Comparison of δ-2AA, δ-4AA, δ-2DOM, δ-4DOM, δ-2VIM, and δ-4VIM for flux (W m−2) at the top and surface by using the mid-latitude winter atmospheric profile with \n
\r\n\tIn this book, the different factors of liquefaction, the field methods and laboratory tests to identify a potentially liquefiable soil aim to be reviewed; in addition with history cases (ground behavior during the occurrence of an earthquake, state of stress, deformation, shear strength, flow, etc.).
\r\n\tA very important aspect of this topic is the presentation of the different constructive techniques used to ground improvement (vibrocompaction, dynamic compaction, jet grouting, chemical injection, replacement, etc.), placing special emphasis on those constructive methods used to solve problems on structures already located in areas of low relative density with liquefaction potential, where the installation of monitoring and control equipment is also required (tiltmeters, piezometers, topographic points, seismographs, pressure cells, etc.).
Spinal cord injury (SCI) is one of the most devastating traumas for an individual and their family because, depending on the level of injury, the complete traumatic SCI leads to paraplegia or tetraplegia [1].
\nImmediate traumatic SCI is the primary mechanical injury caused through the direct injury of the neurons, axons, and blood vessels (compression, laceration, shearing, and even transection of the spinal cord). After the injury event, the secondary injury mechanisms begin immediately and the secondary spinal cord lesions consist of hemorrhages, spinal cord edema, vasospasm, and hypoperfusion of the spinal cord and the damage of the spinal cord continues to progress for several days to weeks, and leads to the death of neurons and the interruption of the axonal tracts [2].
\nMany traumatic spinal cord injuries can be initially incomplete and the secondary damage completes the lesion of the spinal cord. Spinal cord injuries are difficult to treat because of these secondary injuries. Current therapy is unable to act on the primary mechanically lesion, but the secondary injury extension of the spinal cord could be stopped or reduced by an early efficient therapy.
\nIt is necessary to know the type and the evolution of the secondary spinal cord lesions: complete destruction of spinal cord or an injury with potential recovery. To this end it is ideal to have complete and extensive information about the injury and this information must reveal the treatment required to ensure a favorable outcome.
\nIn SCI, the neurological examination brings the first very important information about the lesion and this directs imaging procedures to confirm the lesion. But because of the spinal shock, unstable condition of the patient, attendant injuries, alcohol or drugs etc., the clinical examination immediately following injury, even using the American Spinal Injury Association (ASIA) motor scores or other scales, cannot be considered reliable. These clinical examinations must be repeated, but they offer only static clinical states and no data about possible future development.
\nThe noninvasive imaging techniques used in SCI are the radiographies of the spine, spinal computed tomography (CT) or/and spinal magnetic resonance imaging (MRI), even functional MRI, tractography, or the recent structural volumetric and microstructural MRI protocols of the site of SCI.
\nAll these offer only static images of the stage of the lesion and not an accurate prediction of the severity of SCI and about the development of the secondary spinal cord lesions.
\nWe need a dynamic approach for the lesions; hence, biomarkers were evaluated for their capacity to be sensitive and accurate tools to measure the neuronal injuries and to predict the evolution of these injuries. Biomarkers are measurable features that can be used to confirm the presence or to predict the severity of the disorders. Biomarkers as biochemical indicators in SCI can allow detection of the secondary lesion, can monitor its progress and predict the severity of SCI, and can also indicate the specific treatments required. In SCI, biomarkers detect the severity of injury within the first few hours and can direct the best patient care in a timely manner.
\nIn acute traumatic SCI, the mechanical injuries directly cause axonal destruction in fiber tracts, destruction of the neurons in gray matter, and of the glial cells. Their destruction releases substances—cellular constituents, whose presence, quantity, and dynamics can be lesional biomarkers.
\nThe reactions of partially injured cells and of uninjured cells around the site of injury simultaneously start as response to the biochemical substances released by the destruction. The responses of these cells, that is, the synthesis of new proteins, are secondary to changes in mRNA. Detecting these changes allows us to determine their role at the site of SCI, to stabilize the damaged cells, to stop the spinal cord scar formation, and so on. The correlation of these changes with copied, or transcribed mRNA, from DNA, will establish the responsible genes intervening in the response to the SCI [3].
\nThe assessment of these changes in the damaged spinal cord will allow therapeutic responses, for example:
\n- to stop the mechanisms by which secondary spinal cord lesion occurs and its progression,
- to understand the mechanisms of formation of a spinal cord scar as mechanical barrier that obstructs the axonal regeneration processes,
- even genetically targeted therapy to stimulate the genes in DNA responsible for neuronal regeneration, stimulating mRNA or to use the necessary proteins for SCI healing, and so on.
Detecting these protein changes, their quantity and dynamics may be biomarkers of response, or reaction biomarkers.
\nCorrelating the lesional biomarkers and the reaction biomarkers with the clinical outcome and with the imaging techniques will enable understanding the complexity of the biological response to SCI and the establishment of appropriate therapies. Obtaining cells from the site of SCI is problematic in patients, therefore most research evaluated the lesional biomarkers in human spinal cord injuries. There are new approaches in the management of acute traumatic SCI that could enable obtaining cells from the site of SCI without adverse consequences for the patient.
\nThe lesional biomarkers appear immediately post injury and their dynamics show the extension of the SCI and after several hours there are both lesional biomarkers and reaction biomarkers, involving the secondary cellular response to injury.
\nIn recent years, a number of protein biomarkers have been evaluated to detect neuronal injury and recently there have been studies about their potential diagnostic and predictive value for spinal cord injuries. The concentration of specific proteins in blood or in the cerebrospinal fluid (CSF) must be compared with the nervous tissue injury and these can be biomarkers for the pathologic processes in SCI.
\nThere are numerous experimental studies and a smaller number of clinical studies for determining and validating biomarkers in SCI: c-Tau, myelin basic protein (MBP), neuron-specific enolase (NSE), glial fibrillar acidic protein (GFAP), and so on [4].
\nThe researches have not been systematized and because the studies have been done at different time interval from the moment of the trauma, they did not differentiate between lesional biomarkers and reaction biomarkers and many of them were not followed to verify the clinical usefulness.
\nA brief overview of the evolution of these researches is shown below.
\nvan Dongen et al. [5, 6] correlated the concentration of S-100 protein in the CSF with the results of somatosensory and motor-evoked potential monitoring indicating spinal cord ischemia during and after thoracoabdominal aortic aneurysm (TAAA) surgery and concluded that the S-100 protein in CSF seems to be a marker to detect spinal cord ischemia [5, 6], and in 2001 Kunihara et al. [7] found increased levels of S-100β in patients with post-operative SCI caused by spinal cord ischemia too. Basu et al. [8] presented free radicals as an inflammatory response indicator with the role of biomarker during spinal cord ischemia.
\nIn 2003, Guéz et al. evaluated the CSF concentration of light subunits of neurofilaments (NF-L, 68 kDa) and of glial fibrillary acid protein (GFAP) in after trauma to the cervical spine: patients with acute traumatic cervical SCI and whiplash cases, compared to a control group of normal cases. The CSF concentrations of both light subunits of neurofilaments and GFAPs were significantly higher in all the cases with cervical SCI and pronounced neurological deficits [9].
\nIn 2005, Loy et al. [10] reported that serum levels of NSE and S-100β protein are biomarkers in an animal model of traumatic SCI.
\nKwon et al. [11] studied as possible biomarkers other inflammatory cytokines and structural proteins: S-100β (a glial-specific calcium-binding β protein), glial fibrillary acidic protein (GFAP), and interleukin 8 (IL-8, also known as neutrophil chemotactic factor), in patients within 24 hours post-SCI. Their concentration in the CSF and blood samples in patients with complete and incomplete SCI showed they could be potential biomarkers to diagnose the severity of SCI [11, 12].
\nIn a literature review from 1966 to 2008, Pouw et al. [13] identified the biomarkers S-100β, NSE, neurofilament light chain, and glial fibrillary acidic protein as significantly higher in cases of experimental SCI in animal models.
\nNew potential biomarkers were reported: the neurofilaments, the major cytoskeletal components in axon fibers. The most important are neurofilament subunit proteins (NF) that coassemble forming the cytoskeletal of axon fibers and they consist of five subunits of neurofilaments, named on the basis of molecular weight: heavy or highest (NF-H, 200–220 kDa), medium or middle (NF-M, 145–160 kDa), and light or lowest (NF-L, 68–70 kDa) subunits, also alpha-internexin subunit (NF66) discovered later than NF and the intermediate filament protein subunit peripherin [4].
\nUeno et al. [14] presented a rat model of acute SCI and they showed that the high molecular weight neurofilament subunit levels in plasma could be a biomarker for evaluating the efficacy of therapies for SCI.
\nHayakawa et al. [15] studied the concentration of the phosphorylated neurofilament subunit NF-H (pNF-H) in plasma in patients with acute cervical SCI and concluded pNF-H may be a prognostic biomarker for SCI.
\nIencean et al. [16] measured pNF-H concentration by enzyme-linked immunosorbent assay (ELISA) test in CSF in acute SCI patients and correlated the values of pNF-H with the clinical evolution, also they measured the normal values in samples obtained by lumbar puncture from individuals without neurologic disorders. They showed the phosphorylated form of the neurofilament subunit NF-H (pNF-H) is a biomarker in SCI in humans and its increased values are consistent with an unfavorable outcome. The neurofilament subunit NF-H (pNF-H) is a lesional biomarker, it appears after the mechanical injury by axonal destruction in the fibers tracts [16].
\nBy now these studies have identified some potential biomarkers, but these biomarkers have not been validated and they still cannot be used in the clinical setting, for diagnosis, prognosis, and evaluating therapeutic interventions.
\nThe research in traumatic SCI has been focused on the discovery of lesional biomarkers and lesser for reaction biomarkers. Lesional biomarkers can be studied in patients with acute traumatic SCI immediately after injury; reaction biomarkers occur after a short period post injury and after several hours post injury these two types of biomarkers coexist, and it is difficult to differentiate them. The study of reaction biomarkers involves cells around the lesion, which is not possible in patients with SCI. Therefore research is conducted on nerve cell cultures and there are experimental animal models, but the translation into human medicine is difficult because there are important differences. The most important studies on lesional biomarkers concern the neurofilament subunit proteins (NF).
\nPouw et al. [17] in a prospective cohort study obtained CSF from sixteen acute traumatic SCI patients within 24 hours post injury and found that the concentrations of glial fibrillary acidic protein, NSE, S-100β, tau and neurofilament heavy chain (NFH) in motor complete patients was significantly higher compared with motor incomplete patients.
\nTakahashi et al. [18] conducted a study to evaluate pNF-H levels in the CSF of patients with worsening symptoms of cervical compression myelopathy and their results suggest that pNF-H in CSF can act as a biomarker that reflects the severity of acutely worsening compression myelopathy.
\nIencean et al. measured the phosphorylated neurofilament subunit NF-H (pNF-H) in the CSF of patients with SCI and demonstrated the correlation between the pNF-H levels and the severity of the injury. They studied 15 subjects with acute traumatic SCI who underwent surgery during the first 24 hours post injury (decompression, stabilization): eight patients with complete SCI and seven patients with incomplete SCI. They measured daily the heavy phosphorylated neurofilament subunit (pNF-H) concentration by sandwich ELISA test in CSF in all patients. The level of CSF pNF-H was ten to a hundred times higher in complete SCI than the level of CSF pNF-H in cases with incomplete SCI, where the level of this biomarker was close to normal [19, 20].
\nThe patients with early surgery in complete SCI and with a favorable outcome had a specific pattern of daily values of pNF-H: a sudden increase up to a maximum value then a gradual decrease to normal; the peak was different in each case, from 10 times up to 170 times higher than normal (Figure 1).
\nThe same type of the pattern for the values of pNF-H appears in the incomplete SCI with favorable outcome, but with smaller values of pNF-H.
\nThere are two patterns in cases with unfavorable outcome or neurological stationary after the same early surgery and treatment:
\n- the second unfavorable pattern had a progressive increase up to a peak and then was followed by a progressive decrease to normal values, the peak was a hundred times higher than normal values (Figure 2),
- an increase to a plateau of pNF-H values, with increased values five or ten times higher than normal (Figure 3).
Pattern of daily value of pNF-H in patients with favorable outcome.
Pattern with progressive increase of pNF-H.
Pattern with increase up to a plateau of pNF-H.
The authors found that in patients with favorable development the progressive decrease of pNF-H values after the initial sudden increase, without extension of increased values in plateau or without a second peak, signifies a reduction or even a stop of the secondary lesion with evident effect on the favorable outcome in the SCI (Figure 4).
\nThe three specific and predictive patterns of daily values of pNF-H in traumatic SCI.
Kato et al. [21] investigated the phosphorylated form of the high molecular weight neurofilament subunit (pNF-H) levels in the serum in patients with cervical compressive myelopathy and they found an elevated serum level of pNF-H only in acute worsening of myelopathy and this study confirms that pNF-H is a lesional biomarker.
\nKuhle et al. [22] presented their results on a study of serum neurofilament light chain (pNF-L) in human SCI. They concluded that serum neurofilament light subunit (pNF-L) concentration in SCI patients has a close correlation with acute severity and neurological outcome and it is of predictive value in SCI patients.
\nThe presentation of these studies on biomarkers in SCI highlights that the most important ones and those with significant results relate to lesional biomarkers, and first are the phosphorylated neurofilament subunits, light or heavy (pNF-L or pNF-H), resulting from the axonal neurofilament destruction. The research showed that the phosphorylated neurofilament subunit, light or heavy (pNF-L or pNF-H) in SCI is a specific lesional biomarker for SCI and it can distinguish the severity of SCI (Hayakawa, Iencean, and Kuhle).
\nThe heavy phosphorylated neurofilament subunit (pNF-H) is a predictive lesional biomarker because its values pattern can show the reducing or stopping of the secondary lesions and the favorable outcome. The complete SCI patients with a favorable development had a specific pattern of daily values of pNF-H: a sudden increase up to a maximum value then a progressive decrease to normal. The patients with unfavorable outcome or neurological stabilization had two patterns: an increase to a plateau of pNF-H values or a progressive increase up to a peak followed by a progressive decrease to quasi-normal values.
\nThese studies on biomarkers in spinal cord injuries highlight that the most important lesional biomarkers are the phosphorylated neurofilament subunits, light or heavy (pNF-L or pNF-H). The phosphorylated neurofilament subunits (pNF-L or pNF-H) are specific lesional biomarkers for SCI and they can distinguish the severity of SCI.
\nThe heavy phosphorylated neurofilament subunit (pNF-H) is a predictive lesional biomarker; its values pattern shows the reducing or stopping of the secondary lesions and the favorable outcome.
\nThere is a specific pattern of daily values of pNF-H in complete SCI patients with a favorable outcome: a sudden increase up to a maximum value then a progressive decrease to normal. Also there are two patterns in the patients with unfavorable outcome: an increase to a plateau of pNF-H values or a progressive increase up to a peak followed by a progressive decrease to quasi-normal values.
\nThese specific patterns could be used to aid clinicians with making a diagnosis and establishing a prognosis, and evaluating therapeutic interventions. These studies should continue on larger groups of patients to prove the clinical usefulness.
\nAlso the studies on reaction biomarkers are very important, but obtaining cells from the site of SCI is problematic in humans. A new approach in the management of acute traumatic SCI has been proposed that could enable obtaining cells from the site of SCI without adverse consequences for the patient. In the cases with a predictive pattern of unfavorable outcome or neurological stationary after decompression and stabilization during the first 24 hours, a new approach was proposed based on the predictive pattern of daily values of pNF-H. If the clinical neurologic evolution is unfavorable and imaging techniques (MRI) show a complete SCI and the daily values of NFP-H as lesional biomarker form predictive unfavorable pattern, a second microneurosurgery in the SCI site can create favorable conditions for functional recovery of the remaining spinal cord: opening the spinal cord in the midline and microsurgical debridement of the necrotic tissue. At the same time this second microneurosurgical approach in the SCI site could enable obtaining cells from the site of SCI without adverse consequences for the patient. The use of these cells (neurons and glial cells around the lesion) for cell culture techniques will allow the study of the changes in the spinal cord at the molecular and structural levels in humans.
\nDiagnosis, prognosis, and treatment guidance based on biomarker used as a predictive indicator can determine ethical difficulties by differentiated therapies in patients with SCI.
\nIt is difficult to stop or to limit the treatment of neurological recovery in patients with complete SCI, with paraplegia or tetraplegia, with complete spinal cord lesions on imaging techniques and unfavorable patterns of predictive lesional biomarkers. We do not currently know the value of the lesional predictive biomarkers for the neurological outcome several years after the injury. At the moment, we cannot take a decision limiting the treatment of neurological recovery in patients with complete SCI because we do not know the complexity of the biological response to SCI.
\nThis requires extensive and profound research both on lesional biomarkers and on reaction biomarkers correlated with genetic and molecular response in SCI and we hope further research will deliver effective treatments.
\nThis work was funded by the CNCS– UEFISCDI Romania, grant: “Immediate neuroprotective therapy in acute traumatic spinal cord injury”, grant number: PN-II-IDPCE-2011-3-0569.
\nSolving the radiative transfer (RT) equation is a key issue when dealing with solar/infrared radiative processes related to climate simulations (e.g., radiative flux and heating rate calculation at each level of a climate model). In recent decades, numerous approximation techniques have been proposed to solve the RT equation [1, 2, 3, 4, 5].
\nFor solar radiation, analytical solutions of various two-stream approximations [6] have been widely used in climate models [7]. More accurate methods of four-stream approximation and six-stream approximation have also been proposed (e.g., [8, 9, 10, 11, 12, 13]). Based on the invariance principle [14], an adding method of four-stream discrete ordinates method (DOM) (4DDA) has been proposed. The advantage of the adding method is that it can handle partial cloud in climate models (e.g., [15, 16, 17]). 4DDA is much more accurate than the adding method of two-stream DOM (2DDA) in flux and heating rate calculation, especially under cloudy-sky conditions, where cloud plays an important role in radiative transfer with different dynamics and microphysics [18, 19]. Zhang and Li [20] proposed a new solar RT parameterization (4SDA) which is based on four-stream harmonic expansion to simulate RT in climate modeling.
\nSince scattering is much weaker for infrared radiation than for solar radiation, an absorption approximation (AA) is used in most current climate models [7]. In AA, the scattering phase function is replaced by a δ-function, meaning that scattering is neglected in all but the forward direction. The advantage of AA is that the upward and downward transfer processes are completely separated. If scattering is considered, the upward and downward intensities are coupled. Under this condition, Fu [21] derived an inverse matrix method to deal with multiple scattering in the atmosphere by using the two-stream DOM (2DOM) and the four-stream DOM (4DOM). It is found that 4DOM is more accurate than 2DOM, especially for thin optical depths. Even if scattering is included, TOA errors for 2DOM can still be >2 W m−2 for cirrus clouds [22]. The 4DOM results tend to be very accurate, but the calculation process is complicated. Zhang et al. [23] apply the four-stream adding method to resolve multilayer infrared RT.
\nIn the past two decades, the variational iteration method (VIM) has been used to deal with many nonlinear problems [24, 25, 26, 27, 28, 29, 30, 31]. In addition, VIM has been shown to converge faster to the exact solution than other methods do. An accurate solution can be found for most nonlinear problems in only one or two iterations for small solution domains in VIM [27, 31]. Because of its flexibility, convenience and efficiency, VIM has been applied to various nonlinear equations ([27, 28, 29] and many others). In VIM, a nonlinear problem is separated into linear and nonlinear terms, where the latter are usually difficult to deal with and are initially approximated as a first guess. Subsequently, a correction function is constructed by using a general Lagrange multiplier that can be identified optimally via variational theory. Although the infrared RT equation is not a nonlinear equation, it contains an integral term for the scattering, and this is very complicated to deal with. Therefore, VIM is applied to solve the infrared RT equation.
\nThe objective of the chapter is to introduce 4DDA/4SDA scheme [15] for solar RT, AA, and VIM [32] schemes and apply them to resolve solar/infrared radiative transfer in RT models. The simulation results and their comparison to results from the 128-stream discrete ordinates method radiative transfer scheme [33] are shown in Section 4. Finally, a summary is given in Section 5.
\nThe azimuthally averaged solar RT equation is
\nwhere \n
where \n
The four-stream discrete ordinates scheme (4DDA) use the Gaussian quadrature method to deal with the integration in Eq. (1). Eq. (1) yields
\nwhere the quadrature point \n
After a lengthy and laborious derivation, the solution of (3) is given by
\nwhere G is determined by the boundary conditions, and values of other parameters are shown in [13].
\nThe adding method which is used to deal with multilayer RT is shown in [13].
\nThe purpose of the four-stream spherical harmonic expansion (4SDA) is to separate out the angle-dependent intensity by assuming
\nBy substituting Eqs. (2) and (5) into Eq. (1) and using the orthogonality relation of the Legendre function in \n
The solution of Eqs. (6)–(9) is
\nwhere \n
The 4SDA can also be applied to solve multilayer solar RT. The detailed process can be found at [20].
\nThe infrared RT equation for intensity \n
where \n
In absorption approximation (AA), the scattering phase function \n
We use \n
where \n
A general nonlinear system is used to illustrate the basic idea of variational iteration method (VIM) [24, 25, 26, 27, 28, 29, 30, 31]:
\nwhere \n
Here, \n
Based on VIM, the functional reiteration of the infrared RT equation can be deduced as
\nAccording to VIM theory, under the conditions that \n
For the above correction functional to be stationary [i.e., \n
Therefore the Lagrangian multiplier \n
By substituting Eq. (19) into Eq. (16), we obtain
\nIn two-stream VIM (2VIM), the term of \n
where \n
We consider a surface boundary condition with a surface emissivity \n
where \n
In four-stream VIM (4VIM), the term of\n
where \n
The surface boundary condition with an emissivity \n
To enhance the accuracy of the radiative schemes, a δ-function adjustment is used to adjust the optical parameters following Wiscombe [34]. We refer to the VIM solutions with the δ-function applied as δ-2VIM and δ-4VIM.
\nRT in the atmosphere is a complicated process. It depends not only on the single-layer direct reflection and transmission but also on the diffuse results and the gaseous transmission, cloud-aerosol scattering and absorption, etc. It is important to evaluate errors in radiation under a variety of atmospheric conditions by using a radiation algorithm. The Fu-Liou radiation model [35] is used in this study. This model adopts the correlated-k distribution method for gaseous transmission, including five major greenhouse gases, H2O, CO2, O3, NO2, and CH4.
\nIn benchmark calculations, the discrete ordinates model [33] with a 128-stream scheme (128S) is incorporated with the gaseous transmission scheme of the Fu-Loiu radiation model [35] by replacing the existing radiative transfer algorithm in the model. Also the two-stream discrete ordinates adding method (2DDA), Eddington adding method (2SDA), four-stream discrete ordinates adding method (4DDA), and four stream spherical harmonic adding method (4SDA) schemes are incorporated with the same gaseous transmission scheme. The atmosphere was vertically divided into 280 layers, each of thickness 0.25 km. The mid-latitude winter of atmospheric profile [36] is used with a surface albedo of 0.2 for each band. For a water cloud the optical properties are parameterized in terms of liquid water content (LWC) and effective radius (re) [36]. Two calculations are performed: (1) a low cloud (\n
In Figure 1, the benchmark results of heating rate are shown for three solar zenith angles under low/middle cloud condition (Figure 1a–c for low cloud condition, Figure 1g–i for middle cloud condition) as well as the absolute errors of 2DDA, 2SDA, 4DDA, and 4SDA against the benchmark results (Figure 1d–f for cloud condition, Figure 1j–l for middle cloud condition). When \n
Heating rate profiles computed from 128S and the error profiles from 2DDA, 2SDA, 4DDA, and 4SDA. Three solar zenith angles \n\n\nμ\n0\n\n=\n1\n\n, \n\n\nμ\n0\n\n=\n0.5\n\n, and \n\n\nμ\n0\n\n=\n0.25\n\n are considered.
For the infrared spectra, the accuracy and efficiency of δ-2AA, δ-4AA, δ-2VIM, δ-4VIM, δ-2DOM, and δ-4DOM will be systematically compared. In addition, the discrete ordinates model [28] with a 128-stream scheme (δ-128DOM) is used as the benchmark model.
\nA radiation model [17] is used to study the accuracy of the VIM scheme for multiple layers within a model atmosphere. A correlation-k distribution scheme is used to simulate the gaseous transmission with profiles for H2O, CH4, CO2, NO2, O3, and CFCs. This model is reasonably efficient because it neglects to scatter for certain intervals with very large absorption coefficients and water vapor continuum at high altitudes. Nine infrared bands are adopted in this model in wavenumber ranges 0–340, 340–540, 540–800, 800–980, 980–1100, 1100–1400, 1400–1900, 1900–2200, and 2200–2500 cm−1. The optical properties of ice and water clouds are calculated based on the radiative property parameterization of [37, 38, 39], respectively. The mid-latitude winter atmospheric profiles [36] are used. The atmospheric profile is divided into homogeneous layers with a geometrical thickness of 0.25 km.
\nIn this model, a low cloud with an effective radius \n
In the left column of Figure 2, the benchmark heating rates calculated by δ-128DOM are given under conditions of low clouds (Figure 2a), middle clouds (Figure 2d), high clouds (Figure 2g) and the all-sky case containing a combination of low, middle, and high clouds (Figure 2j). The middle column of Figure 2 shows the errors in the calculated heating rates of δ-2AA, δ-2DOM, and δ-2VIM compared to those of δ-128DOM. Furthermore, the right column of Figure 2 shows the errors in the calculated heating rates of δ-4AA, δ-4DOM, and δ-4VIM compared to those of δ-128DOM.
\nHeating rate profiles calculated by δ-128DOM and the error profiles produced by δ-2AA, δ-4AA, δ-2DOM, δ-4DOM, δ-2VIM, and δ-4VIM in the mid-latitude winter atmospheric profile with surface emissivity \n\n\nε\ns\n\n=\n1\n\n for (a)–(c) low cloud, (d)–(f) middle cloud, (g)–(i) high cloud, and (j)–(l) all three cloud types.
For the low-cloud case, the maximum errors of δ-2AA, δ-2DOM, and δ-2VIM are 20.8, 10.5, and 10.3 K day−1, respectively, at a height corresponding to the top of the cloud (Figure 2b). The maximum errors of δ-4AA, δ-4DOM, and δ-4VIM are 20.9, 10.1, and 10.03 K day−1, respectively, at the same height (Figure 2c). This shows that, for low-level water clouds, δ-2VIM (δ-4VIM) is more accurate than δ-2AA (δ-4AA) and δ-2DOM (δ-4DOM).
\nFor the middle cloud, the maximum errors of δ-2AA, δ-2DOM, and δ-2VIM are approximately 20.9, 11.1, and 10.8 K day−1, respectively, at a height corresponding to the top of the cloud. At the bottom of the cloud, the errors are 10.6 K day−1 for δ-2AA and just 20.05 and 10.05 K day−1 for δ-2DOM and δ-2VIM, respectively (Figure 2e). At the top of the cloud, δ-4AA, δ-4DOM, and δ-4VIM produce biases of approximately 11.1, 10.7, and 10.5 K day−1, respectively (Figure 2f).
\nFor the high-cloud case, the optical depth of the ice cloud is much smaller than that of the water cloud. Based on the results shown in Figure 2, the accuracies of δ-2DOM and δ-2VIM are similar and are better than that of δ-2AA. Furthermore, δ-4AA, δ-4DOM, and δ-4VIM have very similar accuracies. As seen in the third row in Figure 2, δ-2AA produces an error of approximately 10.6 K day−1 at a height corresponding to the top of a high cloud. The accuracies of δ-2DOM and δ-2VIM are similar; the maximum errors of both are approximately 10.5 K day−1 (Figure 2h). The maximum errors of δ-4AA, δ-4DOM, and δ-4VIM are all about 10.1 K day−1(Figure 2i). However, the difference is that the maximum errors occur at top of the high cloud for δ-4AA but on the bottom for δ-4DOM and δ-4VIM.
\nIn the case of all three clouds together, δ-2VIM is more accurate than δ-2AA and δ-2DOM for the low and middle clouds (Figure 2k). In general, δ-4DOM and δ-4VIM are comparable in accuracy for cloud heating rate (Figure 2l).
\nThe results of upward (downward) flux at the TOA (surface) for the six schemes are presented in Table 1 for mid-latitude winter profiles with the surface emissivity \n
Atmospheric condition | \n128DOM | \n2AA | \n2DOM | \n2VIM | \n4AA | \n4DOM | \n4VIM | \n
---|---|---|---|---|---|---|---|
Upward flux (TOA) | \n\n | \n | \n | \n | \n | \n | \n |
Low clouds | \n209.5326 | \n211.2735 (1.7409) | \n208.5206 (−1.012) | \n209.0425 (−0.4901) | \n210.9585 (1.4259) | \n209.1645 (−0.3681) | \n209.5458 (0.0132) | \n
Middle clouds | \n183.5646 | \n185.7200 (2.1554) | \n182.4027 (−1.1619) | \n182.9060 (−0.6586) | \n182.4002 (−1.1644) | \n183.1096 (−0.4550) | \n183.5357 (−0.0289) | \n
High clouds | \n198.1357 | \n200.3545 (2.2188) | \n198.6394 (0.5037) | \n198.8373 (0.7016) | \n199.2441 (1.1084) | \n197.2351 (−0.9006) | \n197.4354 (−0.7003) | \n
Low, middle, and high clouds | \n176.3821 | \n179.7768 (3.3947) | \n175.6421 (−0.7400) | \n176.2356 (−0.1465) | \n178.8788 (2.4967) | \n175.5891 (−0.7930) | \n175.8856 (−0.4965) | \n
Mean square error | \n\n | 6.0309 | \n0.7939 | \n0.2969 | \n2.7128 | \n0.4456 | \n0.1845 | \n
Downward flux (surface) | \n\n | \n | \n | \n | \n | \n | \n |
Low clouds | \n302.7236 | \n302.7267 (0.0031) | \n302.9533 (0.2297) | \n302.9384 (0.2148) | \n302.7429 (0.0193) | \n302.8662 (0.1426) | \n302.8449 (0.1213) | \n
Middle clouds | \n287.4458 | \n286.9155 (−0.5303) | \n287.7658 (0.3200) | \n287.6141 (0.1683) | \n287.7658 (0.3200) | \n287.5678 (0.1220) | \n287.5418 (0.096) | \n
High clouds | \n247.8497 | \n248.0313 (0.1816) | \n248.3309 (0.4812) | \n248.3303 (0.4806) | \n248.2968 (0.4471) | \n248.5608 (0.7111) | \n248.6452 (0.7955) | \n
Low, middle, and high clouds | \n302.1410 | \n302.0497 (−0.0913) | \n302.4413 (0.3003) | \n302.4227 (0.2817) | \n302.0189 (−0.1221) | \n302.3161 (0.1751) | \n302.3151 (0.1741) | \n
Mean square error | \n\n | 0.0806 | \n0.1192 | \n0.0962 | \n0.0794 | \n0.1429 | \n0.1718 | \n
Comparison of δ-2AA, δ-4AA, δ-2DOM, δ-4DOM, δ-2VIM, and δ-4VIM for flux (W m−2) at the top and surface by using the mid-latitude winter atmospheric profile with \n
The flux differences between the six approximate schemes and δ-128DOM are listed in parentheses. All δ symbols are neglected in the table.
For climate modeling, the efficiency of radiative transfer parameterization is also very important. Table 2 lists the computing times required for δ-2AA, δ-4AA, δ-2DOM, δ-4DOM, δ-2VIM, and δ-4VIM, which were computed by HP EliteDesk 880 G1 TWR with 8 Intel(R) Core(TM) i7–4790 CPUs, 32-bit operating system, and 8GB memory. The results are normalized to that of δ-2AA. The computational efficiency of δ-2VIM is slightly better than that of δ-2DOM, which took more than double the time of δ-2AA. However, δ-4VIM is more than twice as fast as δ-4DOM for the radiation algorithm alone and the radiation model.
\n\n | 2AA | \n4AA | \n2DOM | \n4DOM | \n2VIM | \n4VIM | \n
---|---|---|---|---|---|---|
Algorithm only | \n1.0 | \n1.4412 | \n2.2223 | \n14.8195 | \n2.0365 | \n5.3615 | \n
Radiation model | \n1.0 | \n1.1095 | \n1.4114 | \n5.8899 | \n1.3759 | \n2.2739 | \n
Computing times of δ-2AA, δ-4AA, δ-2DOM, δ-4DOM, δ-2DDA, δ-4DDA, δ-2VIM, and δ-4VIM (normalized by the computing time of δ-2AA).
All δ symbols are neglected in the table.
The objective of the paper is to introduce 4DDA/4SDA [13, 20] for solar RT, AA, and VIM [32] for infrared RT and applied them to radiative transfer models.
\n4DDA use Gaussian quadrature method to deal with the integration in the RT equation. 4SDA is based on four-stream harmonic expansion in radiative intensity. By applying 4DDA/4SDA to a realistic atmospheric profile with gaseous transmission considered, it is found that the accuracy of 4DDA/4SDA is superior to Eddington adding method (2SDA) and two-stream discrete ordinates adding method (2DDA), especially for the cloudy conditions. It is shown that the relative errors of 4SDA are generally less than 1% in heating rate, while the relative errors of both 2SDA and 2DDA are over 6%.
\nVIM differs from other analytical methods for solving nonlinear differential equations in that it requires neither linearization nor small perturbations. The optimal result is constructed through variation by a Lagrange multiplier. It was found that the scattering term in the infrared RT equation could be dealt with as a nonlinear operator in VIM. By taking the AA solution as the zeroth-order solution, the scattering effect was properly included in the first-order iterative solution.
\nThe six schemes of δ-2AA, δ-4AA, δ-2DOM, δ-4DOM, δ-2VIM, and δ-4VIM were compared systematically against the benchmark results provided by δ-128DOM for a multilayer case. For a multilayer atmosphere, VIM gave more accurate results than those of DOM and AA for the low and middle clouds in both the two- and four-stream cases. However, the errors from VIM for high clouds were similar to those from AA.
\nComputationally, δ-2VIM and δ-2DDA were slightly faster than δ-2DOM, which took more than double time of δ-2AA. However, δ-4VIM/δ-4DDA was more than twice as fast as δ-4DOM for both the pure radiation algorithm and the radiation model in a layered, cloudy atmosphere. In general, the main benefit of δ-2VIM was an improved accuracy with a computational time similar to that of δ-2DOM. The main benefit of δ-4VIM/δ-4DDA was improved computational efficiency with accuracy similar to that of δ-4DOM.
\nIn view of their overall high accuracy and computational efficiency, the conclusion is that 4DDA, 4SDA, δ-2VIM, and δ-4VIM are well suited for parameterizing the solar/infrared RT in climate models.
\nThe work is supported by the National Key R&D Program of China (2018YFC1507002) and National Natural Science Foundation of China (41675003 and 41675056).
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