Short-Wave Solar Radiation in the Earth’s Atmosphere Part 7 pptx

There are the vertical profile of air temperature TP i, profiles of contents of four gases absorbing radiation QH2OP i , QO3P i , QNO2P i , QNO3P i The O2tent is constant, volume coefficien

Derivative from Values of Solar Irradiance 181The formula of the derivative is specially converted to form (5.12), (5.13).Written in this way, it looks as integral (2.20) directly calculated with theMonte-Carlo method according to (2.21).

Ψa (u)B a (u)W a (u)du= Mξ(Ψa(ξ)W a(ξ)) (5.14)That is to say, the calculation of the derivatives according to (5.14) is reduced

to the multiplying of the value written to the counter by a certain “weight”

function W a(ξ) (Marchuk et al 1980)

To construct the concrete algorithm of calculating W a(ξ) the derivativeexplicit form of the right part of series (5.10) is obtained For that we are usingthe known expression of the derivative of the product through the sum of

logarithm derivatives (xyz .)
=(xyz .)(x
|x + y
|y + z
|z + ) The following

Ψ(ξ) by special weight W a(ξ) during each writing to the counter In addition,

if valueΨ(ξ) depends on the current magnitude of random valueξonly, i e of

the current coordinates of the photon, then W a(ξ) is the sum and depends onthe whole history of its trajectory

Thus, to compute the derivatives of the irradiances, it is enough to ferentiate the explicit expressions of functionsΨa (u), q a (u) and K a (u, u
) withrespect to the retrieved parameters Then the following elementary changes areintroduced to the algorithm of irradiance calculations described in Sect 2.1:

dif-the counting of values W a(for entire set of parameters) at every modeling ofthe element of the photon trajectory with the writing to the special counters

of the derivatives simultaneously with writing to the counters of the ances Although the irradiances are calculated as integrals with respect to

irradi-wavelength (5.4), the irradi-wavelength remains the fixed one while modeling everysingle trajectory Hence, it is enough to consider the monochromatic case onlyduring the differentiation and the derivative of integral (5.4) will be obtainedautomatically It should be emphasized also that the optical thickness itself isthe function of differentiated parameters Thus, the atmospheric pressure is

to be used as a vertical coordinate, while computing the derivatives Nothingchanges in the real modeling but for the derivation of (2.8) the photon free path

probability from altitude level P1(in the pressure scale) to level P is written as:

whereα(P) is the extinction coefficient, then probability density (2.8)

trans-forms to the following:

Now apply the algorithm of the irradiance calculation, described in Sect 2.1

to the algorithm for the calculating of derivatives while taking into account theexplicit form of the functions in (5.16)

Counters W aare introduced for the whole set of parameters Starting every

trajectory of the counter W a:=0 is assumed While modeling every photon freepath, the following value is assigned to the counter while taking into account(5.17):

corre-W a:=W a+ 1

A

∂

Derivative from Values of Solar Irradiance 183The value is written to the counter at every step of modeling the photonscattering in the atmosphere according to (2.9):

where P is the coordinate of the counter.

The obtained algorithm is essentially simplified while taking into accountthat the following sum is calculated simultaneously at the point of the scattering

∂

∂ a[σm (P
)x m(χ) +σa (P
)x a (P
,χ)]

σm (P
)x m(χ) +σa (P
)x a (P
,χ) , (5.23)whereσm , x m,σa , x aare the volume coefficients and phase functions of themolecular and aerosol scattering In addition, remember that the phase func-tion of the molecular scattering determined by (1.25) does not depend onoptical parameters Finally, the only value is written to the counter in thealgorithm of the photon free path modeling:

of the retrieved parameters has been defined in the previous section There

are the vertical profile of air temperature T(P i), profiles of contents of four

gases absorbing radiation QH2O(P i ), QO3(P i ), QNO2(P i ), QNO3(P i) (The O2tent is constant), volume coefficients of the aerosol absorption and scattering

con-σa (P i,λj),κa (P i,λj ), and surface albedo A(λj) The concentrations of the spheric gases will be expressed through the volume-mixing ratio that gives thesimple relation for their counting concentrations:

of the expressions The other approach effectively simplifying the calculations

is application of the expression of the derivative of the product through thelogarithmic derivatives

As intermediate values in the grids P iandλjare computed with the linearinterpolation according to the following:

F(u)=F(u i)u i+1 − u

u i+1 − u i

+ F(u i+1) u − u i

u i+1 − u i

,

the derivative of function∂F(u)|∂F(u i) is obtained as the following process:

After determining number n from condition u n ≤ u ≤ u n+1 the followingequalities are correct:

As the derivative depends on the argument only, specify it as∂F(u)|∂F(u i)≡

L i (u) Then the derivative with respect to the surface albedo is written as:

∂

∂A(λj)(A)=L j(λ)

The photon free path∆τ
(P1P2), as per (2.1)–(2.4), is the quadratic function

of volume extinction coefficientα(P i) Hence the following algorithm is orated for computing derivative∂|∂α(P i)(∆τ
(P1, P2)), where inequity P2< P1

elab-is assumed for the definiteness:

Derivative from Values of Solar Irradiance 185

1 Finding numbers n1and n2from conditions P n1 ≥ P1 ≥ P n1 +1, P n2 ≥

(5.29)

Note that, the volume extinction coefficient in the described algorithm isapplied after recalculating per the pressure unit αP (P i), while it has been

calculated per the altitude unit initially After the differentiation of (5.1) weobtained:

in Chapter 1 and in the previous section are kept

Derivatives with respect to contents of the gases absorbing radiation

(exclud-ing the water vapor) Volume coefficient of the molecular absorptionκm (P i)depends on these contents and the volume extinction coefficient in its turndepends on the volume coefficient of the molecular absorption as per (1.24)

Then specify the concrete gas with subscript k and obtain:

Derivative with respect to water vapor content In addition to the volume

coefficient of the molecular absorption, the volume coefficient of the molecularscattering also depends on H2O content as per (1.27) It yields the followingexpression for the derivative of the free path:

Derivative from Values of Solar Irradiance 187and the following is obtained for the derivative with respect to the coefficient

of the molecular scattering:

The expression for the derivative of the molecular scattering volume cient is obtained as follows:

Derivative with respect to volume coefficient of the aerosol absorption The

volume extinction coefficient only depends on volume coefficient of the aerosolabsorption that directly yields:

Derivative with respect to volume coefficient of the aerosol scattering The

volume coefficients of the absorption and scattering and the phase function

of the aerosol scattering as per (5.9) depend on the volume coefficient of theaerosol scattering Therefore, we obtain:

∂σP,a (P
)

∂σz,a (P i,λj)=L i (P
) ∂σP,a (P i)

At last for the derivative of the phase function the following relations arecorrect:

where

D(P i,χ,λ)=b i(χ,λ) + 2c i(χ,λ) ln(σa,z (P i,λ))

The derivative with respect to air temperature A big quantity of values depends

on temperature Begin from the photon free path and obtain the following forit:

An important feature of calculating the derivatives with respect to temperature

is the necessity of accounting for the temperature dependence in the formula ofthe recalculation of the volume extinction coefficients in terms of atmosphericpressure (5.1) It is obtained as follows:

∂αP (P i)

∂T(P i) = αP (P i)

1

Derivative from Values of Solar Irradiance 189Finally, the problem is reduced to the differentiation of the volume coefficients

of the molecular scattering and absorption The first coefficient is equal to thesum of the coefficients of absorbing gases (all, including O2) by (1.22) Thecorresponding sum is inferred for the derivatives too Specifying the concrete

gas with subscript k, with accounting for (5.18) we get:

The absorption cross-sections of gases NO2, NO3, O3within the range 426–

848 nm don’t depend on temperature, hence, equality∂C a,k|(∂T(P i)) = 0 iscorrect The following is obtained from (5.7) for O3 within the range 330–

– The grid over wavelengths: from 325 to 685 nm with step 20 nm and from

725 to 985 nm with step 40 nm (28 points in a whole)

– The grid over altitude: from 1000 to 800 mbar with step 10 mbar, from

800 to 500 mbar with step 20 mbar, 500 to 110 mbar with step 30 mbar,

90 to 10 mbar with step 10 mbar and levels 5.2 and 0.5 mbar (61 points as

a whole)

The selection of the detailed grid in the lower atmospheric layers is caused bythe irradiance sounding levels and have been measured with a step of 100 mbar.Note that the top of the atmosphere corresponding to 0.5 mbar (about 55 km)

is in a good agreement with the altitude of the standard top atmospheric level,usually used in calculations of the radiative transfer in the shortwave region(Rozanov et al 1995; Kneizis et al 1996)

Consider briefly the specific features of the calculated derivatives of theirradiances and their magnitudes This analysis allows estimating the mech-anisms of the parameter influences on the measured characteristics of solarradiation and concluding the possibility of the retrieval of certain atmosphericparameters

Dependence of the upwelling irradiance upon the surface albedo is well studied (Kondratyev et al 1971, 1977) The inhomogeneous linear function (y=

ax + b) has been proposed for its description, where the multiplicative item is

the part of irradiance directly reflected from the surface proportional to albedo,and the additive item is connected with diffused radiation in the atmosphere.Correspondingly, the greater albedo is the stronger is the upwelling irradiancedependent on it The dependence of the surface albedo is also elucidated in thedownwelling irradiance (Sect 3.4) The corresponding derivative is greater,when the albedo is greater and the scattering in the atmosphere is stronger Itcould reach decimals of percent of the irradiance variation to one percent ofthe albedo variation as it follows from the calculations with the bright surfaceslike snow Thus, the influence of surface albedo on the downwelling irradiancecould exceed the uncertainty of the irradiance observation

Out of O2 and H2O absorption bands, the dependence of the irradiance upon temperature is extremely weak: it conserves close to the value of the

observational uncertainty even if the a priori variations of the temperature aremaximal The same is valid in the case of the ozone absorption bands Thus,the temperature dependence of the irradiances could be ignored out of theabsorption bands and the corresponding derivatives could be assumed equal

to zero At the same time, the temperature dependence is essential within the

O2 and H2O absorption bands including the weak bands also In addition,within some spectral regions, for example in wavelength 932 nm in the center

of the H2O band, it is strong and reaches the percent of the irradiance variation

to one-degree variation of the whole temperature profile

Derivative with respect to water vapor content are also essential only within

its absorption bands, hence the relationship between the volume coefficient ofthe molecular scattering and H2O content could be neglected These derivativesare maximal within the absorption band 910–980 nm, where the irradiancevariation reaches 40% to the a priori variations of the vertical profile of H2Ocontent as a whole

Derivative from Values of Solar Irradiance 191

The derivatives with respect to ozone content reach the maximum in the

stratospheric ozone layer Note that the selection of the upper boundary atlevel 0.5 mbar is determined with the influence of the stratospheric ozone

on the solar irradiance value because the influence of all other components(including the aerosols) is negligibly weak at the high altitudes The maximalirradiance variation at wavelength 330 nm is about 5% to the range of the

a priori ozone variations

The values of the derivatives with respect to N2O content are very low, and,

even with accounting for the possible wide interval of its a priori variations,the retrieval of N2O content is impossible This conclusion does not contradictthe results obtained in the previous section as we have used the extremely highvalues of the absorbing gases content there and have calculated the derivativesfor the averaged model (Rozanov et al 1995; Kneizis et al 1996) The analo-gous situation is arising for NO3gas, although the derivatives with respect to

N2O content within the absorption maximums (bands 524 and 662 nm) areessentially greater and allow principally obtaining certain information about

NO3contents with its high concentration

The derivatives with respect to volume coefficient of the aerosol scattering are

specified with the complicated vertical dependence The volume coefficient ofthe aerosol scattering influences to the solar irradiances owing to two contraryprocesses: the irradiances are decreasing with the aerosol optical thicknessgrowth and are increasing with the aerosol scattering growth Thus, the profiles

of the derivatives in question are sign-invertible: they have a positive maximumaround the observational point, which is decreasing with holding away fromthis point and then they transform to the negative ones Evidently, this obstacle

is connected with the local character of the scattering yield to the irradiances:

it is maximal around the point of the measurement The absolute value ofthe derivatives with respect to volume coefficient of the aerosol scattering

is quite high: the variations of the coefficient even in separate layers couldcause the irradiance variations up to 10% and higher The spectral behavior

of the derivatives in question is weakly expressed There is an approach forretrieval of the altitudinal dependence of the aerosol parameters from theremote measurements within the 760 nm oxygen absorption band (Badaev andMalkevitch 1978; Timofeyev et al 1995) Indeed, there is a certain differencebetween the vertical profiles shape of the derivatives within this band andout of it but it is rather weak that is also provided by the conclusion of thestudy (Timofeyev et al 1995) However, the vertical profile of the retrievedparameters is directly obtained from the airborne observations at differentlevels of the atmosphere

The derivatives with respect to volume coefficient of the aerosol absorption

greatly depend on the type of the selected aerosol model The values are greater

if the aerosol absorption is stronger It is the reason why the retrieval of theaerosol absorption volume coefficient from the data of the observations aboveLadoga Lake has turned out a difficult problem and it has been much morepossible for the observations above the desert The same conclusion is followedfrom the analysis of the irradiances accomplished in Sect 3.3

Results of the Retrieval of Parameters of the Atmosphere and the Surface

The inverse problem of the retrieval of atmospheric and surface parameters wassolved with the method described in Sect 4.3, i e with the method of statisticalregularization as per (4.53) (Vasilyev A and Ivlev 1999) Before discussing theretrieval results, we are pointing at the selection of the a priori and covariancematrices of the desired parameters necessary for the inverse problem solving.The corresponding a priori models of temperature, water vapor, and ozonewere taken from the book by Zuev and Komarov (1986) Two cases: “mid-latitudinal winter” for the observations above the ice and snow and “mid-latitudinal summer” for the observations above the water and sand surfaces.These models were completed with the data from the study by Anderson et al.(1996) to expand them to the top of the atmosphere (0.5 mbar) While com-pleting, the traditional exponential approximation was used for the covariancematrices (Biryulina 1981)

where X is the temperature or content of the atmospheric gas; z i , z j are the

altitudes, where the correlation is calculated, r is the correlation radius and

the only scalar parameter, which the standard altitude of 5 km was used for(Biryulina 1981)

The mean profiles of NO2and NO3were adopted from the text of TRAN computer code (Rozanov et al 1995; Vasilyev A et al 1998) The co-variance matrices were modeled according to (5.49), and the a priori SD wasassumed equal to 100%

GOME-The mean values and the covariance matrices of the albedo of sand, snow,and pure lake water were calculated directly from the observations of the spec-tral brightness coefficient, presented in Sect 3.4 In the approximation of theorthotropic surface, the albedo is equal to the spectral brightness coefficient.Construction of the a priori aerosol models is the most difficult problem,because there is no data about the variations and correlation links of the aerosolparameters in the cited literature in spite of the significant amount of opticalaerosol models In addition, the known models are not intended for applying

to the inverse problems solving and consists of not detailed enough grids overaltitudes and wavelengths Thus, the special aerosol models for the regions andseasons of the observations should be elaborated while taking into account thefeatures of the problem

While elaborating such models, in addition to the cited literature data, theresults of the direct airborne observations of the number concentration andchemical composition of the aerosol particles were used as well These observa-tions were accomplished by the team of the Laboratory for Aerosols Physics ofthe Atmospheric Physics Department of the Physical Institute of the LeningradUniversity above the Kara-Kum Desert and Ladoga Lake (Dmokhovsky et al.1972; Kondratyev and Ter-Markaryants 1976) The following approach, tradi-tional for the modern modeling of the optical properties of the aerosols was

Results of the Retrieval of Parameters of the Atmosphere and the Surface 193used there The aerosols microphysical parameters were specified and the totalset of the desired optical parameters (in our case the aerosol absorption andscattering volume coefficients and the phase function of the aerosol scattering

at the fixed altitude and spectral grids) were calculated with them As the lem of the modeling was obtaining the a priori statistical parameters of theaerosols, they were calculated by the variations of the microphysical parame-ters This methodology of modeling and the aerosol model itself are presented

prob-in detail prob-in the study by Vasilyev and Ivlev (2000)

The considered inverse problem of the retrieval of atmospheric optical rameters from the data of solar irradiance observations has no analogs incontemporary literature Thus, we aim the study at the principal possibil-ity of the retrieval of atmospheric parameters from the data of the irradiancemeasurements and also to the revealing of the methodological algorithm short-

pa-comings Therefore, we are presenting the analysis of all retrieved parameters

of the atmosphere including even the parameters whose obtaining from theobservational data is of no practical interest (the profiles of temperature andhumidity) Moreover, we are presenting some erroneous results, which are ofinterest from the point of elucidating methodological shortcomings of the al-gorithms The results of the retrieval of the aerosol parameters are certainly themost important ones, especially from the aspect of constructing and improv-ing the aerosol models of the atmosphere However, it should be emphasizedthat, if the number of the accomplished experiments with the processed results

is less than ten for every type of surface, it would not be enough for tical analysis of the results and for presenting them as models Nevertheless,

statis-it is possible to limstatis-it our consideration wstatis-ith the most typical results becausethey are the robust (statistically stable) estimation of the mean values of theaerosol parameters for constructing the aerosol models The obtained resultsare presented in Tables A.8–A.11 of Appendix A

Figure 5.4 illustrates the examples of retrieving the temperature verticalprofile The specific features of the profiles, particularly, the strong maximum

at the level 500 mbar, hardly correspond to the real altitudinal temperaturebehavior in the atmosphere, so they have been caused by the essential system-atic uncertainty during the retrieval of the temperature profile It is easy toexplain with the significant temperature dependence of the irradiance withinmolecular absorption bands In particular, it concerns oxygen narrow band

760 nm However, as has been mentioned in Chap 3, while describing the servations with the K-3 spectrometer the large systematic uncertainty couldappear within the oxygen band connected with the shift of the wavelengthscale owing to the mechanical scanning of the K-3 instrument Besides, theinstrumental function obtained from the measurements in the VD spectralregion can (moreover, from the properties of spectral instruments, it has to)show the relationship between its halfwidth and spectral region and can bewider in the NIR region Note that both specific features are clearly seen incomparison of the observations and calculations, illustrated by Fig 5.3 As theoxygen content is fixed, while solving the inverse problem, the temperatureprofile is the only parameter, which links with the absorption band shape and,which could be varied in the algorithm The systematic uncertainties of the

ob-Fig 5.4a,b Results of the retrieval of the vertical temperature profile: a from the data of the

airborne sounding 16th October 1983 above the Kara-Kum Desert, b from the data of the

airborne sounding 29th April 1985 above Ladoga Lake Dotted line indicates the a priori

in the center of the desert (200 km from Krasnovodsk) Therefore, it was cided not to use the data of the direct measurements of the temperature andhumidity profiles in the nearest points to the sites of the observations.Figure 5.5 illustrates the examples of the retrieved water vapor vertical pro-files As follows from the previous analysis of the derivatives, H2O absorptionbands together with the oxygen absorption bands are the only spectral regions,where the essential temperature dependence of the irradiance exists Thus, thesignificant uncertainties mentioned above could affect only H2O content pro-file However, as has been mentioned above, the retrieval of the temperatureand humidity is not of practical interest, and the pointed systematic uncer-

de-Results of the Retrieval of Parameters of the Atmosphere and the Surface 195

Fig 5.5a,b Results of the retrieval of the volume H2O content vertical profile: a from the data of the airborne sounding 16th October 1983 above the Kara-Kum Desert; b from the

data of the airborne sounding 29th April 1985 above Ladoga Lake Dotted line indicates the

The results of the ozone content retrieval are presented in Fig 5.6 It isseen that the retrieved profiles weakly differ from the a priori ones, though O3content above the desert is rather higher than the a priori content

As for the results of N2O and NO3 contents, their uncertainties are close

to the a priori ones, so it is better to discuss the correct accounting of the

a priori indefinites of their content assignments but not the results of thevertical profiles of these gases

Consider the most interesting components of the vector of the retrievedparameters, namely the optical parameters of the atmospheric aerosols Theexamples of retrieving the vertical profiles of the aerosol scattering and absorp-tion volume coefficients are presented in Figs 5.7–5.9 and in Tables A.8–A.11

of Appendix A Note that they are significantly lower than the a priori ones inthe lower troposphere that points out the necessity of correcting the a priorimodels to decrease the aerosol particles content in the corresponding altitu-dinal zones In this connection, the known effect of the strong dependence ofthe results upon zeroth approximation selection should be stressed (Zuev and

Fig 5.6a,b Results of the retrieval of the vertical volume ozone content profile: a from the

data of the airborne sounding 16th October 1983 above the Kara-Kum Desert; b from the

data of the airborne sounding 29th April 1985 above Ladoga Lake Dotted line indicates the

a priori profile

Naats 1990; Vasilyev O and Vasilyev A 1994) Thus, the retrieved results could

be changed after correcting the aerosol model

The systematic uncertainties of the instrument calibration strongly affectthe results of the vertical profiles of the coefficients in question (Vasilyev Aand Ivlev 1999) We illustrate this influence with the simplest example Let themeasured value of the downwelling irradiance at the level 500 mbar be system-atically underestimated only for 1–2% (Sect 3.3) The only way to adjust thedirect problem solution to this observational data is introducing the extinc-tion aerosol layer to the model at the altitude higher than 500 mbar Takinginto account small a priori aerosol content at these altitudes, the introducedaerosol layer must be sufficiently thick to cause the extinction of the down-welling irradiance to 1–2% Thus, even with the low systematic uncertainty inthe observed irradiances the algorithm of the inverse problem solving couldcause the false conclusion about the existence of the aerosol layers in the uppertroposphere and in the stratosphere Hence, the results of the retrieval of theaerosol scattering and absorption volume coefficients obtained in altitudinaldiapason of the airborne observations 500–950 mbar are much more reliable,because only the relative values of the solar irradiances are essential there Thecorresponding profiles are presented in Fig 5.8 The calibrating factor is likely

to be introduced to the vector of the parameters for retrieval though it is makethe retrieval accuracy worse