009 based on the two ocean models and satellite altimetry grids shows good agreement in the tropics and substantial differences in the mid and high latitude regions, and in western boundary current areas as well. The reconstructed spatial variability seems very sensitive to the input spatial information. No clear best case emerges. Thus, using the longest available model-based spatial functions will not necessarily give the most realistic results as it will be much dependent on the quality of the model (and its associated forcing). Altimetry-based reconstructions (with 17-year long input grids) give results somewhat similar to cases with longer model grids. It is likely that better representation of the sea level regional variability by satellite altimetry compensates the shorter input grids length. While waiting for much longer altimetry records, improved past sea level reconstructions may be obtained by averaging an ensemble of different model-based reconstructions, as classically done in climate modelling. Here, we present such a ‘mean’ reconstruction (with associated uncertainty) based on averaging the three individual reconstructions discussed above.     

THE INFLUENCE OF OBLIQUELY DIPPING DISCONTINUITIES ON THE USE OF RAYLEIGH CHANNEL WAVES FOR THE IN-SEAM SEISMIC REFLECTION METHOD *

The geological sequence rock-coal-rock represents a seismic low-velocity channel. Channel waves generated in a coal seam and their reflections from discontinuities can be used for proving the minability of the seam. To investigate the process of reflection, two-dimensional models of the sequence rock-coal-rock have been investigated by means of the ultrasonic transducer technique. Two-dimensional models have the advantage that the wave field can be observed at an arbitrarily chosen point of the model plate. Thus, by means of these models the direct and the reflected Rayleigh channel wave can be observed along their path of propagation. From the geophysical point of view the various types of discontinuities of a coal seam can be divided into two basic types: one is restricted to the seam, the other includes in addition a fractured zone in the adjoining rock. The investigation of the symmetrical Rayleigh channel wave reflected by a discontinuity yields the following results: For dip angles γ between approximately 90° and approximately 60° the reflectivity is virtually independent of the type of discontinuity. This does not hold for the limiting case of γ= 90° (vertical dip) for which the reflectivity increases with increasing influence of the fractured zone. For dip angles γ between approximately 60° and approximately 40° the reflectivity is still independent of the fractured zone but the shape of the reflected wave deteriorates with increasing influence of the fractured zone. For dip angles γ below approximately 40° the reflected wave deteriorates such that the application of the in-seam seismic reflection method will be difficult or even impossible. The conversion of the direct wave of the symmetrical fundamental mode into a reflected wave of the antisymmetrical fundamental mode has been observed.     

Weighing the importance of model uncertainty against parameter uncertainty in earthquake loss assessments

Epistemic uncertainties can be classified into two major categories: parameter and model. While the first one stems from the difficulties in estimating the values of input model parameters, the second comes from the difficulties in selecting the appropriate type of model. Investigating their combined effects and ranking each of them in terms of their influence on the predicted losses can be useful in guiding future investigations. In this context, we propose a strategy relying on variance-based global sensitivity analysis, which is demonstrated using an earthquake loss assessment for Pointe-à-Pitre (Guadeloupe, France). For the considered assumptions, we show: that uncertainty of losses would be greatly reduced if all the models could be unambiguously selected; and that the most influential source of uncertainty (whether of parameter or model type) corresponds to the seismic activity group. Finally, a sampling strategy was proposed to test the influence of the experts’ weights on models and on the assumed coefficients of variation of parameter uncertainty. The former influenced the sensitivity measures of the model uncertainties, whereas the latter could completely change the importance rank of the uncertainties associated to the vulnerability assessment step.     

Hydraulic model calibration for extreme floods in bedrock‐confined channels: case study from northern Thailand

Palaeoflood reconstructions based on stage evidence are typically conducted in data‐poor field settings. Few opportunities exist to calibrate the hydraulic models used to estimate discharge from this evidence. Consequently, an important hydraulic model parameter, the roughness coefficient (e.g. Manning's n), is typically estimated by a range of approximate techniques, such as ‘visual estimation’ and semi‐empirical equations. These techniques contribute uncertainty to resulting discharge estimates, especially where the study reach exhibits sensitivity in the discharge–Manning's n relation. We study this uncertainty within a hydraulic model for a large flood of known discharge on the Mae Chaem River, northern Thailand. Comparison of the ‘calibrated’ Manning's n with that obtained from semi‐empirical equations indicates that these underestimate roughness. Substantial roughness elements in the extra‐channel zone, inundated during large events, contribute significant additional sources of flow resistance that are captured neither by the semi‐empirical equations, nor by existing models predicting stage–roughness variations. This bedrock channel exhibits a complex discharge–Manning's n relation, and reliable estimates of the former are dependent upon realistic assignment of the latter. Our study demonstrates that a large recent flood can provide a valuable opportunity to constrain this parameter, and this is illustrated when we model a palaeoflood event in the same reach, and subsequently examine the magnitude–return period consequences of discharge uncertainty within a flood frequency analysis, which contributes its own source of uncertainty. Copyright © 2005 John Wiley & Sons, Ltd.     

Simulation of long‐term soil redistribution by tillage using a cellular automata model

The process of tillage translocation is well studied and can be described adequately by different existing models. Nevertheless, in complex environments with numerous obstacles, such as olive orchards, the application of conventional tillage erosion models is not straightforward. However, such obstacles have important effects on the spatial pattern of soil redistribution and on resulting soil properties. Cellular automata could provide a valuable alternative here. This study aims at developing a cellular automata model for tillage translocation (CATT) that can take into account such obstacles, exploring its possibilities and limitations. Firstly, model outcome was tested on a traditional field with rolling topography, for which caesium‐137 (¹³⁷Cs) inventories are available. The observed spatial soil redistribution patterns could be adequately represented by the CATT model. Secondly, a global sensitivity analysis was performed to explore the effect of input parameter uncertainty on several selected model outputs. The variance‐based extended Fourier Amplitude Sensitivity Test (FAST) method was used to determine first‐ and total‐order sensitivity indices. Tillage depth was identified as the input parameter that determined most of the output variance, followed respectively by tillage direction and speed. The high difference between the total‐ and first‐order sensitivity indices indicated that, in spite of the simple model structure, the model behaves non‐linearly with respect to some of the model output variables. Higher order interactions were especially important for determining the proportion of eroding and deposition cells. Finally, simulations were performed to analyse the model behaviour in complex landscapes, applying it to a field with protruding obstacles (representing olive trees). The model adequately represented some morphological features observed in actual olive orchards, such as mounds around the olive trees. The results show that cellular automata are an appropriate tool to describe long‐term tillage soil redistribution. Copyright © 2010 John Wiley & Sons, Ltd.     

Influence of Elevation Data Source on 2D Hydraulic Modelling

The aim of this paper is to analyse the influence of the source of various elevation data on hydraulic modelling in open channels. In the research, digital terrain models from different datasets were evaluated and used in two-dimensional hydraulic models. The following aerial and satellite elevation data were used to create the representation of terrain–digital terrain model: airborne laser scanning, image matching, elevation data collected in the LPIS, EuroDEM, and ASTER GDEM. From the results of five 2D hydrodynamic models with different input elevation data, the maximum depth and flow velocity of water were derived and compared with the results of the most accurate ALS data. For such an analysis a statistical evaluation and differences between hydraulic modelling results were prepared. The presented research proved the importance of the quality of elevation data in hydraulic modelling and showed that only ALS and photogrammetric data can be the most reliable elevation data source in accurate 2D hydraulic modelling.     

An extended Bayesian sediment fingerprinting mixing model for the full Bayes treatment of geochemical uncertainties

Recent advances in sediment fingerprinting research have seen Bayesian mixing models being increasingly employed as an effective method to coherently translate component uncertainties into source apportionment results. Here, we advance earlier work by presenting an extended Bayesian mixing model capable of providing a full Bayes treatment of geochemical uncertainties. The performance of the extended full Bayes model was assessed against the equivalent empirical Bayes model and traditional frequentist optimisation. The performance of models coded in different Bayesian software (JAGS and Stan) was also evaluated, alongside an assessment of model sensitivity to reduced source representativeness and nonconservative fingerprint behaviour. Results revealed comparable accuracy and precision for the full and empirical Bayes models across both synthetic and real sediment geochemistry datasets, demonstrating that the empirical treatment of source data here represents a close approximation of the full Bayes treatment. Contrasts in the performance of models coded in JAGS and Stan revealed that the choice of software employed can impact significantly upon source apportionment results. Bayesian models coded in Stan were the least sensitive to both reduced source representativeness and nonconservative fingerprint behaviour, indicating Stan as the preferred software for future Bayesian sediment fingerprinting studies. Whilst the frequentist optimisation generally yielded comparable accuracy to the Bayesian models, uncertainties around apportionment estimates were substantially greater and the frequentist model was less effective at dealing with nonconservative behaviour. Overall, the effective performance of the extended full Bayes mixing model coded in Stan represents a notable advancement in source apportionment modelling relative to previous approaches. Both the mixing model and the software comparisons presented here should provide useful guidelines for future sediment fingerprinting studies.     

Regionalizing mean annual flow and daily flow variability for basin‐scale sediment and nutrient modelling

River discharges vary strongly through time and space, and quantifying this variability is fundamental to understanding and modelling river processes. The river basin is increasingly being used as the unit for natural resource planning and management; to facilitate this, basin‐scale models of material supply and transport are being developed. For many basin‐scale planning activities, detailed rainfall‐runoff modelling is neither necessary nor tractable, and models that capture spatial patterns of material supply and transport averaged over decades are sufficient. Nevertheless, the data to describe the spatial variability of river discharge across large basins for use in such models are often limited, and hence models to predict river discharge at the basin scale are required. We describe models for predicting mean annual flow and a non‐dimensional measure of daily flow variability for every river reach within a drainage network. The models use sparse river gauging data, modelled grid surfaces of mean annual rainfall and mean annual potential evapotranspiration, and a network accumulation algorithm. We demonstrate the parameterization and application of the models using data for the Murrumbidgee basin, in southeast Australia, and describe the use of these predictions in modelling sediment transport through the river network. The regionalizations described contain less uncertainty, and are more sensitive to observed spatial variations in runoff, than regionalizations based on catchment area and rainfall alone. Copyright © 2006 John Wiley & Sons, Ltd.     

Grid parameterization of a conceptual distributed hydrological model through integration of a sub-grid topographic index: necessity and practicability

Abstract

Grid-based distributed models have become popular for describing spatial hydrological processes. However, the influence of non-homogeneity within a grid on streamflow simulation was not adequately addressed in the literature. In this study, we investigated how the statistical characteristics of soil moisture storage within a grid impacts on streamflow simulations. The spatial variation of the topographic index, TI, within a grid was used to determine parameter B of the statistical curve of soil moisture storage in the Xinanjiang model. For comparison of influences of the non-homogeneity within a grid on streamflow simulation, two parameterization schemes of soil moisture storage capacity were developed: a grid-parameterization scheme for a distributed model and a catchment-averaged scheme for a semi-distributed model. The practicability and usefulness of the grid-parameterization method were evaluated through model comparisons. The two models were applied in Jiangwan experimental catchment Zhejiang Province, China. Streamflow discharge data at the catchment outlet from 1971 to 1986 at different temporal resolutions, e.g. 15 min and daily time step, were used for model calibration and validation. Statistical results for different grid scales demonstrated that the mean and variation of TI and B decline significantly as the grid scale increases. The simulated streamflow discharges of the two models were similar and the semi-distributed model outperformed the distributed model slightly when the streamflow at the outlet of the catchment was used as the only basis for comparison. In addition, a relatively larger bias in the predicted discharges between these two models was observed along with an abrupt increase of soil moisture saturation ratio. A further analysis of the simulated soil moisture content distribution revealed that the distributed model can provide a reasonable representation of the variable source area concept, which was justified to some extent by the field experiment data.

Editor D. Koutsoyiannis

Citation Liu, J.T., Chen, X., Wu, J.C., Zhang, X.N., Feng, D.Z. and Xu, C.-Y., 2012. Grid parameterization of a conceptual, distributed hydrological model through integration of a sub-grid topographic index: necessity and practicability. Hydrological Sciences Journal, 57 (2), 282–297.     

SUR LA PUISSANCE DES CRUES AU JAPON A MADAGASCAR ET A LA REUNION

Abstract

Hydrological modelling has faced the problem of ungauged basins for many years: how does one estimate hydrological characteristics for a river for which there are no data? Whatever the kind of model, it needs at least hydroclimatic input data and discharge data for calibration. However, the Yates model does not need any discharge data for calibration: it is a pre-calibrated model from a vegetation—climate classification map. In the specific context of West and Central Africa, where data are often of poor quality and very scarce, it is interesting to compare the performance of such a model with those of calibrated models, and with observed data. For this study, a platform including different semi-global rainfall—runoff models which allow the estimation of monthly runoff at a spatial resolution of 0.5° × 0.5° was used. The performance of the Yates model is very close to those of calibrated models, so that one can say that this simple model, based simply on a vegetation—climate classification, can be a very useful prediction tool in regions of scarce and unreliable data, such as those of interest to the International Association of Hydrological Sciences (IAHS) initiative on prediction in ungauged basins (PUB). Therefore, this model was applied to a period covering the last 30 years, and to a data set covering the first decades of the 21st century, from a climatic scenario of doubling the CO₂ concentration in the atmosphere. The results show that, in West Africa, where drought conditions have now prevailed for 35 years, water resources should still be decreasing in the future, following the general decreasing trend of rainfall projected by the climatic scenarios.     

论地震孕育的层间解缚模式与我国某些大震的余震机制

本文主要讨论了以下几点: 1.在1979年提出的层间解缚模式的基础上进一步论证了在不同的垂直力作用下(地壳下部作下沉,上升和垂直差异运动)在地壳中部软弱面附近两侧岩层(花岗岩层和玄武岩层)间的层间解缚形式,並讨论了层间解缚发生时对大震发生的影响。2.由海城、唐山这两次平推型大震后较大余震平面分布的不对称性论证了层间解缚现象的客观存在性。3.由海城、龙陵、唐山三次平推型大震后余震机制倾滑分量比主震明显增大,余震断层面走向角度在较大范围内变化以及倾滑型余震在整个下沉区的分布论证了大震发生后震中区岩层下塌至解缚区后所引起的岩石碎块间的重力调整过程。     

ON THE ROLE OF PARTIAL RAY EXPANSION IN THE COMPUTATION OF RAY SYNTHETIC SEISMOGRAMS FOR LAYERED STRUCTURES1

This paper presents results of testing an efficient ray generation scheme needed whenever ray synthetic seismograms are to be computed for layered models with more than 10‘ thick’layers. Our ray generation algorithm is based on the concept of kinematically equivalent waves (the kinematic analogs) having identical traveltimes along different ray-paths between the source and the receiver, both located on the surface of the model. These waves, existing in any medium composed of laterally homogeneous parallel layers, interfere at any location along the recording surface, thereby producing a composite wavelet whose amplitude and shape depend directly on the number of kinematic analogs (the multiplicity factor). Hence, explicit knowledge of the multiplicity factor is crucial for any analysis based on the amplitude and shape of individual wavelets, such as wavelet shaping, Q estimation, or linearized wavelet inversion. For unconverted waves, such as those discussed in this paper, the multiplicity factor can be computed analytically using formulae given in the Appendix; for converted waves, the multiplicity factor should be computed numerically, using the algorithm employed for the computation of the seismograms presented in a previous paper by one of the authors.