Grid Point Values Research Articles

APPLICABILITY OF ATMOSPHERIC REANALYSIS DATA FOR THE REPRODUCTION OF TYPHOON-INDUCED STORM SURGE IN JAPAN

The accurate estimation of a typhoon is a key to improving the reproducibility of typhoon-induced storm surges. The grid point value (GPV) data of Local Forecasting Model (LFM), which are provided by the Japan Meteorological Agency (JMA), are shown to be the most accurate data to simulate typhoons in and around Japan. However, due to the limitations of LFM data (limited spatial and temporal coverage), the atmospheric reanalysis data, which are available for the global domain covering a long period, are commonly used in practical applications. According to previous researchers, it is expected to give a better representation of typhoons in terms of structure and intensity with reanalysis datasets although they are in too coarse resolutions to capture the internal dynamics (Malakar et al., 2020), and the reproducibility of typhoons with different reanalysis datasets varies over different oceanic basins (Schenkel and Hart, 2012). Hence, this paper analyses the two major atmospheric reanalysis datasets together with LFM dataset, and confirms their applicability to reproduce typhoon-induced storm surges in Japan.

Post-extrapolation for specified time-step results without interpolation in MOC-based 1D hydraulic transients and gas release computations

The goal of the paper is to present a supplementary step called postextrapolation. When applied to the well-known method of characteristics (MOC), this assures the continuous use of the specified time steps or regular numerical grid without interpolations during computations of transients in 1D 2-phase flow in straight elastic pipes. The new method consists of two steps, the first being a typical MOC step, where the C− and C+ characteristics start from regular nodal points, allowing for the point of intersection to differ from a regular one. After defining the variables there the method transforms it corresponding to the near regular grid point, using the first derivatives contained in the original, nonlinear, governing equations, as evaluated numerically from the variables got earlier in the neighboring nodes. The procedure needs no interpolations; it deals with grid-point values only. Instead of the Courant-type stability conditions, shock-wave catching and smoothing techniques help to assure numerical stability between broad limits of parameters like the closing time of a valve and the initial gas content of the fluid. Comparison by runs with traditional codes under itemized boundary conditions and measurements on a simple TPV (tank-pipe-valve) setup show acceptable scatter.

Intercomparisons of methods for extracting the internal climate variability from the observed records over the Indo‐Pacific sector

AbstractA careful isolation of the externally forced component (EFC) and the internal climate variability (ICV) embedded in the observed records as well as the climate simulations is critical to investigate an actual response to the external radiative forcing and/or background dynamics in the ICV. Employing three different methods, we evaluate the EFCs contained in the observed sea surface temperature (SST) and sea level pressure (SLP) fields in the Indo‐Pacific region. After removing these EFCs, we obtain the respective ICVs as the remaining anomalies. The remaining SST and SLP anomalies are then evaluated on decadal time scales in a combined empirical orthogonal function (EOF) analysis of different spatial portions: the tropical Pacific, the Indian Ocean and the whole Indo‐Pacific region. After making statistical intercomparisons of the spatial patterns and associated time series of the EOF analyses, we found that the EFCs of the individual grid point values (GPVs) were appropriately estimated by regressing onto the multi‐model ensemble global mean surface temperature (GMSTMME) and were less well approximated by the conventional linear trend and the multi‐model ensemble mean of the simulated GPVs. The regressed SST anomalies of the individual historical simulations onto the GMSTMME were much larger than the observed anomalies, illustrating that the ICV‐to‐EFC variance ratio is a performance‐improvement indicator of climate models.

Applicability of a nationwide flood forecasting system for Typhoon Hagibis 2019

Floods can be devastating in densely populated regions along rivers, so attaining a longer forecast lead time with high accuracy is essential for protecting people and property. Although many techniques are used to forecast floods, sufficient validation of the use of a forecast system for operational alert purposes is lacking. In this study, we validated the flooding locations and times of dike breaking that had occurred during Typhoon Hagibis, which caused severe flooding in Japan in 2019. To achieve the goal of the study, we combined a hydrodynamic model with statistical analysis under forcing by a 39-h prediction of the Japan Meteorological Agency's Meso-scale model Grid Point Value (MSM-GPV) and obtained dike-break times for all flooded locations for validation. The results showed that this method was accurate in predicting floods at 130 locations, approximately 91.6% of the total of 142 flooded locations, with a lead time of approximately 32.75 h. In terms of precision, these successfully predicted locations accounted for 24.0% of the total of 542 locations under a flood warning, and on average, the predicted flood time was approximately 8.53 h earlier than a given dike-break time. More warnings were issued for major rivers with severe flooding, indicating that the system is sensitive to extreme flood events and can issue warnings for rivers subject to high risk of flooding.

Skill and Potential Economic Value of Forecasts of Ice Accretion on Wind Turbines

AbstractIn this paper, a verification study of the skill and potential economic value of forecasts of ice accretion on wind turbines is presented. The phase of active ice formation on turbine blades has been associated with the strongest wind power production losses in cold climates; however, skillful icing forecasts could permit taking protective measures using anti-icing systems. Coarse- and high-resolution forecasts for the range up to day 3 from global (IFS and GFS) and limited-area (WRF) models are coupled to the Makkonen icing model. Surface and upper-air observations and icing measurements at turbine hub height at two wind farms in central Europe are used for model verification over two winters. Two case studies contrasting a correct and an incorrect forecast highlight the difficulty of correctly predicting individual icing events. A meaningful assessment of model skill is possible only after bias correction of icing-related parameters and selection of model-dependent optimal thresholds for ice growth rate. The skill of bias-corrected forecasts of freezing and humid conditions is virtually identical for all models. Hourly forecasts of active ice accretion generally show limited skill; however, results strongly suggest the superiority of high-resolution WRF forecasts relative to other model variants. Predictions of the occurrence of icing within a period of 6 h are found to have substantially better accuracy. Probabilistic forecasts of icing that are based on gridpoint neighborhood ensembles show slightly higher potential economic value than forecasts that are based on individual gridpoint values, in particular at low cost-loss ratios, that is, when anti-icing measures are comparatively inexpensive.

Possibility of Wave Prediction by Deep Learning

Numerical wave prediction models require a large amount of computational power to timely complete the required calculations. Artificial Neural Networks (ANN) have been introduced to perform predictions at a lesser computational cost and increased processing speed. Deep learning and specifically Convolutional Neural Networks (CNN) have become accepted for various image recognition applications. Motivation for the examination of wave prediction by deep learning came from the success of CNN in vision applications and the similarity of meteorological weather grid data to visual images. This study investigates a deep learning technique using the Japan Meteorological Agency’s Grid Point Value Mesoscale Model to predict wave height and period. In particular, this study uses the Xception deep learning architecture with depthwise separable convolution to obtain improved wave height and period prediction over artificial neural networks, and gets overall success results.

An objective definition of seasons for the Mediterranean region

AbstractAn objective definition of seasons for the Mediterranean region is performed using the mean intra‐annual variations of 12 climatological parameters. The aim is to achieve a climatologically appropriate determination of the onset and cessation dates, as well as the duration, of each season, better than the conventional seasons (winter, spring, summer and autumn). Moreover, possible changes of the defined seasons during the 70‐year study period 1949–2018 are investigated over the climatologically sensitive Mediterranean region. The data used are daily NCEP/NCAR Reanalysis grid point values of precipitation rate, convective precipitation rate, 2 m temperature, total cloud cover, 2 m zonal and meridional wind, 500 and 1,000 hPa geopotential height, 500 and 850 hPa temperature, 850 hPa specific humidity and precipitable water over the Mediterranean region, for the period 1949–2018. Firstly, Principal Component Analysis is applied to the mean intra‐annual variations of the above parameters in order to reduce the dimensionality and then k‐means Cluster Analysis is applied to the resultant Principal Components in order to group dates with similar regimes of the above parameters. This methodology is applied for the 70‐year period 1949–2018 as well as for the five overlapping 30‐year subperiods 1949–1978, 1959–1988, 1969–1998, 1979–2008, and 1989–2018. According to the results, four seasons are defined for all periods. Although these seasons correspond to the four conventional seasons, there are differences regarding the onset and cessation dates and the duration. In general, it is found that winter lasts about 4 months, spring and summer are a little shorter than 3 months and autumn lasts about 2.5 months. The most remarkable long‐term changes are: (a) the warming during the last 30‐year period, which agrees with the ongoing climate change, (b) the shortening of winter and spring due to the delay of their onset date, and (c) the extension of autumn due to the delay of its cessation date.

Study on Storm Surge Using Parametric Model with Geographical Characteristics

The coastal area of Japan has been damaged yearly by storm surges and flooding disasters in the past, including those associated with typhoons. In addition, the scale of damage is increasing rapidly due to the changing global climate and environment. As disasters due to storm surges become increasingly unpredictable, more measures should be taken to prevent serious damage and casualties. The Japanese government published a hazard map manual in 2015 and obligates the creation of a hazard map based on a parametric model as a measure to reduce high-scale storm surges. Parametric model (typhoon model) accounting for the topographical influences of the surroundings is essential for calculating the wind field of a typhoon. In particular, it is necessary to calculate the wind field using a parametric model in order to simulate a virtual typhoon (the largest typhoon) and to improve the reproducibility. Therefore, in this study, the aim was to establish a hazard map by assuming storm surges of the largest scale and to propose a parametric model that considers the changing shape of typhoons due to topography. The main objectives of this study were to analyze the characteristics of typhoons due to pass through Japan, to develop a parametric model using a combination of Holland’s and Myers’s models that is appropriate for the largest scale of typhoon, and to analyze the parameters of Holland’s model using grid point values (GPVs). Finally, we aimed to propose a method that considers the changing shape of typhoons due to topography. The modeling outcomes of tide levels and storm surge heights show that the reproduced results obtained by the analysis method proposed in this study are more accurate than those obtained using GPVs. In addition, the reproducibility of the proposed model was evaluated showing the high and excellent reproducibility of storm surge height according to the geographic characteristics.

Comparative assessment of NCEP and ECMWF global datasets and numerical approaches on rough sea ship navigation based on numerical simulation and shipboard measurements

Long-term shipboard measurements of vessel performance in actual seas and practical guidance for proper ship operations in rough sea conditions are of great importance for a safe, economical and environment-friendly optimum ship routing. Data was collected from a bulk carrier during 2010–2016 covering both the Southern and Northern Hemispheres with high seasonal variability. Using the meteorological and wave models, the Weather Research and Forecasting Model (WRF) and WAVEWATCH III (WW3), numerical simulations of the weather and sea state were conducted for eight rough sea cases using the grid point value (GPV) datasets NCEP-FNL and ERA-Interim. Model results were validated by shipboard measurements and ship motion calculations. Results show a better performance of NCEP-FNL on generating high ocean surface wind speed of heavy weather than that of the ERA-Interim, while the ERA-Interim tends to perform better for generation of wind direction. Besides, utilization of a high-resolution WRF model tends to increase the accuracy of wind simulations for extremely high and low wind speed cases when compared to the linear interpolation of GPV datasets, which tends to perform better for relatively modest wind speed conditions. Results also suggest that, to achieve an optimum ship routing considering safety, cost and emission reduction; it should use the high-resolution WRF model with NCEP-FNL initial and boundary conditions to generate ocean surface winds, followed by the WW3 wave model to generate ocean surface waves for the most accurate performance in heavy weather.

Comparative study on WRF model simulations from the viewpoint of optimum ship routing

In addition to other studies focused on the WRF performance of rainfall, temperature, and precipitation, the present study provides practical suggestions for ship operators on choosing grid point value (GPV) datasets and WRF modeling approaches to help make better decisions on weather routing by estimating strong wind with a higher accuracy. Two different GPV datasets, the NCEP-FNL and ERA-Interim, are used for WRF modeling. Various modeling approaches, which include applying different physics options and choosing whether to adopt the four-dimensional-data-assimilation, are employed for numerical simulation. Then, on-board measurements of a bulk carrier has been used to validate the calculated wind of eight rough-sea navigational cases. This occurs over a wide spatial range, covering ocean regions such as the Northwest Pacific Ocean, South of Australia, Tasman Sea, Southeast of Africa, and Southeast of America, as well as a wide temporal range of four years. Results of 64 WRF simulations show that ERA-Interim is suitable for longer periods, while NCEP-FNL can analyze extremely high winds. Additionally, grid nudging can also reduce the wind strength due to its smoothing effect. Finally, selections of these different GPV datasets and modeling approaches for ship weather routing are discussed, particularly for rough-sea navigations.

Accuracy Comparison of Coastal Wind Speeds between WRF Simulations Using Different Input Datasets in Japan

In order to improve the accuracy of the wind speed simulated by a mesoscale model for the wind resource assessment in coastal areas, this study evaluated the effectiveness of using the Japan Meteorological Agency (JMA)’s latest and finest (2 km × 2 km) grid point value (GPV) data, produced from the local forecast model (LFM) as input data to the mesoscale model. The evaluation was performed using wind lidar measurements at two sites located on the coasts of the Sea of Japan and Pacific Ocean. The accuracy of the LFM–GPV was first compared with that of two products from the JMA Meso Scale Model (MSM) (5 km × 5 km): MSM-GPV and mesoscale analysis (MANAL). Consequently, it was shown that LFM–GPV exhibited the most accurate wind speeds against lidar measurements. Next, dynamical downscaling simulations were performed using the weather research and forecasting model (WRF) forced by the three datasets above, and their results were compared. As compared to the GPVs, it was found that the WRF dynamical downscaling simulation using them as input can improve the accuracy of the coastal wind speeds. This was attributed to the advantage of the WRF simulation to improve the negative bias from the input data, especially for the winds blowing from the sea sectors. It was also found that even if the LFM–GPV is used as an input to the WRF simulation, it does not always reproduce more accurate wind speeds, as compared to the simulations using the other two datasets. This result is partly owing to the tendency of WRF to overestimate the wind speed over land, thus obscuring the higher accuracy of the LFM–GPV. It was also shown that the overestimation tendency cannot be improved by only changing the nudging methods or the planetary boundary layer schemes in WRF. These results indicate that it may be difficult to utilize the LFM–GPV in the WRF wind simulation, unless the overestimation tendency of WRF itself is improved first.

A Novel DNA and Dengue Virus Protein Binding of Antipyridine Based Copper and Zinc Metal Complexes through Molecular Docking Studies

The Cu(II) and Zn(II) complexes were synthesized using Schiff base ligand by refluxing with CuCl2 and ZnCl2. A Schiff base ligand (L) was synthesized by condensing o-vanillin and 4-aminoantipyrine. The synthesized ligand and metal complexes were characterized by IR and mass spectrometry. The synthesized Cu(II) and Zn(II) metal complexes were especially docked with human DNA (PDB I.D: 1BNA) and dengue protein virus (PDB ID: 2VBC) using auto dock software tools version 1.5.6 and pymol. The binding of the ligand and biomolecule in grid point value of x × y × z directions of 90 × 90 × 90 and a grid space group value of 0.380 Å. The binding energy values of Cu(II) and Zn(II) complexes were respectively -7.1 and -7.4 kcal mol–1 towards NS3 protease-helicase, while the binding energy values of Cu(II) and Zn(II) complexes were found to be -8.5 and -8.2 kcal mol–1, respectively towards B-DNA.

A hierarchical analysis of the impact of methodological decisions on statistical downscaling of daily precipitation and air temperatures

Despite the widespread application of statistical downscaling tools, uncertainty remains regarding the role of model formulation in determining model skill for daily maximum and minimum temperature (Tmax and Tmin), and precipitation occurrence and intensity. Impacts of several key aspects of statistical transfer function form on model skill are evaluated using a framework resistant to model overspecification. We focus on: (a) model structure: simple (generalized linear models, GLMs) versus complex (artificial neural networks, ANNs) models. (b) Predictor selection: Fixed number of predictors chosen a priori versus stepwise selection of predictors and inclusion of grid point values versus predictors derived from application of principal components analysis (PCA) to spatial fields. We also examine the influence of domain size on model performance. For precipitation downscaling, we consider the role of the threshold used to characterize a wet day and apply three approaches (Poisson and Gamma distributions in GLM and ANN) to downscale wet‐day precipitation amounts. While no downscaling formulation is optimal for all predictands and at 10 locations representing diverse U.S. climates, and due to the exclusion of variance inflation all of the downscaling formulations fail to reproduce the range of observed variability, models with larger suites of prospective predictors generally have higher skill. For temperature downscaling, ANNs generally outperform GLM, with greater improvements for Tmin than Tmax. Use of PCA‐derived predictors does not systematically improve model skill, but does improve skill for temperature extremes. Model skill for precipitation occurrence generally increases as the wet‐day threshold increases and models using PCA‐derived predictors tend to outperform those based on grid cell predictors. Each model for wet‐day precipitation intensity overestimates annual total precipitation and underestimates the proportion derived from extreme precipitation events, but ANN‐based models and those with larger predictor suites tend to have the smallest bias.

Test of a cubic spline interface for physical processes with a 1-D third-order spectral element model

A common way to introduce physical processes into numerical models of the atmosphere is to call the parameterization at every grid point. This can lead to considerable errors. A simple 1-D example is proposed to illustrate that when a physical process occurs at one grid point only, a considerable sampling error may occur, with the result that only a fraction of the true impact of this process is seen. The interface to the physical parameterization in numerical weather prediction model using a third-order 1-D spectral element method (SEM3) model is investigated by homogeneous advection. In SEM3, the grid points, called principal nodes, are at boundaries of computational intervals and two more collocation points in the interior of each cell. This article argues that it is sufficient to do the physical parameterization for principal nodes only that creating the interior grid-point values of physics schemes by linear interpolation. This is called the spline interface method. A simple condensation model of water is taken as an example. Compared to the standard paramaterization, which computes the physical processes at every grid point, the spline interface method is more accurate and has a potential to save computer time. It turns out that the standard method creates a noisy wave which can easily be filtered by hyperviscosity. In the spline interface to the condensation physics, the condensation is done at every third grid point only. Third-order spline methods are used to represent the condensation at other points. The method using a smaller grid to compute condensation represented the condensation process more accurately and produced less of the computational noise. This version could be run without hyperviscosity, as no significant computational noise mode was generated by condensation. By doing physical processes only at every third grid point computer time may be saved.

A New Proposal for Vertical Extrapolation of Offshore Wind Speed and an Assessment of Offshore Wind Energy Potential for the Hibikinada Area, Kitakyushu, Japan

The author’s research group has been conducting research on applications of various meteorological Grid Point Value (GPV) data offered by the Japan Meteorological Agency (JMA) to the field of wind power generation. In particular, the group’s research has been focusing on the following areas: 1) the use of GPV data from the JMA Meso-Scale Model (MSM-S; horizontal resolution: 5 km) and the JMA Local Forecast Model (LFM-S; horizontal resolution: 2 km), and 2) examinations of the prediction accuracy of local wind assessment with the use of these data. In both the MSM-S and the LFM-S, grid points are fixed at 10 m above the sea (ground) surface. The purpose of the present study is to establish a method in which the values of the MSM-S and LFM-S wind speed data from the 10 m height are used as the reference wind speed and are, using a power law, vertically extrapolated to 80 to 90 m heights, typical hub-heights of offshore wind turbines. For this purpose, the present study examined time-averaged vertical profiles of wind speed over the ocean based on the MSM-S data and in-situ data in the Hibikinada area, Kitakyushu City, Fukuoka Prefecture, Japan. As a result, it was revealed that a strong wind shear existed close to the sea surface, between the 10 and 30 m heights. In order to address the above-mentioned wind shear, a two-step vertical extrapolation method was proposed in the present study. In this method, two values of N, specifically for low and high altitudes (below and above approximately 30 m, respectively), were calculated and used. The data were created for the five years between 2012 and 2016. Similarly to previous analyses, the analysis of the created data set indicated that the potential of offshore wind power generation in the Hibikinada area, Kitakyushu City is quite high.

Estimation of optimal dispersion model source parameters using satellite detections of volcanic ash

AbstractIn this paper we demonstrate how parameters describing the geometry of the volcanic ash source for a particular volcanic ash dispersion model (Hybrid Single‐Particle Lagrangian Integrated Trajectory (HYSPLIT)) may be inferred by the use of satellite data and multiple trial simulations. The areas of space likely to be contaminated by ash are identified with the aid of various remote sensing techniques, and polygons are drawn around these areas as they would be in an operational setting. Dispersion model simulations are initialized by either a cylindrical source or a specified ash distribution depending on the context. Parameters of interest such as the base and top height, diameter, and optimal release time of the cylindrical source or the height of the specified ash distribution are inferred by forming a parameter grid and running multiple simulations for each parameter grid point value. Optimal values of the parameter values are identified by calculating spatial correlations between the model simulations and observations. We demonstrate that the methodology can be used to correctly infer various model parameters and improve volcanic ash forecasts in various eruption case studies.

Daily precipitation grids for Austria since 1961—development and evaluation of a spatial dataset for hydroclimatic monitoring and modelling

Spatial precipitation datasets that are long-term consistent, highly resolved and extend over several decades are an increasingly popular basis for modelling and monitoring environmental processes and planning tasks in hydrology, agriculture, energy resources management, etc. Here, we present a grid dataset of daily precipitation for Austria meant to promote such applications. It has a grid spacing of 1 km, extends back till 1961 and is continuously updated. It is constructed with the classical two-tier analysis, involving separate interpolations for mean monthly precipitation and daily relative anomalies. The former was accomplished by kriging with topographic predictors as external drift utilising 1249 stations. The latter is based on angular distance weighting and uses 523 stations. The input station network was kept largely stationary over time to avoid artefacts on long-term consistency. Example cases suggest that the new analysis is at least as plausible as previously existing datasets. Cross-validation and comparison against experimental high-resolution observations (WegenerNet) suggest that the accuracy of the dataset depends on interpretation. Users interpreting grid point values as point estimates must expect systematic overestimates for light and underestimates for heavy precipitation as well as substantial random errors. Grid point estimates are typically within a factor of 1.5 from in situ observations. Interpreting grid point values as area mean values, conditional biases are reduced and the magnitude of random errors is considerably smaller. Together with a similar dataset of temperature, the new dataset (SPARTACUS) is an interesting basis for modelling environmental processes, studying climate change impacts and monitoring the climate of Austria.

Improving accuracy in the MPM method using a null space filter

The material point method (MPM) has been very successful in providing solutions to many challenging problems involving large deformations. Nevertheless there are some important issues that remain to be resolved with regard to its analysis. One key challenge applies to both MPM and particle-in-cell (PIC) methods and arises from the difference between the number of particles and the number of the nodal grid points to which the particles are mapped. This difference between the number of particles and the number of grid points gives rise to a non-trivial null space of the linear operator that maps particle values onto nodal grid point values. In other words, there are non-zero particle values that when mapped to the grid point nodes result in a zero value there. Moreover, when the nodal values at the grid points are mapped back to particles, part of those particle values may be in that same null space. Given positive mapping weights from particles to nodes such null space values are oscillatory in nature. While this problem has been observed almost since the beginning of PIC methods there are still elements of it that are problematical today as well as methods that transcend it. The null space may be viewed as being connected to the ringing instability identified by Brackbill for PIC methods. It will be shown that it is possible to remove these null space values from the solution using a null space filter. This filter improves the accuracy of the MPM methods using an approach that is based upon a local singular value decomposition (SVD) calculation. This local SVD approach is compared against the global SVD approach previously considered by the authors and to a recent MPM method by Zhang and colleagues.

Thermal Anomalies in Relation to Earthquakes in India and its Neighbourhood

Thermal anomalies based on satellite and surface meteorological data in the epicentral region prior to a few earthquakes in India and elsewhere have raised a question whether such inferences can be relied upon keeping in view larger meteorological variability associated with synoptic weather changes. This article examines outgoing longwave radiation (OLR) data by INSAT over Sikkim region to find whether any anomaly developed prior to the earthquake of September 2011 (M w 6.9). Similar study was made using surface meteorological data (weather observatories or NCAR grid point values) in the meizoseismal areas of Bhadrachalam (1969, m s 5.7), Kinnaur (1975, m s 6.9), Bihar-Nepal (1988, m s 6.9), Uttarkashi (1991, m s 6.8), Latur (1993, m s 6.1), Jabalpur (1997, m s 5.6), Chamoli (1999, m b 6.5), Bhuj (2001, M w 7.6) and Muzaffarabad (2005, M w 7.6) earthquakes. The OLR data from INSAT were also examined during heat-wave conditions prevailing over northern India in summer during non-seismic conditions. It was found that there was no significant thermal anomaly prior to any earthquake in the Indian region. It has been shown that the precursory thermal anomaly reported in earlier studies for Bhuj (2001), Muzaffarabad (2005), Jabalpur (1997) and Latur (1993) earthquakes was misinterpreted since the rise in surface temperature (if any) was found to be well within the meteorological variability caused by synoptic weather conditions. Persistent heat-wave conditions causing very large thermal anomaly for several days in northern India raise false alarms. The synthesis of the results presented in this study should desist geoscientists from considering thermal anomalies as earthquake precursors.

SOLVING BOUNDARY VALUE PROBLEMS FOR SECOND ORDER SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS BY Ε-APPROXIMATE FIXED-POINT METHOD

In this paper, the boundary value problem for second order singularly perturbed delay differential equation is reduced to a fixed-point problem v = Av with a properly chosen (generally nonlinear) operator A. The unknown fixed-point v is approximated by cubic spline v h defined by its values v i = v h (t i ) at grid points t i , i = 0, 1, . . . , N. The necessary for construction the cubic spline and missing the first derivatives at the boundary are replaced by the derivatives of the corresponding interpolating polynomials matching the grid points values nearest to the boundary points. An approximation of the solution is obtained by minimization techniques applied to a function whose arguments are the grid point values of the sought spline. The results of numerical experiments with two boundary value problems for the second order singularly perturbed delay differential equations as well as their comparison with the results of other methods employed by other authors are also provided.