220
C. Weitkamp
for a measurement of the local extinction coefficient 0'0. Unfortunately, extinction coefficients
determined from the regression formula (10.6) depend critically on the choice of 0'0; in fact,
the omnipresent noise on the measured signals P is sufficient to get the denominator on the
right-hand side of Eq. (10.6) so close to zero that absurd or (meaningless) negative absorption
coefficients occur, unless unrealistically large values of the constants F or K, are used. This
procedure usually called forward integration because Eq. (10.6) is solved by beginning at the
near end of the range is therefore applicable to special cases only. Now the boundary values
(10.8) can be defined at any point within the !idar range, including the most remote point Xm
with
(10.9)
then entering into Eq. (10.6). This small modification turns the minus sign into a plus sign and
thus removes the singularity of Eq. (10.6). This backward integration is today the procedure
of choice. It is carried out in such a way that the !idar range is chosen long enough to reach
into a region where aerosol extinction is negligible relative to molecular extinction. Molecular
extinction is easily calculated from Rayleigh theory and is given by
8lTsr
amol(x) = -3-/3mol(lT, >'),
(10.10)
with the wavelength-dependent 180 0 molecular scattering coefficient (Ref. 3)
( 550nm) 4 -6 -1 -1
/3mol( IT, >.) = 1,47 ->.- 10 m sr .
(10.11 )
The parameter K, in Eq. (10.5) is for all practical purposes set equal to 1, and from empirical
data a value for the ratio 0'//3 is chosen so that backward integration can be carried out. In
doing so, care must be taken to avoid numerous pitfalls.
For one thing, backward integration has a tendency to flatten the data and to "iron out"
structures clearly visible in forward integration and perceptible in the atmosphere with the
naked eye (Kunz, 1987). Second, independent measurements of a and /3 (Ansmann et al.,
1991) show that the ratio is not spatially constant even in one type of cloud. As can be seen
from Fig. 10.2, lidar ratios of about 2 Sf prevail in the upper part of a cirrus above 8 km,
whereas values around 12 sr are measured in the lower parts between 7 and 8 km. Thirdly,
lidar ratios in clouds such as cirrus change with time. In six consecutive measurements of 2.5
minutes duration each, only the second, third, and fourth measurement of a cirrus between 9.3
and 11.7 km height were correctly represented by an average lidar ratio of 13 sr, whereas for
measurement no. 1 a value of 16 and for measurement no. 5 a value of 20 sr is more appropriate;
for measurement no. 6 the lidar ratio is even larger and was not determined (Fig. 10.3).
A convenient way for the determination of the average Ii dar ratio of a cloud or other mass of
aerosol is the consecutive use of forward and backward integration and variation of the lidar
ratio until both results agree. To avoid the problems associated with forward integration the
procedure should be started with low 0'//3 ratios. Figure 10.4 shows a sequence of a profiles
generated in this way. However, and fourthly, the results obtained must not be mistaken as
the real profile of the atmospheric values of a. Figure 1O.5a shows a case in which the Klett
forward and backward profiles agree quite well with the result of an independent measurement
of a and /3. In Fig. 10.5b this is not the case: the independently measured profile (dotted line)
differs markedly from the Klett result, although the forward (solid line) and backward result
(dashed line) agree so well that they can hardly be distinguished. It must be kept in mind that
the same measurement data were used for the generation of the three curves of Fig. 10.5b, with
an additional measurement of /3 available for the dotted profile. In summary, forward-backward
integration appear to be a viable means of determination of the average lidar ratio provided he
following conditions are met:
C. Weitkamp
for a measurement of the local extinction coefficient 0'0. Unfortunately, extinction coefficients
determined from the regression formula (10.6) depend critically on the choice of 0'0; in fact,
the omnipresent noise on the measured signals P is sufficient to get the denominator on the
right-hand side of Eq. (10.6) so close to zero that absurd or (meaningless) negative absorption
coefficients occur, unless unrealistically large values of the constants F or K, are used. This
procedure usually called forward integration because Eq. (10.6) is solved by beginning at the
near end of the range is therefore applicable to special cases only. Now the boundary values
(10.8) can be defined at any point within the !idar range, including the most remote point Xm
with
(10.9)
then entering into Eq. (10.6). This small modification turns the minus sign into a plus sign and
thus removes the singularity of Eq. (10.6). This backward integration is today the procedure
of choice. It is carried out in such a way that the !idar range is chosen long enough to reach
into a region where aerosol extinction is negligible relative to molecular extinction. Molecular
extinction is easily calculated from Rayleigh theory and is given by
8lTsr
amol(x) = -3-/3mol(lT, >'),
(10.10)
with the wavelength-dependent 180 0 molecular scattering coefficient (Ref. 3)
( 550nm) 4 -6 -1 -1
/3mol( IT, >.) = 1,47 ->.- 10 m sr .
(10.11 )
The parameter K, in Eq. (10.5) is for all practical purposes set equal to 1, and from empirical
data a value for the ratio 0'//3 is chosen so that backward integration can be carried out. In
doing so, care must be taken to avoid numerous pitfalls.
For one thing, backward integration has a tendency to flatten the data and to "iron out"
structures clearly visible in forward integration and perceptible in the atmosphere with the
naked eye (Kunz, 1987). Second, independent measurements of a and /3 (Ansmann et al.,
1991) show that the ratio is not spatially constant even in one type of cloud. As can be seen
from Fig. 10.2, lidar ratios of about 2 Sf prevail in the upper part of a cirrus above 8 km,
whereas values around 12 sr are measured in the lower parts between 7 and 8 km. Thirdly,
lidar ratios in clouds such as cirrus change with time. In six consecutive measurements of 2.5
minutes duration each, only the second, third, and fourth measurement of a cirrus between 9.3
and 11.7 km height were correctly represented by an average lidar ratio of 13 sr, whereas for
measurement no. 1 a value of 16 and for measurement no. 5 a value of 20 sr is more appropriate;
for measurement no. 6 the lidar ratio is even larger and was not determined (Fig. 10.3).
A convenient way for the determination of the average Ii dar ratio of a cloud or other mass of
aerosol is the consecutive use of forward and backward integration and variation of the lidar
ratio until both results agree. To avoid the problems associated with forward integration the
procedure should be started with low 0'//3 ratios. Figure 10.4 shows a sequence of a profiles
generated in this way. However, and fourthly, the results obtained must not be mistaken as
the real profile of the atmospheric values of a. Figure 1O.5a shows a case in which the Klett
forward and backward profiles agree quite well with the result of an independent measurement
of a and /3. In Fig. 10.5b this is not the case: the independently measured profile (dotted line)
differs markedly from the Klett result, although the forward (solid line) and backward result
(dashed line) agree so well that they can hardly be distinguished. It must be kept in mind that
the same measurement data were used for the generation of the three curves of Fig. 10.5b, with
an additional measurement of /3 available for the dotted profile. In summary, forward-backward
integration appear to be a viable means of determination of the average lidar ratio provided he
following conditions are met: