An improved altimeter-derived gravity anomaly from shipborne gravity based on the mean sea surface height constraint factors method

Coastal marine gravity modeling faces challenges due to the degradation of the quality and poor coverage of altimeter data in coastal regions. The effective fusion of shipborne gravity data and altimeter-derived marine gravity data can make shipborne gravity data more useful for the accurate estimation of altimeter-derived coastal marine gravity. A mean sea surface height constraint factor (MSSHCF) method based on the ordinary kriging method and the remove-restore technique is proposed to fuse altimeter-derived gravity model with shipborne gravity data. In this method, all data are standardized during the interpolation process to reduce the error and mean sea surface as a vertical variable is added to the semi-variance function in ordinary kriging to obtain the residual shipborne gravity as corrected data source. The coastal marine gravity models V2.1 and V3.1 which fused altimeter-derived gravity data with shipborne gravity data and V1.1 without shipborne gravity data at a spatial resolution of 1′×1′ can be obtained. Validation experiments show that the accuracy of the gravity model V3.1 obtained by the MSSHCF method more closely agrees with the validated gravity model DTU17 and SS V31 than the model V2.1 obtained by the ordinary kriging interpolation method and the V1.1 model. Our results were validated against shipborne gravity data; the accuracy of model V3.1 was 4.95 % higher than the model V1.1 in South China Sea area A and 2.48 % higher in South China Sea area B. Meanwhile, the accuracy of model V3.1 was 2.07 % higher than model V2.1 in South China Sea area A and 2.42 % higher in South China Sea area B. The effects of distance from the coast and sea depth on the marine gravity model were also evaluated. The results show that the gravity model V3.1 has higher accuracy with the change in ocean distance and depth than the V2.1 and V1.1 gravity models. Thus, our study shows that the MSSHCF method effectively refines coastal altimeter-derived gravity using shipborne gravity data.     

On the selection of optimal global geopotential model for geoid modeling: A case study in Pakistan

A statistical comparison has been made between gravity field parameters derived from different global geopotential models (GGMs) and observed gravity anomalies, gravimetric geoid and GPS-Leveling data. The motivation behind this study is the selection of best possible global geopotential model that best matches statistically with the local observed data in Pakistan. This will facilitate in decreasing the load on observed data for the development of regional gravimetric geoid in remove-compute-restore technique when used in the Stokes’s integral for computation of the residual part. It is observed that combined geopotential models such as EGM96 and PGM200A, EIGEN-GL04C and EIGEN-CG03C reflect the better match in the total spectral range of gravity and GPS-Leveling data. Results of the precise local geoid model also indicate similar characteristics. A very-high-degree model “EGM2008” (degree/order 2160) exhibits relatively superior statistical fit with observed ground data in Pakistan region. For satellite-only models an increasing trend in the standard deviation can be seen with maximum of about ∼4 m in difference between GPS-Leveling and corresponding GGM’s geoid with increase in the order from 50 to 120 and then it decreases afterwards. However, for the EIGEN-CHAMP03SP, standard deviation saturates to a value of 3.4 m. This is an indication of contamination in the long to medium wavelength part, i.e. 50–100° for the satellite-only models. Moreover, the models DEOS-CHAMP-01C, GGM02C and then ITG-GRACE03 appear to have better fit for medium to long wavelength and can possibly be recommended for use as long wavelength part with the local observed data. While a hybrid geopotential model selection can be achieved through the selection from either of DEOS-CHAMP-01C, GGM02C, GGM02S, EIGEN-GRACE02S or ITG-GRACE03 in the long wavelength (to degree and order 40) and EGM96, PGM200A, EIGEN-GL04C, EIGEN-CG03C or even EGM2008 in medium to short wavelength, i.e. from degree 41 to maximum degree and order.     

Prediction of geosynchronous electron fluxes using an artificial neural network driven by solar wind parameters

There are hundreds of satellites operating at the geosynchronous (GEO) orbit where relativistic electrons can cause severe damage. Thus, predicting relativistic electron fluxes is significant for spacecraft safety. In this study, using GOES satellite data during 2011–2020, we propose two neural network models with two hidden layers to predict geosynchronous relativistic electron fluxes at two energy channels (>0.8 MeV and > 2 MeV). The number of input neurons of the two channels (>0.8 MeV and > 2 MeV) are determined to be 36 and 44, respectively. The > 0.8 MeV model has 22 and 9 neurons in the hidden layers, while the > 2 MeV model has 25 and 15 neurons in the hidden layers. The input parameters include the north–south component of the interplanetary magnetic field, solar wind speed, solar wind dynamic pressure and solar wind proton density. Through the analysis of different time delays, we determine that the optimal time delays of two energy channels (>0.8 MeV and > 2 MeV) are 8 days and 10 days, respectively. The training set and validation set (Jan 2011-Dec 2018) are divided by the 10-fold cross-validation method, and the remaining data (Jan 2019-Feb 2020) is used to analyze the model performance as a test set. The prediction results of both energy channels show good agreement with satellite observations indicated by low RMSE (～0.3 cm^-2sr^-1s^?1), high PE (～0.8) and CC (～0.9). These results suggest that only using solar wind parameters is capable of obtaining reasonable predictions of geosynchronous relativistic electron fluxes.     

Mapping mean lake surface from satellite altimetry and GPS kinematic surveys

Lake water height is a key variable in water cycle and climate change studies, which is achievable using satellite altimetry constellation. A method based on data processing of altimetry from several satellites has been developed to interpolate mean lake surface (MLS) over a set of 22 big lakes distributed on the Earth. It has been applied on nadir radar altimeters in Low Resolution Mode (LRM: Jason-3, Saral/AltiKa, CryoSat-2) in Synthetic Aperture Radar (SAR) mode (Sentinel-3A), and in SAR interferometric (SARin) mode (CryoSat-2), and on laser altimetry (ICESat). Validation of the method has been performed using a set of kinematic GPS height profiles from 18 field campaigns over the lake Issykkul, by comparison of altimetry’s height at crossover points for the other lakes and using the laser altimetry on ICESat-2 mission. The precision reached ranges from 3 to 7 cm RMS (Root Mean Square) depending on the lakes. Currently, lake water level inferred from satellite altimetry is provided with respect to an ellipsoid. Ellipsoidal heights are converted into orthométric heights using geoid models interpolated along the satellite tracks. These global geoid models were inferred from geodetic satellite missions coupled with absolute and regional anomaly gravity data sets spread over the Earth. However, the spatial resolution of the current geoid models does not allow capturing short wavelength undulations that may reach decimeters in mountaineering regions or for rift lakes (Baikal, Issykkul, Malawi, Tanganika). We interpolate in this work the geoid height anomalies with three recent geoid models, the EGM2008, XGM2016 and EIGEN-6C4d, and compare them with the Mean Surface of 22 lakes calculated using satellite altimetry. Assuming that MLS mimics the local undulations of the geoid, our study shows that over a large set of lakes (in East Africa, Andean mountain and Central Asia), short wavelength undulations of the geoid in poorly sampled areas can be derived using satellite altimetry. The models used in this study present very similar geographical patterns when compared to MLS. The precision of the models largely depends on the location of the lakes and is about 18 cm, in average over the Earth. MLS can serve as a validation dataset for any future geoid model. It will also be useful for validation of the future mission SWOT (Surface Water and Ocean Topography) which will measure and map water heights over the lakes with a high horizontal resolution of 250 by 250 m.     

Assessment of depth to magnetic sources using high resolution aeromagnetic data of some parts of the Lower Benue Trough and adjoining areas,Southeast Nigeria

High resolution airborne magnetic data acquired between 2005 and 2010 were used to determine depth to shallow and deep magnetic sources in some parts of Southeastern Nigeria. Various depth estimation methods such as standard Euler deconvolution (SED), source parameter imaging (SPI), spectral depth analysis (SDA) and two dimensional (2-D) forward modeling were applied. Results obtained from SED, SPI and models of profiles 1 and 2 indicate that the Abakaliki Anticlinorium (AA) and Ikom-Mamfe Rift (IMR) regions are dominated by short wavelength magnetic anomalies caused by extensive tectonic events. The SED map showed depth to shallow and deep magnetic sources ranging from ~ 16.6 to ~ 338.3 m and ~ 394.3 to ~ 5748.1 m respectively. Likewise, depth estimates from the SPI map varies from ~ 147.1 to ~ 554.2 m (shallow magnetic sources) and ~ 644.2 to ~ 6141.6 m (deep magnetic sources). The result obtained from SDA revealed depths to deep magnetic basement in the range of ~ 769 to ~ 6666 m with an average of ~ 3449 m. Also, it showed that depth to shallow magnetic sources vary between ~ 119 and ~ 434 m with mean of ~ 269 m. The 2-D forward modelling showed maximum depth values of ~ 4700, ~4600 and ~ 6500 m in the models of profiles 1, 2 and 3 within the Anambra Basin (AB), Afikpo Syncline (AS) and Calabar Flank (CF) respectively. Generally, from all the various methods applied the results indicate that AB, AS and CF are dominated by long wavelength anomalies. The 2-D models indicated that the basement framework is undulant. Also, depth estimates involving the various methods used in this study correlate strongly with each other in the AB, AS and CF geological regions.     

Simulation study for the determination of the lunar gravity field from PRARE-L tracking onboard the German LEO mission

A simulation study has been performed at GFZ Potsdam, which shows the anticipated improvement of the lunar gravity field model with respect to current (LP150Q model) or near-future (SELENE) knowledge in the framework of the planned German Lunar Explorations Orbiter (LEO) mission, based on PRARE-L (Precise Range And Range-rate Equipment – Lunar version) Satellite-to-Satellite (SST) and Satellite-Earth-Satellite (SEST) tracking observations. It is shown that the global mean error of the lunar gravity field can be reduced to less than 0.1 mGal at a spatial resolution of 50 km. In the spectral domain, this means a factor of 10 (long wavelengths) and some 100 (mid to short wavelengths) improvement as compared to predictions for SELENE or a factor of 1000 with respect to LP150Q. Furthermore, a higher spatial resolution of up to 28 km seems feasible and would correspond to a factor of 2–3 improvement of SELENE results. Moreover, PRARE-L is expected to derive the low-degree coefficients of the lunar gravity field with unprecedented accuracy. Considering long mission duration (at least 1 year is planned) this would allow for the first time a precise direct determination of the low-degree tidal Love numbers of the Moon and, in combination with high precision SEST, would provide an experimental basis to study relativistic effects such as the periselenium advance in the Earth–Moon system.     

The effect of spatial truncation error on integral inversion of satellite gravity gradiometry data

The satellite gravity gradiometry (SGG) data can be used for local modelling of the Earth’s gravity field. In this study, the SGG data in the local north-oriented and orbital frames are inverted to the gravity anomaly at sea level using the second-order partial derivatives of the extended Stokes formula. The emphasis is on the spatial truncation error and the kernel behaviour of the integral formulas in the aforementioned frames. The paper will show that only the diagonal elements of gravitational tensor at satellite level are suitable for recovering the gravity anomaly at sea level. Numerical studies show that the gravity anomaly can be recovered in Fennoscandia with an accuracy of about 6 mGal directly from on-orbit SGG data.     

The effect of spatial truncation error on variance of gravity anomalies derived from inversion of satellite orbital and gradiometric data

The spatial truncation error (STE) is a significant systematic error in the integral inversion of satellite gradiometric and orbital data to gravity anomalies at sea level. In order to reduce the effect of STE, a larger area than the desired one is considered in the inversion process, but the anomalies located in its central part are selected as the final results. The STE influences the variance of the results as well because the residual vector, which is contaminated with STE, is used for its estimation. The situation is even more complicated in variance component estimation because of its iterative nature. In this paper, we present a strategy to reduce the effect of STE on the a posteriori   variance factor and the variance components for inversion of satellite orbital and gradiometric data to gravity anomalies at sea level. The idea is to define two windowing matrices for reducing this error from the estimated residuals and anomalies. Our simulation studies over Fennoscandia show that the differences between the 0.5°×0.5°$0.5°\times 0.5°$ gravity anomalies obtained from orbital data and an existing gravity model have standard deviation (STD) and root mean squared error (RMSE) of 10.9 and 12.1 mGal, respectively, and those obtained from gradiometric data have 7.9 and 10.1 in the same units. In the case that they are combined using windowed variance components the STD and RMSE become 6.1 and 8.4 mGal. Also, the mean value of the estimated RMSE after using the windowed variances is in agreement with the RMSE of the differences between the estimated anomalies and those obtained from the gravity model.     

Inversion of satellite gradiometry data using statistically modified integral formulas for local gravity field recovery

The satellite gravity gradiometric data can be used directly to recover the gravity anomaly at sea level using inversion of integral formulas. This approach suffers by the spatial truncation errors of the integrals, but these errors can be reduced by modifying the formulas. It allows us to consider smaller coverage of the satellite data over the region of recovery. In this study, we consider the second-order radial derivative (SORD) of disturbing potential (T_rr) and determine the gravity anomaly with a resolution of 1° × 1° at sea level by inverting the statistically modified version of SORD of extended Stokes’ formula. Also we investigate the effect of the spatial truncation error on the quality of inversion considering noise of T_rr. The numerical investigations show satisfactory results when the area of T_rr coverage is the same with that of the gravity anomaly and the integral formula is modified by the biased least-squares modification. The error of recovery will be about 6 mGal after removing the regularization bias in the presence of 1 mE noise in T_rr measured on the orbit.     

An improved cyclic multi model-eXtreme gradient boosting (CMM-XGBoost) forecasting algorithm on the GNSS vertical time series

The study of GNSS vertical coordinate time series forecasting is helpful for monitoring the crustal plate movement, dam or bridge deformation monitoring, and global or regional coordinate system maintenance. The eXtreme Gradient Boosting (XGBoost) algorithm is a machine learning algorithm that can evaluate features, and it has a great potential and stability for long-span time series forecasting. This study proposes a multi-model combined forecasting method based on the XGBoost algorithm. The method constitutes a new time series as features through the fitting and forecasting results of the forecasting model. The XGBoost model is then used for forecasting. In addition, this method can obtain higher precision forecasting results through circulation. To verify the performance of the forecasting method, 1095 epochs of data in the Up coordinate of 16 GNSS stations are selected for the forecasting test. Compared with the CNN-LSTM model, the experimental results of our forecasting method show that the mean absolute error (MAE) values are reduced by 30.23 %～52.50 % and the root mean square error (RMSE) values are reduced by 31.92 %～54.33 %. The forecasting results have higher accuracy and are highly correlated to the original time series, which can better forecast the vertical movement of the GNSS stations. Therefore, the forecasting method can be applied to the up component of the GNSS coordinate time series.     

Performance evaluation of some rain rate conversion models for microwave propagation studies

An important characteristic of rainfall levels at a particular place is the statistical distribution of rainfall rate. In this paper, 5-min integration time rainfall data for the Northcentral region of Nigeria was obtained from the Tropospheric Data Acquisition Network (TRODAN), Anyigba, Nigeria. Also, 1-min integration time rainfall was measured at Minna, Nigeria. In order to obtain the optimal rain rate model suitable for this region, two globally recognised rain rate models were critically evaluated and compared with the 1-min measurements. These are the ITU-R P.837-7 and Lavergnat-Gole (L-G) models. The results obtained showed that the ITU-R P.837-7 and L-G models respectively underestimated the measured rain rate by 7.3 mm/h and 9 mm/h at time percentage exceedance of 0.1%, while they underestimated the measured rain rate by 23.4 mm/h and 13 mm/h respectively at 0.01%. At 0.001%, the measured rain rate was overestimated by the ITU-R P.837-7 and L-G models by 27.4 mm/h and 3 mm/h respectively. Further performance evaluation of the predefined models was carried out using different error metrics such as sum of absolute error (SAE), mean absolute error (MAE), root mean square error (RMSE), standard deviation (STDEV) and Spearman’s rank correlation. The results obtained adjudged the Lavergnat-Gole model as the best rain rate prediction model for this region.     

A multilayer perceptron and multiclass support vector machine based high accuracy technique for daily rainfall estimation from MSG SEVIRI data

The current paper introduces a new multilayer perceptron (MLP) and support vector machine (SVM) based approach to improve daily rainfall estimation from the Meteosat Second Generation (MSG) data. In this study, the precipitation is first detected and classified into convective and stratiform rain by two MLP models, and then four multi-class SVM algorithms were used for daily rainfall estimation. Relevant spectral and textural input features of the developed algorithms were derived from the spectral MSG SEVIRI radiometer channels. The models were trained using radar rainfall data set colected over north Algeria. Validation of the proposed daily rainfall estimation technique was performed by rain gauge network data set recorded over north Algeria. Thus, several statistical scores were calculated, such as correlation coefficient (r), root mean square error (RMSE), mean error (Bias), and mean absolute error (MAE). The findings given by: (r = 0.97, bias = 0.31 mm, RMSE = 2.20 mm and MAE = 1.07 mm), showed a quite satisfactory relationship between the estimation and the respective observed daily precipitation. Moreover, the comparison of the results with those of two advanced techniques based on random forests (RF) and weighted ‘k’ nearest neighbor (WkNN) showed higher accuracy obtained by the proposed model.     

The temporal variation of mean gravity of atmosphere

One of the precise widely used global Zenith Hydrostatic Delay (ZHD) model is based on the gravity value at the centroid of the atmospheric column at the station of observation and gravity value at the centroid is constant in this model for a specific location throughout the year. However, as the content and extent of atmosphere varies temporally, its centroid and consequently gravity value at the centroid also varies. Apart from this, the actual atmospheric condition of different region is not alike. Therefore, there is a need to develop a regional mean gravity model and development of such model has been discussed in this paper. To obtain the mean gravity model, first a regional model of centroid height of atmospheric column was developed as a function of the surface pressure and temperature. It was developed by multiple regressions between estimated centroid of the atmosphere and surface pressure, surface temperature using radiosonde data of five radiosonde stations spread over the Indian subcontinent. Three years radiosonde data from 2006 to 2008 was used for each station. The root mean square error in estimating centroid of the atmospheric column is about ±326 m, which is negligible considering the variability of the atmosphere and its content. The centroid height model has been used to formulate the mean gravity model, considering uniform lapse rate in gravity with height. It is found that proposed mean gravity model provides temporal variation of mean gravity values at the centroid and thus matches with the reality. The interesting advantage of the developed model is that the model shows diurnal variation of mean gravity. The accuracy of ZHD has shown of the order of about 0.3 mm using the developed regional mean gravity model. However, already developed ZHD model has shown a slight inferior result compared to the developed model. These models have shown accuracy of about 0.8 mm and 0.6 mm.     

On the usefulness of atmospheric and oceanic angular momentum in recovering polar motion and gravity field variations in a unified process

Earth rotation parameters (ERPs) are excited by variations in the mass distribution on the Earth’s surface and the exchange of angular momentum between the atmosphere and oceans and the solid Earth. The same mass redistribution causes temporal changes in the gravity field coefficients with the second degree harmonics related to the rotational deformation and hence to changes in the Earth’s inertial tensor. If precise models of the atmospheric and oceanic angular momentum (AM) are available solution for polar motion and degree 2 Stokes harmonics can be unified. In this study we utilize SLR tracking of LAGEOS to compare (i) degree 2 harmonics from ERPs and gravitation, and (ii) LAGEOS excitation functions and geophysical data (mass + motion). Finally, we investigate to what extent a unified approach is possible with current models for AM data and gravity mass change estimated from ERP within orbit determinations.     

Temporal extrapolation of TEC using WNN during 2007–2018 and comparison against GIM,IRI2016 and Kriging

In this paper, a new method of temporal extrapolation of the ionosphere total electron content (TEC) is proposed. Using 3-layer wavelet neural networks (WNNs) and particle swarm optimization (PSO) training algorithm, TEC time series are modeled. The TEC temporal variations for next times are extrapolated with the help of training model. To evaluate the proposed model, observations of Tehran GNSS station (35.69°N, 51.33°E) from 2007 to 2018 are used. The efficiency of the proposed model has been evaluated in both low and high solar activity periods. All observations of the 2015 and 2018 have been removed from the training step to test the proposed model. On the other hand, observations of these 2 years are not used in network training. According to the F10.7, the 2015 has high solar activity and the 2018 has quiet conditions. The results of the proposed model are compared with the global ionosphere maps (GIMs) as a traditional ionosphere model, international reference ionosphere 2016 (IRI2016), Kriging and artificial neural network (ANN) models. The root mean square error (RMSE), bias, dVTEC = |VTEC_GPS ? VTEC_Model| and correlation coefficient are used to assess the accuracy of the proposed method. Also, for more accurate evaluation, a single-frequency precise point positioning (PPP) approach is used. According to the results of 2015, the maximum values of the RMSE for the WNN, ANN, Kriging, GIM and IRI2016 models are 5.49, 6.02, 6.34, 6.19 and 13.60 TECU, respectively. Also, the maximum values of the RMSE at 2018 for the WNN, ANN, Kriging, GIM and IRI2016 models are 2.47, 2.49, 2.50, 4.36 and 6.01 TECU, respectively. Comparing the results of the bias and correlation coefficient shows the higher accuracy of the proposed model in quiet and severe solar activity periods. The PPP analysis with the WNN model also shows an improvement of 1 to 12 mm in coordinate components. The results of the analyzes of this paper show that the WNN is a reliable, accurate and fast model for predicting the behavior of the ionosphere in different solar conditions.     

The simulation of lunar gravity field recovery from D-VLBI of Chang’E-1 and SELENE lunar orbiters

The lunar gravity field is a foundation to study the lunar interior structure, and to recover the evolution history of the Moon. It is still an open and key topic for lunar science. For above mentioned reasons, it becomes one of the important scientific objectives of recent lunar missions, such as KAGUYA (SELENE) the Japanese lunar mission and Chang’E-1, the Chinese lunar mission. The Chang’E-1 and the SELENE were successfully launched in 2007. It is estimated that these two missions can fly around the Moon longer than 6 months simultaneously. In these two missions, the Chinese new VLBI (Very Long Baseline Interferometry) network will be applied for precise orbit determination (POD) by using a differential VLBI (D-VLBI) method during the mission period. The same-beam D-VLBI technique will contribute to recover the lunar gravity field together with other conventional observables, i.e. R&RR (Range and Range Rate) and multi-way Doppler. Taking VLBI tracking conditions into consideration and using the GEODYNII/SOVLE software of GSFC/NASA/USA [8 and 10], we simulated the lunar gravity field recovering ability with and without D-VLBI between the Chang’E-1 and SELENE main satellite. The cases of overlapped flying and tracking period of 30 days, 60 days and 90 days have been analyzed, respectively. The results show that D-VLBI tracking between two lunar satellites can improve the gravity field recovery remarkably. The results and methods introduced in this paper will benefit the actual missions.     

Estimates of mesospheric gravity wave activity over convection from a global model

A new convective gravity wave source spectrum parameterization has been implemented in the Whole Atmosphere Community Climate Model version 2 (WACCM2). This parameterization specifies the momentum flux phase speed spectrum of gravity waves in the Tropics based on the properties of underlying convection; Hence, this parameterization provides realistic global estimates of gravity wave activity. In this paper, we show the estimated gravity wave phase speed spectra in the Tropics from a WACCM2 simulation, at the source level and at 85 km. Spatial distribution of gravity wave activity at 85 km is also presented. Subsequently, we discuss the factors that are primarily responsible for the estimated differences in gravity wave distribution across phase speeds with latitude and asymmetries in direction of gravity wave propagation in the mesosphere. We also examine which of the model assumptions can lead to uncertainties in our estimates of mesospheric gravity wave activity and we discuss how these assumptions provide challenges for comparison with observations of gravity waves in the mesosphere.     

基于解析规则的舰艇区域防空作战能力需求生成

首先,在能力需求生成中,通过对舰艇面临的威胁任务分解,构建作战能力指标体系,分析"元任务-能力指标"之间的解析规则进而得到舰艇作战所需的各项能力指标;其次,以单舰舰空导弹区域防空为背景,根据作战流程的各个阶段建立相应的数学模型,以作战需求以及敌我双方的相关因素为变量,结合关系图对杀伤区远近界、发射区远近界、预警机前出距离、高度和必需预警距离等作战能力指标进行了仿真验证;最后,分析了所需探测距离及速度对作战能力指标的影响关系,定量得到相应所需的区间数值解,为舰艇区域防空作战能力需求生成提供了依据。      

Gravity field models from kinematic orbits of CHAMP,GRACE and GOCE satellites

The aim of our work is to generate Earth’s gravity field models from GPS positions of low Earth orbiters. Our inversion method is based on Newton’s second law, which relates the observed acceleration of the satellite with forces acting on it. The observed acceleration is obtained as numerical second derivative of kinematic positions. Observation equations are formulated using the gradient of the spherical harmonic expansion of the geopotential. Other forces are either modelled (lunisolar perturbations, tides) or provided by onboard measurements (nongravitational perturbations). From this linear regression model the geopotential harmonic coefficients are obtained.