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NASA NTRS1986

Preliminary evaluation of the SIR-B response to soil moisture, surface roughness, and crop canopy cover

Two predawn ascending data-takes by the Shuttle Imaging Radar-B (SIR-B) were used to evaluate the effects of surface roughness, crop canopy, and soil moisture on radar backscatter. The two images, separated by three days, were both obtained at 30-deg local angle of incidence, but with opposite azimuth viewing directions. The imagery was externally calibrated with respect to the radar backscattering coefficient sigma(0) via response to arrays of point and area-extended targets of known radar cross section. Three land-cover classes: (1) corn, (2) corn stubble and plowed bare soil, and (3) disked bare soil, soybeans, soybean stubble, alfalfa, and clover could be readily separated for either observation date on the basis of image tone alone. The dependence of sigma(0) on the surface roughness and canopy brightness inhibits the capability of SIR to globally estimate the near-surface soil moisture from the value of sigma(0) for single date observations, unless the surface roughness or canopy cover conditions are accounted for. However, within given ranges of these conditions, the sigma(0) was found to be highly correlated with the soil moisture.

Dobson, M. C.↗

NASA NTRS1997

Retrieval of Soil Moisture and Roughness from the Polarimetric Radar Response

The main objective of this investigation was the characterization of soil moisture using imaging radars. In order to accomplish this task, a number of intermediate steps had to be undertaken. In this proposal, the theoretical, numerical, and experimental aspects of electromagnetic scattering from natural surfaces was considered with emphasis on remote sensing of soil moisture. In the general case, the microwave backscatter from natural surfaces is mainly influenced by three major factors: (1) the roughness statistics of the soil surface, (2) soil moisture content, and (3) soil surface cover. First the scattering problem from bare-soil surfaces was considered and a hybrid model that relates the radar backscattering coefficient to soil moisture and surface roughness was developed. This model is based on extensive experimental measurements of the radar polarimetric backscatter response of bare soil surfaces at microwave frequencies over a wide range of moisture conditions and roughness scales in conjunction with existing theoretical surface scattering models in limiting cases (small perturbation, physical optics, and geometrical optics models). Also a simple inversion algorithm capable of providing accurate estimates of soil moisture content and surface rms height from single-frequency multi-polarization radar observations was developed. The accuracy of the model and its inversion algorithm is demonstrated using independent data sets. Next the hybrid model for bare-soil surfaces is made fully polarimetric by incorporating the parameters of the co- and cross-polarized phase difference into the model. Experimental data in conjunction with numerical simulations are used to relate the soil moisture content and surface roughness to the phase difference statistics. For this purpose, a novel numerical scattering simulation for inhomogeneous dielectric random surfaces was developed. Finally the scattering problem of short vegetation cover above a rough soil surface was considered. A general scattering model for grass-blades of arbitrary cross section was developed and incorporated in a first order random media model. The vegetation model and the bare-soil model are combined and the accuracy of the combined model is evaluated against experimental observations from a wheat field over the entire growing season. A complete set of ground-truth data and polarimetric backscatter data were collected. Also an inversion algorithm for estimating soil moisture and surface roughness from multi-polarized multi-frequency observations of vegetation-covered ground is developed.

Sarabandi, Kamal↗

NASA NTRS2005

Using Remote Sensing Data to Evaluate Surface Soil Properties in Alabama Ultisols

Evaluation of surface soil properties via remote sensing could facilitate soil survey mapping, erosion prediction and allocation of agrochemicals for precision management. The objective of this study was to evaluate the relationship between soil spectral signature and surface soil properties in conventionally managed row crop systems. High-resolution RS data were acquired over bare fields in the Coastal Plain, Appalachian Plateau, and Ridge and Valley provinces of Alabama using the Airborne Terrestrial Applications Sensor multispectral scanner. Soils ranged from sandy Kandiudults to fine textured Rhodudults. Surface soil samples (0-1 cm) were collected from 163 sampling points for soil organic carbon, particle size distribution, and citrate dithionite extractable iron content. Surface roughness, soil water content, and crusting were also measured during sampling. Two methods of analysis were evaluated: 1) multiple linear regression using common spectral band ratios, and 2) partial least squares regression. Our data show that thermal infrared spectra are highly, linearly related to soil organic carbon, sand and clay content. Soil organic carbon content was the most difficult to quantify in these highly weathered systems, where soil organic carbon was generally less than 1.2%. Estimates of sand and clay content were best using partial least squares regression at the Valley site, explaining 42-59% of the variability. In the Coastal Plain, sandy surfaces prone to crusting limited estimates of sand and clay content via partial least squares and regression with common band ratios. Estimates of iron oxide content were a function of mineralogy and best accomplished using specific band ratios, with regression explaining 36-65% of the variability at the Valley and Coastal Plain sites, respectively.

Sullivan, Dana G.↗

NASA NTRSDate not supplied

Improved Peatland Hydrology and L-Band Microwave Radiative Transfer Modeling in Version 7 of the SMAP Level-4 Soil Moisture Data Assimilation Product

SMAP↗

NASA NTRS2003

Using Remote Sensing Platforms to Estimate Near-Surface Soil Properties

Evaluation of near-surface soil properties via remote sensing (RS) could facilitate soil survey mapping, erosion prediction, fertilization regimes, and allocation of agrochemicals. The objective of this study was to evaluate the relationship between soil spectral signature and near surface soil properties in conventionally managed row crop systems. High resolution RS data were acquired over bare fields in the Coastal Plain, Appalachian Plateau, and Ridge and Valley provinces of Alabama using the Airborne Terrestrial Applications Sensor (ATLAS) multispectral scanner. Soils ranged from sandy Kandiudults to fine textured Rhodudults. Surface soil samples (0-1 cm) were collected from 163 sampling points for soil water content, soil organic carbon (SOC), particle size distribution (PSD), and citrate dithionite extractable iron (Fed) content. Surface roughness, soil water content, and crusting were also measured at sampling. Results showed RS data acquired from lands with less than 4 % surface soil water content best approximated near-surface soil properties at the Coastal Plain site where loamy sand textured surfaces were predominant. Utilizing a combination of band ratios in stepwise regression, Fed (r2 = 0.61), SOC (r2 = 0.36), sand (r2 = 0.52), and clay (r2 = 0.76) were related to RS data at the Coastal Plain site. In contrast, the more clayey Ridge and Valley soils had r-squares of 0.50, 0.36, 0.17, and 0.57. for Fed, SOC, sand and clay, respectively. Use of estimated eEmissivity did not generally improve estimates of near-surface soil attributes.

Sullivan, D. G.↗

NASA NTRSDate not supplied

Version 8 of the SMAP Level-4 Soil Moisture Data Assimilation Product

The NASA Soil Moisture Active Passive (SMAP) mission Level-4 Soil Moisture (L4_SM) product provides global, 9-km resolution, 3-hourly surface (0-5 cm) and root-zone (0-100 cm) soil moisture from April 2015 to present with a mean latency of 2.5 days from the time of observation. The product is based on the assimilation of SMAP L-band (1.4 GHz) brightness temperature (Tb) observations into the NASA Catchment land surface model as the model is driven with observations-based precipitation forcing. This presentation discusses the improvements in the forthcoming Version 8 of L4_SM, including updates in the precipitation forcing, the Catchment model parameters, and the L-band microwave radiative transfer model (mwRTM). The precipitation observations used in L4_SM Version 8 outside of North America and the high latitudes are from the latest (Version 7) NASA Integrated Multi-satellitE Retrievals for the Global Precipitation Measurement mission (IMERG) products. Moreover, occasionally excessive precipitation rates in earlier versions of L4_SM along certain longitudes in North America were eliminated by a bug fix in the precipitation corrections algorithm. The Catchment model in L4_SM Version 8 uses climatological snow albedo values based on observations from the Moderate Resolution Imaging Spectroradiometer, replacing the look-up table parameterization of earlier versions. Additionally, corrected soil parameters were implemented for a small region in Argentina that had erroneously been classified as peat because of an error in the ancillary soil data. Finally, the mwRTM in L4_SM Version 8 uses the Mironov soil mixing approach and updated values of the L-band scattering albedo, soil roughness, and vegetation opacity climatology obtained from the latest (Release 19) SMAP Level-2 dual-channel soil moisture retrieval product. During the development of L4_SM Version 8, the change in the mwRTM parameterization resulted in a reduced unbiased RMSE of surface soil moisture when verified against in situ measurements. It also reduced the standard deviation of the Tb observation-minus-forecast residuals by ~0.15 K, highlighting the importance of the mwRTM for successful data assimilation. The bug fixes in the precipitation corrections algorithm and the Catchment model soil parameters in the Argentina region result in locally large improvements of the simulated land surface states. In summary, the ongoing refinements of the L4_SM product continue to improve its science quality and performance for global soil moisture monitoring

Rolf Reichle↗

NASA NTRS1992

Botswana water and surface energy balance research program. Part 2: Large scale moisture and passive microwaves

The Botswana water and surface energy balance research program was developed to study and evaluate the integrated use of multispectral satellite remote sensing for monitoring the hydrological status of the Earth's surface. The research program consisted of two major, mutually related components: a surface energy balance modeling component, built around an extensive field campaign; and a passive microwave research component which consisted of a retrospective study of large scale moisture conditions and Nimbus scanning multichannel microwave radiometer microwave signatures. The integrated approach of both components are explained in general and activities performed within the passive microwave research component are summarized. The microwave theory is discussed taking into account: soil dielectric constant, emissivity, soil roughness effects, vegetation effects, optical depth, single scattering albedo, and wavelength effects. The study site is described. The soil moisture data and its processing are considered. The relation between observed large scale soil moisture and normalized brightness temperatures is discussed. Vegetation characteristics and inverse modeling of soil emissivity is considered.

Vandegriend, A. A.↗

NASA NTRS1984

Reflectance Modeling

A classical description of the one dimensional radiative transfer treatment of vegetation canopies was completed and the results were tested against measured prairie (blue grama) and agricultural canopies (soybean). Phase functions are calculated in terms of directly measurable biophysical characteristics of the canopy medium. While the phase functions tend to exhibit backscattering anisotropy, their exact behavior is somewhat more complex and wavelength dependent. A Monte Carlo model was developed that treats soil surfaces with large periodic variations in three dimensions. A photon-ray tracing technology is used. Currently, the rough soil surface is described by analytic functions and appropriate geometric calculations performed. A bidirectional reflectance distribution function is calculated and, hence, available for other atmospheric or canopy reflectance models as a lower boundary condition. This technique is used together with an adding model to calculate several cases where Lambertian leaves possessing anisotropic leaf angle distributions yield non-Lambertian reflectance; similar behavior is exhibited for simulated soil surfaces.

Smith, J. A.↗

NASA NTRS1985

Simple equation to approximate the bidirectional reflectance from vegetative canopies and bare soil surfaces

A simple equation has been developed for describing the bidirectional reflectance of some vegetative canopies and bare soil surfaces. The equation describes directional reflectance as a function of zenith and azimuth view angles and solar azimuth angle. The equation works for simulated and field measured red and IR reflectance under clear sky conditions. Hemispherical reflectance can be calculated as a function of the simple equation coefficients by integrating the equation over the hemisphere of view angles. A single equation for estimating soil bidirectional reflectance was obtained using the relationships between solar zenith angles and the simple equation coefficients for medium and rough soil distributions. The equation has many useful applications such as providing a lower level boundary condition in complex plant canopy models and providing an additional tool for studying bidirectional effects on pointable sensors.

Walthall, C. L.↗

NASA NTRS1973

Static penetration resistance of soils

Model test results were used to define the failure mechanism associated with the static penetration resistance of cohesionless and low-cohesion soils. Knowledge of this mechanism has permitted the development of a new analytical method for calculating the ultimate penetration resistance which explicitly accounts for penetrometer base apex angle and roughness, soil friction angle, and the ratio of penetration depth to base width. Curves relating the bearing capacity factors to the soil friction angle are presented for failure in general shear. Strength parameters and penetrometer interaction properties of a fine sand were determined and used as the basis for prediction of the penetration resistance encountered by wedge, cone, and flat-ended penetrometers of different surface roughness using the proposed analytical method. Because of the close agreement between predicted values and values measured in laboratory tests, it appears possible to deduce in-situ soil strength parameters and their variation with depth from the results of static penetration tests.

Durgunoglu, H. T.↗

NASA NTRS2012

Using Polarimetric SAR Data to Infer Soil Moisture from Surfaces with Varying Subsurface Moisture Profiles

A time-series approach is used to estimate the moisture content-based on polarimetric SAR data. It is found that under the assumption of constant soil moisture, empirically observed relationships between radar backscatter and moisture are only half as sensitive to moisture as compared to actual radar data. A numerical finite element method is used to calculate the radar backscatter for rough soils with arbitrarily varying soil moisture as a function of depth. Several instance of drying and wetting moisture profiles are considered and the radar backscatter is calculated in each case. Radar backscatter is found to crucially depend on the soil moisture variation in the top half wavelength of soil.

subsurface sensing↗

NASA NTRSDate not supplied

Examination of L-Band Brightness Temperature Forecasts in the SMAP Level-4 Soil Moisture Analysis

The NASA Soil Moisture Active Passive (SMAP) mission [4] has been providing L-band (1.4 GHz) passive microwave brightness temperature (Tb) observations since April 2015. By assimilating the Tb observations into the NASA Catchment land surface model [5] using a spatially distributed ensemble Kalman filter, the NASA Global Modeling and Assimilation Office generates the SMAP Level-4 Soil Moisture (L4_SM) product, which provides global, 3-hourly, 9-km resolution estimates of surface (0-5 cm) and root-zone (0-100 cm) soil moisture with ~2.5-day latency for use in research and applications [6]. The L4_SM product also includes estimates of soil temperature, land surface fluxes, and assimilation diagnostics such as the model forecast and observed Tb values [7]. The output from the L4_SM system is routinely monitored by the L4_SM team. Such monitoring provides valuable information; instances of unusually large Tb observation-minus-forecast (O-F) residuals can indicate events during which soil moisture conditions are poorly described in the land modeling system [7,8]. For example, repeated occurrences of very large Tb O-F values in central Australia were traced back to deficiencies in the gauge-based precipitation product used in the land modeling system through L4_SM Version 5; this discovery prompted the use, beginning in Version 6, of satellite- and gauge-based precipitation observations outside of North America [9]. Recently, a statistical analysis of the Tb O-F residuals revealed systematic errors in the “tau-omega” L-band radiative transfer model that converts the land-model simulated soil moisture and temperature into the forecast Tb prior to the L4_SM analysis. The L4_SM analysis is built on the ensemble Kalman filter and assumes unbiased forecast errors. The presence of systematic Tb forecast errors could thus adversely impact the quality of the analyzed soil moisture. In this paper, we examine the Tb O-F residuals of the latest Version 7 L4_SM data (Science Version ID Vv7030 and Vv7032) [10,11,12]. In the L4_SM Version 7 algorithm, key parameters of the L-band radiative transfer model, including soil roughness, scattering albedo, and a (seasonally varying) climatology of vegetation opacity, are obtained from the SMAP Level-2 Radiometer retrieval product (Version 5) [1]. Additionally, we also use ground measurements of surface soil moisture and soil temperature from the SMAP core validation sites as in situ reference of soil conditions [2,3]. As part of the L4_SM system calibration, the seasonally-varying bias between the model forecast Tb and the observed values is removed prior to the assimilation of the SMAP Tb observations [7]. That is, the L4_SM system is designed to only correct errors in synoptic-scale and interannual variations from the long-term mean seasonal cycle while maintaining the model’s (potentially erroneous) climatology. The Tb O-F residuals examined here are thus computed using SMAP Tb observations after they are rescaled to the mean seasonal cycle of Tb from the modeling system without SMAP data assimilation. Consequently, the long-term average of the Tb O-F residuals has a global mean of only 0.13 K and locally small values, ranging from -1 to 3 K. Despite the small time-average values of the Tb O-F residuals, the model forecast Tb was nevertheless found to exhibit undesirable systematic errors. At some locations, the time-average Tb O-F values strongly depend on surface soil moisture (SM). At the Yanco SMAP core validation site, for example, the Tb O-F residuals typically range from 5 to 15 K under dry soil moisture conditions (SM < 0.15 m3 m-3) yet are predominantly negative under wet soil moisture conditions (SM > 0.25 m3 m-3), with values ranging from 0 to -40 K. This results in soil moisture analysis increments that persistently make the soil drier under dry SM conditions and persistently make the soil wetter under wet SM conditions, suggesting an error in the dynamic range of the simulated Tb, soil moisture or soil temperature. In this paper, we describe the higher-order systematic Tb forecast errors in more detail, examine their impact on the L4_SM product quality, and explore potential avenues to further improve the L4_SM algorithm.

Rolf Reichle↗

NASA NTRS2010

Application of the Tor Vergata Scattering Model to L Band Backscatter During the Corn Growth Cycle

At the USDA's Optimizing Production Inputs for Economic and Environmental Enhancement (OPE3) experimental site in Beltsville, Maryland, USA) a field campaign took place throughout the 2002 corn growth cycle from May 10th (emergence of corn crops) to October 2nd (harvest). One of the microwave instruments deployed was the multi-frequency (X-, C- and L-band) quad-polarized (HH, HV, VV, VH) NASA GSFC/George Washington University (GWU) truck mounted radar. During the field campaign, this radar system provided once a week fully polarized C- and L-band (4.75 and 1.6 GHz) backscatter measurements from incidence angle of 15, 35, and 55 degrees. In support of microwave observations, an extensive ground characterization took place, which included measurements of surface roughness, soil moisture, vegetation biomass and morphology. The field conditions during the campaign are characterized by several dry downs with a period of drought in the month of August. Peak biomass the corn canopies was reached on July 24th with a total biomass of approximately 6.5 kg/sq m. This dynamic range in both soil moisture and vegetation conditions within the data set is ideal for the validation of discrete medium vegetation scattering models. In this study, we compare the L band backscatter measurements with simulations by the Tor Vergata model (ferrazzoli and Guerriero 1996). The measured soil moisture, vegetation biomass and most reliably measured vegetation morphological parameters (e.g. number of leaves, number of stems and stem height) were used as input for the Tor Vergata model. The more uncertain model parameters (e.g. surface roughness, leaf thickness) and the stem diameter were optimized using a parameter estimation routine based on the Levenberg-Marquardt algorithm. As cost function for this optimization, the HH and VV polarized backscatter measured and stimulated by the TOR Vergata model for incidence angle of 15, 35, and 55 degrees were used (6 measurements in total). The calibrated Tor Vergata model simulations are in excellent agreement with the measurements of Root Mean Squared Differences (RMSD's) of 0.8, 0.9 and 1.4 dB for incidences of 15, 35 and 55 degrees, respectively. The results from this study that a physically based scattering model with the appropriate parameterization can accurately simulate backscatter measurements and, as such, have the potential of being used for the retrieval of biophysical variables (e.g. soil moisture and vegetation biomass).

Joseph, A. T.↗

NASA NTRS1983

Effects of vegetation canopy on the radar backscattering coefficient

Airborne L- and C-band scatterometer data, taken over both vegetation-covered and bare fields, were systematically analyzed and theoretically reproduced, using a recently developed model for calculating radar backscattering coefficients of rough soil surfaces. The results show that the model can reproduce the observed angular variations of radar backscattering coefficient quite well via a least-squares fit method. Best fits to the data provide estimates of the statistical properties of the surface roughness, which is characterized by two parameters: the standard deviation of surface height, and the surface correlation length. In addition, the processes of vegetation attenuation and volume scattering require two canopy parameters, the canopy optical thickness and a volume scattering factor. Canopy parameter values for individual vegetation types, including alfalfa, milo and corn, were also determined from the best-fit results. The uncertainties in the scatterometer data were also explored.

Mo, T.↗

NASA NTRS1984

Bidirectional Reflectance Modeling of Non-homogeneous Plant Canopies

Efforts to develop a three dimensional model to predict canopy, bidirectional reflectance for heterogenous plant stands using incident radiation and canopy structural descriptions as inputs are described. Utility programs were developed to cope with the complex output from the 3 dimensional model. In addition an attempt was made to define leaf and soil properties, which are appropriate to the mode, by measuring leaf and soil bidirectional reflectance distribution functions; since almost no data exist on these distributions. In the process it was realized that most models probably are using the wrong leaf spectral properties, and that off-nadir reflectance measurements are difficult to make because of non-Lambertian properties of reference surfaces. Also, in the visible wavebands, rough soil may not be distinguishable from canopies when viewed from above.

Norman, J. M.↗

NASA NTRS1987

Retrieval of surface roughness parameters from dual-frequency measurements of radar backscattering coefficients

The use of dual-frequency data of backscattering coefficients at a fixed angle to estimate surface roughness parameters is evaluated. Radar backscattering coefficients at 1.5 and 4.25 GHz are calculated using a model based on Kirchhoff approximation of electromagnetic wave scatttering from a rough soil surface. Plots of the calculated backscattering coefficients for Kansas soil moisture contents at the C- and L-band frequencies and HH polarization are analyzed. The effects of changes in correlation length on the backscattering coefficients are investigated. The calculated backscattering coefficients are compared with scatterometer data collected at 1.5 and 4.25 GHz, and it is detected that the model and field data correlate well. The data reveal that it is possible to retrieve the surface roughness parameters from measured radar data.

Mo, Tsan↗

NASA NTRS2012

Inferring Land Surface Model Parameters for the Assimilation of Satellite-Based L-Band Brightness Temperature Observations into a Soil Moisture Analysis System

The Soil Moisture and Ocean Salinity (SMOS) satellite mission provides global measurements of L-band brightness temperatures at horizontal and vertical polarization and a variety of incidence angles that are sensitive to moisture and temperature conditions in the top few centimeters of the soil. These L-band observations can therefore be assimilated into a land surface model to obtain surface and root zone soil moisture estimates. As part of the observation operator, such an assimilation system requires a radiative transfer model (RTM) that converts geophysical fields (including soil moisture and soil temperature) into modeled L-band brightness temperatures. At the global scale, the RTM parameters and the climatological soil moisture conditions are still poorly known. Using look-up tables from the literature to estimate the RTM parameters usually results in modeled L-band brightness temperatures that are strongly biased against the SMOS observations, with biases varying regionally and seasonally. Such biases must be addressed within the land data assimilation system. In this presentation, the estimation of the RTM parameters is discussed for the NASA GEOS-5 land data assimilation system, which is based on the ensemble Kalman filter (EnKF) and the Catchment land surface model. In the GEOS-5 land data assimilation system, soil moisture and brightness temperature biases are addressed in three stages. First, the global soil properties and soil hydraulic parameters that are used in the Catchment model were revised to minimize the bias in the modeled soil moisture, as verified against available in situ soil moisture measurements. Second, key parameters of the "tau-omega" RTM were calibrated prior to data assimilation using an objective function that minimizes the climatological differences between the modeled L-band brightness temperatures and the corresponding SMOS observations. Calibrated parameters include soil roughness parameters, vegetation structure parameters, and the single scattering albedo. After this climatological calibration, the modeling system can provide L-band brightness temperatures with a global mean absolute bias of less than 10K against SMOS observations, across multiple incidence angles and for horizontal and vertical polarization. Third, seasonal and regional variations in the residual biases are addressed by estimating the vegetation optical depth through state augmentation during the assimilation of the L-band brightness temperatures. This strategy, tested here with SMOS data, is part of the baseline approach for the Level 4 Surface and Root Zone Soil Moisture data product from the planned Soil Moisture Active Passive (SMAP) satellite mission.

Reichle, Rolf H.↗

NASA NTRS2013

Uncertainty Quantification of GEOS-5 L-band Radiative Transfer Model Parameters Using Bayesian Inference and SMOS Observations

Uncertainties in L-band (1.4 GHz) radiative transfer modeling (RTM) affect the simulation of brightness temperatures (Tb) over land and the inversion of satellite-observed Tb into soil moisture retrievals. In particular, accurate estimates of the microwave soil roughness, vegetation opacity and scattering albedo for large-scale applications are difficult to obtain from field studies and often lack an uncertainty estimate. Here, a Markov Chain Monte Carlo (MCMC) simulation method is used to determine satellite-scale estimates of RTM parameters and their posterior uncertainty by minimizing the misfit between long-term averages and standard deviations of simulated and observed Tb at a range of incidence angles, at horizontal and vertical polarization, and for morning and evening overpasses. Tb simulations are generated with the Goddard Earth Observing System (GEOS-5) and confronted with Tb observations from the Soil Moisture Ocean Salinity (SMOS) mission. The MCMC algorithm suggests that the relative uncertainty of the RTM parameter estimates is typically less than 25 of the maximum a posteriori density (MAP) parameter value. Furthermore, the actual root-mean-square-differences in long-term Tb averages and standard deviations are found consistent with the respective estimated total simulation and observation error standard deviations of m3.1K and s2.4K. It is also shown that the MAP parameter values estimated through MCMC simulation are in close agreement with those obtained with Particle Swarm Optimization (PSO).

MCMC↗