Groundbased measurements

Ground-based measurements of characteristics of the ocean-atmosphere interaction are taken out using several types of carriers which serve as a platform for mounting measuring equipment such as commercial, fishing and research vessels, anchor and drifting buoys, fixed foundations (particularly, drilling platforms), pilot balloons, captive balloons, airships and planes. The most commonly used carriers are commercial and fishing ships. There were about 7690 in 1984. Thus, the total number of daily meteorological measurements carried out on these ships is 7690 x 0.3 x 4 = 9228, where the second figure on the left-hand side is the time fraction during which the ships are at sea and the third figure is the number of reports received each day. In practice, Bracknell Regional Centre of meteorological information, for example, receives about 3700 messages every day, that is, 40% of the expected number.

One could have accepted this loss of information if its other part had been distributed more or less uniformly over the ocean. Unfortunately, this is not the case, as is obvious from Table 1.2, which lists information on the percentage of 5-degree squares having more than 100 and those having more than 30 samples per month for the tropical zone of the ocean. As may be seen, the availability of observational data in the Atlantic, Indian and Pacific Oceans leaves much to be desired. Moreover, it turns out that these data are not absolutely independent and are not distributed at random in time and space (they cluster in the vicinity of islands and ship routes). Clearly, even a slight increase in density and/or improvement in the quality of measurements

Table 1.2. Percentage of 5-degree squares with > 100 and > 30 samples per month from the tropical zone of the Atlantic, Indian and Pacific Oceans (according to Taylor, 1985)









Sampling area







September, 1967







September, 1977







December, 1978







December, 1980







Note: values for December, 1980, include island and buoy reports.

Note: values for December, 1980, include island and buoy reports.

beyond these areas can result in noticeable changes in average values of the characteristics sought.

After these preliminary remarks we give a brief outline of some of the methods available for ground-based measurements of the parameters of the marine surface atmospheric layer, and their accuracy.

Sea surface temperature For sea surface temperature measurements, use is made of the non-insolated bucket, condenser or cooling system intake, and hull conduit or through hull sensors. Applications of these methods on voluntary observing ships from different countries (a percentage of the total number of ships for each national fleet) are presented in Table 1.3. This suggests that in the mid-1980s the most common method of sea surface temperature measurement was the cooling system intake method. But the bucket method, adopted at the dawn of the industrial epoch, is still widely used.

Both methods yield different temperature values: bucket values are lower by 0.3 °C than intake values. If one takes into account that a mass transition from the first to the second method occurred in the late 1930s, then this might be one cause, among others, of the global rise in sea surface temperature obtained from measurement data in the 1940s. But this is not all. A subsequent check has shown (see Taylor, 1985) that the difference between bucket and non-bucket temperature values varies in time and space. Particularly in a high heat flux situation (e.g. the Gulf Stream, Kuroshio or Agulhas areas in winter) this can mean a difference of 1 °C or more.

Now, as 50 years ago, there is no agreement as to which method is more efficient. Each method has its own disadvantages. The disadvantages of

Table 1.3. Percentage of ships using each method of temperature measurement for the six largest national fleets and for all voluntary observation ships (WMO, 1984)


Table 1.3. Percentage of ships using each method of temperature measurement for the six largest national fleets and for all voluntary observation ships (WMO, 1984)





























All ships




the bucket method are related to non-unified measurement techniques and the determination, strictly speaking, not of the sea surface temperature but of the temperature of some near-surface water layer of finite depth. The disadvantages of the non-bucket methods are similar. We note in this connection the strong dependence of temperature measurements on ship size, depth of intake and position of the intake thermometer.

Thus, we have to state that the error of ship sea surface temperature measurements is a fortiori more than +0.1 °C.

Air temperature and humidity Conventional ship measurements of air temperature use either thermometers mounted on screens or take dry-bulb measurements from a portable Psychrometer located on the top of the foremast or on the windward side of the top bridge, that is, from a position where the distorting influence of the ship's hull is minimized.

Screen values of air temperature may be biased high due to the radiative heating of the thermometer screen, or due to heat radiation of the hull. The error of such measurements, on average, is ±0.5 °C and not less than ±0.1 °C, according to Taylor (1985). Air temperatures from Psychrometers subjected to radiation and ventilation errors are necessarily higher than screen values. Moreover, the accuracy of psychrometric measurements depends critically on the location and method of exposure. In this respect well-sited screens are preferable since errors can be more readily assessed. Results of air temperature measurements obtained on USSR ships in GATE confirm this conclusion. As it turns out (see Taylor, 1985), when measurements are produced on the end of a 2.5 m boom extended from the bridge the mean difference between ship and profile buoy measurements is 0.04 + 0.12 °C.

Near-surface humidity is usually determined by data from dry- and wet-bulb psychrometry. It is clear that distortions created by radiative heating of the ship and screen should be taken into account in this case. But because an increase in air temperature and a decrease in relative humidity partly compensate each other, specific humidity is determined more or less accurately. Obviously, the most serious errors arise from contamination of the wet-bulb wick (say, by salt deposition) and from evaporation of rain or spray from a damp screen. Today it is a matter of common knowledge that ship measurements tend to overestimate air specific humidity and that the mean bias is 0.5 + 0.2 g/kg (see Taylor, 1985).

Surface wind velocity According to WMO data relating to 1984, the anemometric measurements of wind velocity are carried out on only one-third of all ships; on other ships wind velocity is estimated by data from visual observations of sea state. Both methods of wind velocity determination are not without problems. The second method in particular, based on the assumption that there is a one-to-one correspondence between wind force and sea state and also on the assumption that the wind equivalent connecting the Beaufort estimation and wind velocity is known in all cases, has at least two disadvantages: subjectivity and application limited to daytime. Errors of wind velocity determined by this method vary from ± 2 m/s when wind force is equal to one point on the Beaufort scale to ± 5 m/s when wind force is equal to five points on the Beaufort scale, according to Taylor (1983).

Problems arising from the first method are caused by the distorting influence of the ship's hull. It is usually assumed that the most representative site for anemometer mounting is the top of the foremast or the bow boom. Thorough analysis of the wind velocity field in the vicinity of the ship confirms this conclusion (see Figure 1.3). But even here distortions are still considerable. Moreover, it turns out that errors of anemometric measurements of wind velocity depend significantly on the ratio between the height of anemometer exposure and the height of the superstructures on the top bridge, as well as on the relative wind direction from the ship's course. Data presented in Table 1.4 bear witness to this. As can be seen, for bow winds the accuracy of anemometric measurements on top of a foremast varies from 1 to 8% of wind velocity. For other winds, measured wind velocities can differ from the reference value by 50% and more.

Table 1.4. Relative error (in %) of anemometric ship measurements of wind velocity

Relative wind direction

from 45 to 115°

Location of

Wind velocity

from —45 to 45°

and from 255 to

from 115 to 255°


bench measurements


(bow wind)

315° (beam wind)

(stern wind)

Augstein et ah (1974)

Meteorological buoy







from —6 to —21


Ching (1976)

Bow boom


from —3 to 5

from — 11 to 45


Kidwall and Seguin (1978)

Bow boom


from —2 to —6

from — 1 to 8

from -30 to -65

Large and Pond (1982)

Océanographie buoy





Romanov et al. (1983)

Beyond influence of


from 1 to —4

from —4 to 6

from —6 to 2

the domain of ship


Note: Positive values correspond to overestimation; negative values correspond to underestimation of wind velocity.

Note: Positive values correspond to overestimation; negative values correspond to underestimation of wind velocity.

height of the anemometer and the height of superstructures on the top bridge is equal to 1.9:1 for a case of bow wind direction.
Figure 1.4 Comparison of monthly mean values of wind velocity obtained from standard ship measurement data and measurements on research meteorological platforms, according to Taylor (1983).

The estimates presented above apply to practically instantaneous (averaged over a small time interval) values of wind velocities. It is interesting to determine by how much ship measurements differ from appropriate measurements from buoys and fixed platforms for monthly averaging periods. Results of comparisons are shown in Figure 1.4, from which it is obvious that ship measurements systematically overrate monthly averaged wind velocity. This fact, if confirmed by further research, may imply important and far-reaching climatic consequences.

To determine wind velocity (and temperature, and also humidity) from different heights in the atmosphere radiosonde, airship, airplane and similar measurements are used.

Net radiation flux at the ocean surface

The net radiation flux (radiation balance) at the underlying surface represents the difference between the solar radiation flux assimilated by the underlying surface and the effective emission of the underlying surface, defined as the difference between long-wave upward and downward thermal radiation fluxes.

Note that the Sun radiates energy as a black body (body in a state of thermodynamic equilibrium) at a temperature of 6000 K. Its radiation intensity is defined by the Planck formula Ix = (2hc2/X5)[exp(hc/kXT) — 1]~\ where h = 6.62 x 10"20Js is Planck's constant; k= 1.38 x 10~9J/K is Boltzmann's constant; c = 3 x 108 m/s is the velocity of light in a vacuum; X is the wavelength; T is the absolute temperature. From this formula it follows that Ix = 0 at 1 = 0 and X = oo. But since lk cannot take on negative values within the wavelength range from X = 0 to X = oo, at least one maximum of function JA has to exist. Differentiating /A over X and equating the resulting expression to zero we obtain the necessary condition of the extremum, that is, [(Xc/5kXT) + 1 - exp(hc/kXT)] = 0.

The above equation has one root at the point hc/kXt = 4.9651, from which XmT = const = 0.2897 cm K, where Xm is the wavelength conforming to the maximum of Ix. Finding the second derivative and determining its sign we assure ourselves that the function Ix has to have a maximum at the point indicated above. The equality Xm T = const expressing the Wien law demonstrates that a rise in temperature is accompanied by a shift in the maximum in the direction of shorter wavelengths. As is known, the temperature of the underlying surface and the atmosphere is much less than the temperature of the Sun. Therefore, solar radiation is concentrated in the short-wave (from 0.3 to 5 |im, that is, within the visible) spectral range, and emission of the underlying surface and the atmosphere is concentrated in the long-wave (from 4 to 100 nm) spectral range.

Actinometric measurements are carried out solely on research and ocean weather ships. It is clear that this information is not sufficient to reconstruct the spatial structure of the separate components of radiation balance, and hence there is no other option but to draw on indirect methods. Their practical realization is reduced to searching for empirical relationships associating the components of radiation balance with the parameters determining them (solar radiation at the upper atmospheric boundary, solar altitude, amount of total and low-level cloud, water and air temperature), and consequent reconstruction of information missing from standard ship measurement data. Comparison with independent measurements shows (see Strokina, 1989) that, at least for monthly average periods, the discrepancies between calculated and observed values of radiation balance components in tropical and temperate latitudes of the ocean amount to 2-4% in relative units.

Vertical eddy fluxes of momentum, heat and moisture at the ocean-atmosphere interface

Vertical eddy fluxes of momentum (t), heat (H), and moisture (E) are defined by equalities z = —pu'w', H = pcpw'T', and E — pw'q', where w' and u' are vertical and horizontal (oriented along the direction of surface wind) velocity fluctuations; 7" and q' are fluctuations of temperature and specific humidity; p and cp are the air density and specific heat at constant pressure; the overbar designates time averaging. To estimate eddy fluxes according to these definitions it is necessary, firstly, to measure fluctuations of wind velocity, temperature and humidity; secondly, to obtain instantaneous values of co-variations u'w', w'T', w'q'; and, thirdly, to average them within the limits of appropriately selected time intervals.

Measurements of velocity wind fluctuations are made using hot-wire anemometers, acoustic anemometers, anemometers with spherical meters responding to pressure change, cup anemometers, vane anemometers and flag anemometers registering wind load on the axes. Within this group of measuring equipment, acoustic anemometers based on the use of the linear dependence between time of sound propagation along the fixed section of a path guided in a certain direction and wind velocity in the same direction are the most popular. Acoustic anemometers differ favourably from other types of anemometer by the linear dependence of the output signal on wind velocity fluctuations, by the simplicity of detection of the necessary wind velocity component and by the small time lag. These advantages more than compensate for their disadvantages - averaging over the length of the base which is of the order of 10 cm.

Measurements of temperature fluctuations are effected by thermocouples, thermistors and resistance thermometers. All these sensors are characterized by high precision and small time lag. The first attempts to measure humidity fluctuations were undertaken by Dyer in 1961 (see Dyer, 1974). Dry-bulb and wet-bulb thermometers were used for this purpose. Over time, these were replaced by more sensitive optical hygrometers, the general principle of operation being based on the dependence of light absorption in a certain

(usually infra-red or ultra-violet) wavelength band on water vapour concentration in the air. Examples of such hygrometers are a hygrometer with an ultra-violet light source (so-called lightman-alpha hygrometer), a microwave refractometer-hygrometer, a gyrometer with a quartz oscillator sensor and an infra-red narrow band hygrometer.

The main requirements to be met by sensors designed to measure fluctuations in wind velocity, temperature and humidity, and their covariances are as follows. First, they should be high-sensitivity (low-inertia) sensors. Otherwise, determination of eddy fluxes will be attended by substantial errors. For example, the use of vane anemometers with a time lag of 0.3 s when measuring vertical velocity fluctuations at a height of four metres from the underlying surface results in an underestimation of eddy fluxes of momentum and heat by 20-30%. The same tendency, but for eddy fluxes of moisture, occurs when using dry-bulb and wet-bulb thermometers (see Kuharets et ai, 1980).

Second, the averaging period necessary to obtain statistically stable results should be sufficiently small, on the one hand, to exclude the influence of low-frequency oscillations of synoptic origin, and sufficiently large, on the other hand, to cover the whole spectrum of turbulent fluctuations wherever possible. According to modern concepts the turbulent fluctuations of wind velocity and humidity in the marine surface atmospheric layer have spatial scales ranging from fractions of a millimetre to tens, or even hundreds, of metres, and in the low-frequency range (range of large spatial scales) their spectra essentially differ from each other: temperature and humidity spectra, and especially horizontal velocity spectra, extend to lower frequencies than do vertical velocity spectra. This poses problems when selecting the spectral minimum range separating turbulent fluctuations from disturbances of the synoptic scale.

For vertical eddy fluxes of momentum, heat and moisture the case is simpler because the function a>Sww(co) (where Sww(a>) is the spectral density of the vertical velocity fluctuations; co is the frequency) drops to practically zero in the range of spatial scales of the order of several hundred metres in the surface boundary layer and several kilometres in the planetary boundary layer. Accordingly, cospectra coSww(co), a>SwT(co), coSwq(co), in the range of approximately the same spatial scales should also drop down to zero, so that covariances uV = J Suw(co) da>, w'T' = J SwT(a>) dco, w'q'= f Swq(co) dco make clear physical sense - these are integrals over the spectrum from scales of about 10 cm (base length of an instrument) to a low-frequency minimum of the function coSww(co). Thus, in order to provide reliable estimates of vertical eddy fluxes the frequency characteristics of measuring equipment have to be cut off (reduced to zero) in the range of the minimum of coSww(co). This condition is fulfilled if measurements cover the range of 10"3 < ojz/u < 5 or even 1CT3 < coz/u <10, where z is the height at which the measurement is taken; a> is frequency in Hz. It follows from this that at z = 5 m and u = 5 m/s the lowest frequency must be equal to 10 ~ 3 Hz, and the period of averaging must be not less than 15 minutes.

Finally, the reliability of the information is determined by the degree of mobility of the base on which the equipment is mounted. If the base is not stabilized, its movements will result in distortions of vertical fluctuations of the wind velocity and eddy fluxes on wave frequencies. These distortions can be eliminated by registering equipment movements and amending the data.

We consider the simple procedure to account for the oscillating motion effect proposed by Yolkov and Koprov (1974).

Let £ represent the vertical displacement of a device with respect to its average location and let w'0 = w' + u sin 6 + d£/dt, u'0 = u' 4- £ du/dz, T'0 = T + £ 8T/8z, e'0 = £ de/dz be fluctuations of wind velocity, temperature and absolute humidity distorted by pitching effects. Here u, w, T and e are average values of wind velocity, temperature and humidity; u', w', T' and e' are turbulent fluctuations of wind velocity, temperature and absolute humidity in the absence of equipment base motion (at £ = 0); the second term on the right-hand side of the first equality describes the effect of slope of the underlying surface; 6 is the angle of inclination (angle of pitching).

Let us compile products u'0w'0, w'0T'0, w'0e'0 and then average them over time. Then taking into account that £ and 6 do not correlate with u', w', T and e', and also that £ d£/8t = 0 and sin 6x0 we have u'w' = u'0w'0 — uOl; du/dz, wT = - uM 8T/dz,

To estimate the second terms on the right-hand sides of these expressions we assume that £ = 1 m, 6 = 1°, u = 10 m/s, 8T/8z = 0.01 °C/m, 8u/dz = 0.1 8e/8z = 0.005 g/m4. Substitution of these values yields u6t 8u/8z k 150 cm2/s2, u9£ 8T/8z % 0.20 °C cm/s, ud^ de/dz x, 0.75 x 10~6 g/cm2 s. Typical values of u'w', w'T' and w'e' are equal to 400cm2/s2, 1 °C cm/s and 3 x 10"6 g/cm2 s, respectively, so that the eddy flux error caused by pitching can be up to 30%. Thus, one more necessary condition for reliability of information on turbulence characteristics in the surface boundary layer is the precise allowance for displacements or stabilization of the height from which measurements are taken.

These requirements are quite restrictive. Perhaps this (plus the sophisticated nature and high cost of the measuring equipment) explains the fact that, nowadays, lack of data on eddy fluxes of momentum, heat and moisture is felt as much as it was 30 years ago when experimental research of ocean-atmosphere interaction was only just beginning.

1.4.2 Satellite measurements

In discussing satellite measurements of surface atmospheric layer characteristics, we make the reservation that equipment mounted on satellites registers not atmospheric layer characteristics themselves but, rather, different characteristics (such as intensity of thermal radiation or intensity of electromagnetic wave scattering) that are indirectly related to those in which we are interested. Moreover, the characteristics registered are functions of several parameters of the Earth's surface and the atmosphere. Naturally, this complicates the interpretation of satellite data, resulting in the need to take measurements in several spectral intervals in order to separate the effects of different parameters, and, mainly, to enable fall-back on calibration of satellite data by ground-based measurement data. It is clear that the precision of satellite measurements is limited by that of ground-based data and, hence, the advantage of satellite information is not its quality but, rather, its quantity, providing the possibility of examining ocean-atmosphere interaction on global scales.

We next consider the principal features of satellite measurements of the surface layer characteristics and their capacities.

Sea surface temperature As is well known, the spectrum of terrestrial emission represents the irregular alternation of bands of transmission (so-called transparency spectral windows) and absorption of thermal radiation. In the former, emission of the ocean surface, in its passage through the atmosphere, is attenuated to a minimal degree, and in the latter to a maximal degree. Thus, to indicate the temperature of the ocean surface it seems to be sufficient to measure the intensity of emission in one or another spectral window and then determine the temperature appropriate to the emission intensity as registered by a radiometer.

Let us examine, for example, the emission of a black body. According to Planck's formula (see p. 26), the intensity of emission of such a body is Ix = (2hc2/A5)[exp(hc/kAT) — 1]_1, from which it follows that for small wavelengths when Qxp( — hc/kXT) » 1 Planck's formula reduces to the Wienn formula Ix = (2hc2/X5) exp( — hc/kXT) describing an abrupt drop of Ix in the violet spectral range; on the other hand, for larger wavelengths, when Qxp(hc/kXT) « 1 + (hc/kXT), Planck's formula transforms into the Rayleigh-Jeans formula Ix = (2c/a4) k T describing a much slower drop of Ix in infra-red and radiowave spectral ranges. Thus, all other things being equal, use of a transparency window from the near-infra-red range (wavelengths ranging from 3.4 to 4.2 pm) would be preferable to those from the middle-infra-red range (wavelengths ranging from 8.1 to 14.0 pm), and especially from the microwave range1 (wavelengths ranging from 0.1 to 21.0 cm), if there were no effect from patches of sunlight, which results in an increase in the area of mirror reflection of solar radiation and, hence, the comparability of fluxes of emission and reflected solar radiation. Because of this the selection of a transparency window from the near-IR range means that the ability to receive noiseless information is restricted to night time, while for a transparency window from the medium-IR range there are no such restrictions.

The main disadvantages of radiometric measurements in the IR range (wavelengths ranging from 1 to 15 pm) is the fact that ocean emission is significantly attenuated by the influence of water vapour, aerosols and cloud when passing through the atmosphere. Their influence can be excluded if, instead of thermal emission measurements in the IR range, one takes measurements of radio thermal emission in the microwave range. Here, the absorption of microwave emission occurs only in clouds containing large raindrops. But things are not quite so simple in this case either because of the relatively low sensitivity of the emission intensity to variations in sea surface temperature and the strong dependence of emissivity on the local properties of waves and on the presence of froth, ice and surface-active substances.

Finally, one more source of uncertainty in radiometric measurements is the so-called skin-effect (temperature inversion in a thin near-surface ocean layer with a thickness of several millimetres). Note that data on radiometric measurements in the infra-red and microwave ranges describe radiative and radio-brightness temperature, respectively, defined as the temperature of the black body whose emission is equal to the actual emission. Because the latter refers to the layer with thickness of the order of a millimetre, inversion of temperature in this layer, created mainly by evaporation, should result in underestimation of the sea surface temperature compared with its value as determined by conventional methods in a layer with thickness from several tens of centimetres up to several metres. If we include the fact that the temperature difference in the near-surface inversion layer is not constant (see Table 1.5), then the possibility of establishing a universal dependence between radiative or radio-brightness temperature and sea surface temperature as measured by conventional methods is out of the question since this introduces additional errors.

Nowadays, the root-mean-square error of satellite measurements of the sea surface temperature is no less than 0.6 °C in the IR range, and 1.0 °C in the microwave range.

1 This range is defined according to the classification of radiowaves. Waves belonging to this range have much longer wavelengths than those from the range of long-wave emission.

Table 1.5. Temperature difference in the near-surface inversion layer



Source data





Paulson and Parker


Small fetch, no waves



Hill (1972)


Dynamic wind velocity less

than 35 cm/s


Dynamic wind velocity

more than 35 cm/s, waves

appear and develop

Katsaros etal. (1977)


No wind


Woodcock and


Nighttime, calm


Stommel (1947)

Ewing and McAlister


Nighttime, wind velocity


less than 0.5 m/s



Wind velocity less than


10 m/s, most probable


Klauss et al. (1970)





Bortkovskii et al.


Open ocean; wind velocity


less than 4 m/s


Wind velocity varies from

6 to 8 m/s

Grassel (1976)


Wind velocity varies from 1

to 10 m/s

Katsaros etal. (1977)

up to 3.4

Wind velocity less than

1 m/s, no waves

Schooley (1967)


Wind velocity about

2.5 m/s, cloudless sky


Same, but with clouds

Hunjua and


Wind velocity up to

Andreyev (1974)

6-8 m/s, waves up to

3 points

Paulson and


Wind velocity from 5.5 to

Simpson (1981)

9.2 m/s

Panin (1985)


Wind velocity less than

3 m/s

Air temperature and humidity The determination of temperature at any height in the atmosphere (including the surface boundary layer) reduces to registration of the Earth's radiation in different parts of the C02 absorption band (wavelengths 4.3 and 15 |im), that is, in the spectral interval where the intensity of emission is controlled mainly by the vertical distribution of temperature and concentration of C02, but not by humidity and aerosols. A primary prerequisite is indicated by the well-known physical fact that the emission at various parts of the C02 absorption band is formed in different atmospheric layers, and therefore for prescribed and invariable vertical distribution of C02 concentration the correspondence between the emission and the temperature of these layers must be single-valued.

In a cloudless sky the dependence of emission intensity Ix conforming to wavelength k on temperature T of the atmosphere should have the form (see Malkevich, 1973)

where 9 is the zenith angle; 5X is the emissivity of the underlying surface; BX(T) = (2hc/k3)[exp(hc/kT) — l]1 is Planck's function; Px(£, 9) is a function of radiation transmission (ratio of radiation passing through the layer with thickness £ to incoming radiation at the boundary of this layer) in a direction forming an angle 9 with the vertical; Px(£,) is the transmission function averaged over all 9 from the upper half-space of directions; £ = p/ps is a vertical variable in the isobaric coordinate system; p is the atmospheric pressure; c = 3 x 108 m/s is the velocity of light; h = 6.62 x 10"20 J s is Planck's constant; k = 1.38 x 10"9 J/K is Boltzmann's constant; subscript s indicates belonging to the underlying surface. The first term on the right-hand side of (1.4.1) describes the emission of the underlying surface, the second term describes the emission of the atmospheric layer contained between the underlying surface (£ = 1) and level & and the third term describes the thermal emission reflected from the underlying surface.

If we assume that the ocean to a first approximation is a perfect black body (<5a =1), and that measurements of Ix are carried out from a satellite in the nadir point (9 = 0, <5 = 0), then on the basis of (1.4.1) we have

This expression representing the Fredholm equation of first order with respect to T(0 allows determination of the vertical profile of temperature if the values of radiation intensity Ix and transmission function Px are given. The same is possible in principle if the angular distribution of Ix(9) is known. But in

9) = dxBx(Ts)Px(l, 9) - Bx\_Tm f1 bp xpx(i,9) B.inm-^dz,

reality this will entail large errors due to horizontal non-uniformity of the temperature field and the presence of cloudiness.

Thus, we realize the necessity of solving the inverse problem of radiation theory - to determine the properties of the environment from data on its emission. This problem is ill-posed in the sense that small errors in signal measurements can entail large errors in estimating environmental parameters.

Let us illustrate this by a simple example. Let us assume that a solution x of the equation f(x) — a has to be found. It is obvious that x = f~1(a), where /_1 is the inverse function of f Using the theorem on derivatives of inverse functions we have dx/x = (p(a) da/a. This expression sets the relationship between errors at the input (da/a) and at the output (dx/x) of the system. Here <p(a) = a/[f~ l(a)f'(a)] is an error amplification coefficient. Based on the preceding expression for dx/x we conclude that output errors will be large at large (p{a) or, otherwise, at small f'(a), that is, at those values of a which are close to the roots of the derivative f'(x).

To achieve acceptable accuracy it is necessary to either minimize errors at the input da/a, or, if this is impossible, to regularize the problem. The latter means that the function /(x) is replaced by a closer function ft(x) such that, in the range of interest, the new error amplification coefficient (p^a) is not beyond the limits of a certain prescribed value.

Thus, to obtain the solution of an ill-posed problem it is necessary to have a priori information about the solution which allows exclusion of large errors in the experimental data. These problems are doubled by the fact that, as applied to the atmosphere, the required and measured parameters are random functions, and so in searching for a solution that is appropriate to an integral equation of type (1.4.2), one has to apply statistical regularization with the statistical characteristics of the radiative field of the Earth and fields of atmospheric parameters serving as a priori information. Experience in its use indicates that the standard deviation between vertical temperature distributions in the lower 24-kilometre layer of the atmosphere reconstructed from data of satellite and radiosonde measurements is usually about 1.5 °C.

In contrast to the vertical profile of temperature, the vertical profile of humidity is determined by measurements of emission /A at different parts of the water vapour absorption band (wavelengths 6.3 and 20-25 |im). In this case, all other things being equal, Equation (1.4.2) is rewritten as where wA(£) = (pjg) jo-1"' df is the moisture content of the atmospheric column between the upper atmospheric boundary (^ = 0) and level 4 WA is the integral moisture content of the atmosphere; q is the air specific humidity.

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