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A note on the justification of landsat data transformations
Pattern Recognition Vol. 17. No. 4, pp. 429 432, 1984.
0031 3203/84 ~;3.00+ .00 Pergamon Press Ltd. q 1984 Pattern Recognition Society
Printed in Great Britain.
A NOTE ON THE JUSTIFICATION OF LANDSAT DATA TRANSFORMATIONS J. D. TUBBS Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, U.S.A.
(Received 12 Auoust 1983; in reuisedform 4 October 1983; receivedfor publication 24 October 1983) Abstract--This paper discusses the mathematical justification for Landsat data transformations. The results are based upon an atmospheric radiation transfer equation given by Turner (5). Several proposed data transformations are discussed and evaluated in light of these justifications. Landsat multispectral scanner data Tasseled cap transformation
Radiation transfer m o d e l
1. I N T R O D U C T I O N
Researchers in the area of pattern recognition of remotely sensed agricultural regions using Landsat multispectral scanner data are often confronted with the problem of explaining empirical probabilities of misclassification (PMC) which are higher than their expected theoretical PMC. This is particularly true in signature extension, where the training signatures obtained in one region are used to classify crops in another region differing in either time or geographical location. The shortcomings of these signature extension procedures are often blamed upon: inappropriate models, either physical or statistical; violation of underlying assumptions; sensor limitations. Several papers have been written over the years addressing these shortcomings and proposing alternative procedures or models. For example, it has been shown that atmospheric conditions affect the signature extension classification results, hence, there have been several proposed atmospheric models and correction procedures. However, before many of these new procedures were defined, a commonly used step in these classification procedures involved the use of a data Sensor
Transformations
transformation. For example, the so-called tasselled cap transformation proposed by Kauth and Thomas m was used in LACIE and was an integral part of more extensive recognition procedures in the AgRISTARS project. Since many of these early data transformations have a way of reappearing in all recognition procedures, it seems worthwhile to evaluate data transformations in general and to develop the framework for any transformation, The purpose of this paper is to develop b.oth the mathematical and statistical justification for any Landsat data transformation. This formulation is based upon an atmospheric radiation transfer model presented by Turner. ~2~ 2. T U R N E R ' S R A D I A T I O N T R A N S F E R M O D E L
Turner 12) demonstrated that a general solution to the radiation-transfer equation is given by L = L o T + Lp
(2.1)
where L is the total spectral radiance at the sensor, Lo is the radiance at the target in the direction of the sensor, T is the transmittance between the target and sensor and Lp is the path radiance. This equation is greatly simplified for the purposes of this paper. However, the explicit dependence is as follows. L(2, t, b, ~b) = Lo(2, q~)S()., t) + Lp(2, t, b, ~),
(2.2)
where ). is the wavelength, t denotes the nature of the target, q~ contains the many physical and spectral parameters, such as optical thickness, nadar view angle, altitude of sensor, etc., S(2, t) is the actual signal transmitted by the target t in wavelength 2 and b represents the background to the target. The path radiance can be expressed as Lp(2, t, b, q~) = Lp.,(~,, q~) + Lp,(2, t, ~) Torget Fig. 1. R a d i a n c e c o m p o n e n t s
Background in a s c a t t e r i n g a t m o s p h e r e .
+ Lp~(2, b, 0). These components are illustrated in Fig. 1. 429
(2.3)
430
J.D. TuBns
/~'~
Puregross ~ 0
E=L
E=L
i
~9
I
Jz v~
o~
g ro
o
i
Wheot surrounded
E 7
E 8
re" 5
I
I
255
0 65
I
I
0 ?'5 0 85 Wovetength I/~rn)
I
0 95
Fig. 2. Total spectral radiance as a function of a variable wheat-grass background for a wheat target and a grass target. Visual range = 10 km, solar zenith angle = 31 °.
I
0 55
3. STATISTICAL IMPLICATIONS F r o m equation (2.2), it is not surprising that many signature extensions failed whenever certain physical parameters, such as atmospheric conditions, were ignored. For the sake of simplicity, assume that two targets, tl and t 2, are measured under identical atmospheric and sensor conditions. The question of interest is, what effect does (2.2) have upon statistical classification results? Rather than consider a specific classification procedure, consider the following general procedure: two targets are said to be from the same population if they are close in some sense. That is, t t and t2 are assigned to the same population if
p(s(tt), s(t2) ) < k
(3.1)
for some specified k, S(tl) and s(t2) are the actual spectral signal generated by targets tt and t2, respectively, and
p(s(t,), s(t2) ) = [ s ( t ~ ) - s(t2)]' W-'[s(t~) - s(t2)]
(3.2)
where W is a known matrix. Since s(t) is the actual signal rather than the observed signal, one would like (3.1) to be equivalent to
p(L(t 1), L(t2) ) < k*
(3.3)
whenever tl and t 2 are from the population and L(t) is the observed spectral signal for target t and k* a specified value. However, if model (2.2) holds, then
I
0 65
I
,
I
0 75 0 85 Wavel.ength (/~m )
I
0 95
Fig. 3. Variation of wheat radiance for a black, wheat and grass background. Visual range = 10 km, solar zenith angle = 31%
It is not clear that (3.1) and (3.3) are equivalent unless either Lp(t, b) = 0 for all t and backgrounds b or that Lp(tl, b) - Lp(t2, b) = 0 for all b, whenever tl is close to t 2. Indeed, if the background scatter effect is significant then one would not necessarily be able to group, say, wheat surrounded by grass with wheat surrounded by bare soil. This problem would be insurmountable. The purpose of the remainder of the paper is to determine the nature of a transformation A such that
ALp(t, b) = 0
(3.5)
for all values of t and b. If (3.5) holds, then
p(AL(tl), AL(t2) ) = p(ALos(tl), ALas(t2) ), (3.6) which would allow one to modify the value of k* to insure that (3.1) is equivalent to (3.3) using the transformed signal.
4. SOLUTION TO ALp(T, b) = 0 Let I denote a vector whose components are the value of Lp(t, b) evaluated at file wavelength 2. Since I is a vector, it can be shown that A = (I -
I + i)
(4.1)
satisfies (3.5) where I + is the Moore-Penrose inverse of L Furthermore, A can be written as
p(L(t,), L(t2) ) = L'op(s(t,), s(t2))L o
A = (I - I(1'I)-t I'),
+ (s(tl) - s(t2)) L'o W - l ( L p ( t l , b)
- Lp(t:, b)) + p(Lp(t I, b), Lp(t 2, b)). (3.4)
(4.2)
since I is a vector. However, unless I is specified, then A is unknown. In order to determine the effect that
Landsat data transformations background has upon the spectral radiance of a specified target, Turner (2~ simulated several examples where a target is surrounded by varying types of targets. Some of his results are seen in Figs. 2 and 3. In Fig. 2 the spectral radiance of wheat and grass is plotted against 2 for different locations in the field. Figure 3 presents the spectral wheat radiance with varying backgrounds. From Fig. 2 one observes that the spectral radiance level is a function of the target type and background. In comparing wheat alone vs all wheat in Fig. 3, one observes that the two curves almost differ by a constant for all 2. Hence, by assuming that 1 = cj, where c is a constant and j is a vector of ones, it follows that
431
to note that the background effect b is not contained in the r.h.s, of equation (5.2). Furthermore, if (5.2) holds, then it follows that 3
L(2, t, b) = ~ cj(t)2tj(2) + g(t),
(5.4)
j-I
where 9(t) is a function of the target t. By comparing (5.4) with (5.1) we have that 3
Lo(2)S(t, 2) + Lp(2, t, b) = ~ cj(t)21i(2) + 9(t). (5.5) j=l
By letting 3
Lo(2)S(2, t) = ~ q(t)lj(2), 3=t
l j,j),
(4.3)
A = (lk-- k
where k is the dimension of i (k = 4 in Landsat). It should be mentioned that A has many equivalent forms which can be found by modifying A under elementary row or column operations. I n particular, A could be written as
then a reasonable approximation to Lp(2, t, b) is given by g(t). Hence, one could assume that Lp(2, t, b) is not dependent upon either b or 2 and that A given by equation (4.3) is correct. It should also be mentioned that, since L(2, t, b) is not continuous in 2, then 0L(2, t, b) . L(2i' t, b)
OL 1
A=
-1
0
1
0
0
-
(4.4)
which is the transformation used by Engvall et al.. TM in defining their multitemporal "Delta classifier". In summary, if one assumes that the value of Lp(t, b, 2) is constant in 2 for t and all b, then A given by (4.3) or (4.4) satisfies (3.5). Hence, for constant atmospheric conditions, equation (3.3) is equivalent to (3.1). However, it may not hold that Lg(t, b) is constant over 2. In the next section this problem is addressed from another approach.
-
-
L ( 2 i + l , t,
b)
2i _ 2i+1
(5.6)
for i = 1, 2, 3. By letting 2~ = Landsat channel i, it follows that the derivative in (5.6) is equivalent to the transformation defined by A in (4.4). Indeed, the above observations and assumptions were the basis for Engvalrs "Delta classifier". It should be mentioned that (5.5) is reasonable whenever the atmospheric conditions are fixed. If these conditions vary or are unknown, as in the case of Engvall's analysis, the discrepancy in the observed spectral radiance was offset by knowing the additional multi-temporal patterns.
6. ADDITIONAL TRANSFORMATIONS 5. ANAI,YTICAL DETERMINATION OF Lp(2, t, b)
If one assumes that (2.2) holds and that Lp(,;, t, b)
= 9(0, then A given by (4.3) satisfies the desired constraint that AL~(2, t, b) = 0. The literature contains
From the previous sections we have that
L(2, t, b) = Lo(2)S(2, t) + Lp(2, t, b),
(5.1)
where the parameter q~ is fixed and the atmospheric conditions are constant. By considering the plots in Figs. 2 and 3 to be continuous in 2 and taking the derivative with respect to 2 it can be shown that c7L(2, t, b)
- -
02
3
- ~ cj(t) lj(2) j:t
(5.2)
where c~(t) are constants which are target dependent and I~(2) are indicator functions for 2, i.e. l j(2) =
1 if 2 is in intervalj 0 if not.
(5.3)
From these plots one observes that [cj(tl) - cj(t2) i is small whenever t~ and t2 are from the same population type. Hence the function ?L(2, t, b)/?2 could possibly be used to Separate different crop types. It is important
numerous data transformations which have been adapted to Landsat agricultural recognition. A wellknown and widely used transformation is the Kauth-Thomas tasselled-cap transformation. "~ Although its formulation was motivated by entirely different reasons than the guidelines suggested in this paper, it is easy to evaluate the procedure against the results in the previous sections. One version of the tasselled cap transformation is given by Lambeck and Rice. c4) [- 0.43257 I-0.28972 A=]-0.82943 1._ 0.22303
0.63248 -0.56199 0.52244 0.01170
0.58572 0.26414"-] 0.59953 0.49070] -0.03899 0.19386 I 0.54250 0.809821
(6.1) It is fairly obvious that cAj ~ 0 for any c ~ 0. Recently, Dave ~5}demonstrated that the characteristics of
432
J.D. TuBBS
the tasselled cap are significantly affected by the geometric, as well as the atmospheric, parameters. The author is not aware of any studies concerning this transformation's robustness to Lp(2, t, b), but one suspects that its characteristics would vary significantly.
remove the additive effect, hence, its characteristics could be affected by background scatter.
7. CONCLUSIONSAND SUMMARY
8. REFERENCES 1. R. J. Kauth and G. S. Thomas. The tasselled cap---a graphical description of spectral-temporal development of agricultural crops as seen by Landsat. LARS Sym-
The paper considers a general radiation transfer model proposed by Turnert2~ and seeks to determine the effect that path radiance would have upon recognition procedures. By assuming that the term Lp(2, t, b) = g(t), it was shown that this effect could be removed by a data transformation. Furthermore, the form of the transformation could be specified. The assumption that Lp(2, t, b) is independent of both 2 and b was derived from two results given by Turner,12~ however this assumption has not been studied sufficiently in order to determine its validity. It was also shown that the tasselled cap transformation did not
2. R.E. Turner, Atmosphericeffects in multispectral remote sensor data, ERIM Final NASA report, NASA CRERIM 10960(015-F(1975). 3. J. L. Engvall, J. D. Tubbs and Q. A, Holmes, Pattern recognition of Landsat data based upon temporal trend analysis. Remote Sensing Envir. 6, 303-314 (1977). 4. P.E. Lambeck and D. P. Rice, Signature extension using transformed cluster statistics and related techniques, ERIM Final NASAreport, NASACR-ERIM 109600-70F (1966). 5. J. V. Dave, Influence of illumination and viewing geometry ar~j atmospheric composition in the "tasselled cap" transformation of Landsat mss data. Remote Sensing Envir. 11, 37-55 (1981).
posium on Machine Processing of Remotely Sensed Data, Purdue University, Indiana (1976).
About the Author--J. D. TtJmts received the B.S. in Mathematics from Eastern New Mexico University in 1970and the Ph.D. in Statistics from Texas Tech Universityin 1974. He is currently an Associate Professor of Mathematics at the Universityof Arkansas. Before coming to the University,he spent two years (1974-1976) at the Johnson Space Center in Houston, Texas, under a National Research Council postdoctoral fellowship. It was during this period that he first became interested in pattern recognition and remote sensingunder the NASA Large Area Crop Inventory Experiment (LACIE).The author is currently the principal investigatorof a NASA contract investigating multivariate atmospheric variables and their impact upon the NASA space shuttle.
0031 3203/84 ~;3.00+ .00 Pergamon Press Ltd. q 1984 Pattern Recognition Society
Printed in Great Britain.
A NOTE ON THE JUSTIFICATION OF LANDSAT DATA TRANSFORMATIONS J. D. TUBBS Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, U.S.A.
(Received 12 Auoust 1983; in reuisedform 4 October 1983; receivedfor publication 24 October 1983) Abstract--This paper discusses the mathematical justification for Landsat data transformations. The results are based upon an atmospheric radiation transfer equation given by Turner (5). Several proposed data transformations are discussed and evaluated in light of these justifications. Landsat multispectral scanner data Tasseled cap transformation
Radiation transfer m o d e l
1. I N T R O D U C T I O N
Researchers in the area of pattern recognition of remotely sensed agricultural regions using Landsat multispectral scanner data are often confronted with the problem of explaining empirical probabilities of misclassification (PMC) which are higher than their expected theoretical PMC. This is particularly true in signature extension, where the training signatures obtained in one region are used to classify crops in another region differing in either time or geographical location. The shortcomings of these signature extension procedures are often blamed upon: inappropriate models, either physical or statistical; violation of underlying assumptions; sensor limitations. Several papers have been written over the years addressing these shortcomings and proposing alternative procedures or models. For example, it has been shown that atmospheric conditions affect the signature extension classification results, hence, there have been several proposed atmospheric models and correction procedures. However, before many of these new procedures were defined, a commonly used step in these classification procedures involved the use of a data Sensor
Transformations
transformation. For example, the so-called tasselled cap transformation proposed by Kauth and Thomas m was used in LACIE and was an integral part of more extensive recognition procedures in the AgRISTARS project. Since many of these early data transformations have a way of reappearing in all recognition procedures, it seems worthwhile to evaluate data transformations in general and to develop the framework for any transformation, The purpose of this paper is to develop b.oth the mathematical and statistical justification for any Landsat data transformation. This formulation is based upon an atmospheric radiation transfer model presented by Turner. ~2~ 2. T U R N E R ' S R A D I A T I O N T R A N S F E R M O D E L
Turner 12) demonstrated that a general solution to the radiation-transfer equation is given by L = L o T + Lp
(2.1)
where L is the total spectral radiance at the sensor, Lo is the radiance at the target in the direction of the sensor, T is the transmittance between the target and sensor and Lp is the path radiance. This equation is greatly simplified for the purposes of this paper. However, the explicit dependence is as follows. L(2, t, b, ~b) = Lo(2, q~)S()., t) + Lp(2, t, b, ~),
(2.2)
where ). is the wavelength, t denotes the nature of the target, q~ contains the many physical and spectral parameters, such as optical thickness, nadar view angle, altitude of sensor, etc., S(2, t) is the actual signal transmitted by the target t in wavelength 2 and b represents the background to the target. The path radiance can be expressed as Lp(2, t, b, q~) = Lp.,(~,, q~) + Lp,(2, t, ~) Torget Fig. 1. R a d i a n c e c o m p o n e n t s
Background in a s c a t t e r i n g a t m o s p h e r e .
+ Lp~(2, b, 0). These components are illustrated in Fig. 1. 429
(2.3)
430
J.D. TuBns
/~'~
Puregross ~ 0
E=L
E=L
i
~9
I
Jz v~
o~
g ro
o
i
Wheot surrounded
E 7
E 8
re" 5
I
I
255
0 65
I
I
0 ?'5 0 85 Wovetength I/~rn)
I
0 95
Fig. 2. Total spectral radiance as a function of a variable wheat-grass background for a wheat target and a grass target. Visual range = 10 km, solar zenith angle = 31 °.
I
0 55
3. STATISTICAL IMPLICATIONS F r o m equation (2.2), it is not surprising that many signature extensions failed whenever certain physical parameters, such as atmospheric conditions, were ignored. For the sake of simplicity, assume that two targets, tl and t 2, are measured under identical atmospheric and sensor conditions. The question of interest is, what effect does (2.2) have upon statistical classification results? Rather than consider a specific classification procedure, consider the following general procedure: two targets are said to be from the same population if they are close in some sense. That is, t t and t2 are assigned to the same population if
p(s(tt), s(t2) ) < k
(3.1)
for some specified k, S(tl) and s(t2) are the actual spectral signal generated by targets tt and t2, respectively, and
p(s(t,), s(t2) ) = [ s ( t ~ ) - s(t2)]' W-'[s(t~) - s(t2)]
(3.2)
where W is a known matrix. Since s(t) is the actual signal rather than the observed signal, one would like (3.1) to be equivalent to
p(L(t 1), L(t2) ) < k*
(3.3)
whenever tl and t 2 are from the population and L(t) is the observed spectral signal for target t and k* a specified value. However, if model (2.2) holds, then
I
0 65
I
,
I
0 75 0 85 Wavel.ength (/~m )
I
0 95
Fig. 3. Variation of wheat radiance for a black, wheat and grass background. Visual range = 10 km, solar zenith angle = 31%
It is not clear that (3.1) and (3.3) are equivalent unless either Lp(t, b) = 0 for all t and backgrounds b or that Lp(tl, b) - Lp(t2, b) = 0 for all b, whenever tl is close to t 2. Indeed, if the background scatter effect is significant then one would not necessarily be able to group, say, wheat surrounded by grass with wheat surrounded by bare soil. This problem would be insurmountable. The purpose of the remainder of the paper is to determine the nature of a transformation A such that
ALp(t, b) = 0
(3.5)
for all values of t and b. If (3.5) holds, then
p(AL(tl), AL(t2) ) = p(ALos(tl), ALas(t2) ), (3.6) which would allow one to modify the value of k* to insure that (3.1) is equivalent to (3.3) using the transformed signal.
4. SOLUTION TO ALp(T, b) = 0 Let I denote a vector whose components are the value of Lp(t, b) evaluated at file wavelength 2. Since I is a vector, it can be shown that A = (I -
I + i)
(4.1)
satisfies (3.5) where I + is the Moore-Penrose inverse of L Furthermore, A can be written as
p(L(t,), L(t2) ) = L'op(s(t,), s(t2))L o
A = (I - I(1'I)-t I'),
+ (s(tl) - s(t2)) L'o W - l ( L p ( t l , b)
- Lp(t:, b)) + p(Lp(t I, b), Lp(t 2, b)). (3.4)
(4.2)
since I is a vector. However, unless I is specified, then A is unknown. In order to determine the effect that
Landsat data transformations background has upon the spectral radiance of a specified target, Turner (2~ simulated several examples where a target is surrounded by varying types of targets. Some of his results are seen in Figs. 2 and 3. In Fig. 2 the spectral radiance of wheat and grass is plotted against 2 for different locations in the field. Figure 3 presents the spectral wheat radiance with varying backgrounds. From Fig. 2 one observes that the spectral radiance level is a function of the target type and background. In comparing wheat alone vs all wheat in Fig. 3, one observes that the two curves almost differ by a constant for all 2. Hence, by assuming that 1 = cj, where c is a constant and j is a vector of ones, it follows that
431
to note that the background effect b is not contained in the r.h.s, of equation (5.2). Furthermore, if (5.2) holds, then it follows that 3
L(2, t, b) = ~ cj(t)2tj(2) + g(t),
(5.4)
j-I
where 9(t) is a function of the target t. By comparing (5.4) with (5.1) we have that 3
Lo(2)S(t, 2) + Lp(2, t, b) = ~ cj(t)21i(2) + 9(t). (5.5) j=l
By letting 3
Lo(2)S(2, t) = ~ q(t)lj(2), 3=t
l j,j),
(4.3)
A = (lk-- k
where k is the dimension of i (k = 4 in Landsat). It should be mentioned that A has many equivalent forms which can be found by modifying A under elementary row or column operations. I n particular, A could be written as
then a reasonable approximation to Lp(2, t, b) is given by g(t). Hence, one could assume that Lp(2, t, b) is not dependent upon either b or 2 and that A given by equation (4.3) is correct. It should also be mentioned that, since L(2, t, b) is not continuous in 2, then 0L(2, t, b) . L(2i' t, b)
OL 1
A=
-1
0
1
0
0
-
(4.4)
which is the transformation used by Engvall et al.. TM in defining their multitemporal "Delta classifier". In summary, if one assumes that the value of Lp(t, b, 2) is constant in 2 for t and all b, then A given by (4.3) or (4.4) satisfies (3.5). Hence, for constant atmospheric conditions, equation (3.3) is equivalent to (3.1). However, it may not hold that Lg(t, b) is constant over 2. In the next section this problem is addressed from another approach.
-
-
L ( 2 i + l , t,
b)
2i _ 2i+1
(5.6)
for i = 1, 2, 3. By letting 2~ = Landsat channel i, it follows that the derivative in (5.6) is equivalent to the transformation defined by A in (4.4). Indeed, the above observations and assumptions were the basis for Engvalrs "Delta classifier". It should be mentioned that (5.5) is reasonable whenever the atmospheric conditions are fixed. If these conditions vary or are unknown, as in the case of Engvall's analysis, the discrepancy in the observed spectral radiance was offset by knowing the additional multi-temporal patterns.
6. ADDITIONAL TRANSFORMATIONS 5. ANAI,YTICAL DETERMINATION OF Lp(2, t, b)
If one assumes that (2.2) holds and that Lp(,;, t, b)
= 9(0, then A given by (4.3) satisfies the desired constraint that AL~(2, t, b) = 0. The literature contains
From the previous sections we have that
L(2, t, b) = Lo(2)S(2, t) + Lp(2, t, b),
(5.1)
where the parameter q~ is fixed and the atmospheric conditions are constant. By considering the plots in Figs. 2 and 3 to be continuous in 2 and taking the derivative with respect to 2 it can be shown that c7L(2, t, b)
- -
02
3
- ~ cj(t) lj(2) j:t
(5.2)
where c~(t) are constants which are target dependent and I~(2) are indicator functions for 2, i.e. l j(2) =
1 if 2 is in intervalj 0 if not.
(5.3)
From these plots one observes that [cj(tl) - cj(t2) i is small whenever t~ and t2 are from the same population type. Hence the function ?L(2, t, b)/?2 could possibly be used to Separate different crop types. It is important
numerous data transformations which have been adapted to Landsat agricultural recognition. A wellknown and widely used transformation is the Kauth-Thomas tasselled-cap transformation. "~ Although its formulation was motivated by entirely different reasons than the guidelines suggested in this paper, it is easy to evaluate the procedure against the results in the previous sections. One version of the tasselled cap transformation is given by Lambeck and Rice. c4) [- 0.43257 I-0.28972 A=]-0.82943 1._ 0.22303
0.63248 -0.56199 0.52244 0.01170
0.58572 0.26414"-] 0.59953 0.49070] -0.03899 0.19386 I 0.54250 0.809821
(6.1) It is fairly obvious that cAj ~ 0 for any c ~ 0. Recently, Dave ~5}demonstrated that the characteristics of
432
J.D. TuBBS
the tasselled cap are significantly affected by the geometric, as well as the atmospheric, parameters. The author is not aware of any studies concerning this transformation's robustness to Lp(2, t, b), but one suspects that its characteristics would vary significantly.
remove the additive effect, hence, its characteristics could be affected by background scatter.
7. CONCLUSIONSAND SUMMARY
8. REFERENCES 1. R. J. Kauth and G. S. Thomas. The tasselled cap---a graphical description of spectral-temporal development of agricultural crops as seen by Landsat. LARS Sym-
The paper considers a general radiation transfer model proposed by Turnert2~ and seeks to determine the effect that path radiance would have upon recognition procedures. By assuming that the term Lp(2, t, b) = g(t), it was shown that this effect could be removed by a data transformation. Furthermore, the form of the transformation could be specified. The assumption that Lp(2, t, b) is independent of both 2 and b was derived from two results given by Turner,12~ however this assumption has not been studied sufficiently in order to determine its validity. It was also shown that the tasselled cap transformation did not
2. R.E. Turner, Atmosphericeffects in multispectral remote sensor data, ERIM Final NASA report, NASA CRERIM 10960(015-F(1975). 3. J. L. Engvall, J. D. Tubbs and Q. A, Holmes, Pattern recognition of Landsat data based upon temporal trend analysis. Remote Sensing Envir. 6, 303-314 (1977). 4. P.E. Lambeck and D. P. Rice, Signature extension using transformed cluster statistics and related techniques, ERIM Final NASAreport, NASACR-ERIM 109600-70F (1966). 5. J. V. Dave, Influence of illumination and viewing geometry ar~j atmospheric composition in the "tasselled cap" transformation of Landsat mss data. Remote Sensing Envir. 11, 37-55 (1981).
posium on Machine Processing of Remotely Sensed Data, Purdue University, Indiana (1976).
About the Author--J. D. TtJmts received the B.S. in Mathematics from Eastern New Mexico University in 1970and the Ph.D. in Statistics from Texas Tech Universityin 1974. He is currently an Associate Professor of Mathematics at the Universityof Arkansas. Before coming to the University,he spent two years (1974-1976) at the Johnson Space Center in Houston, Texas, under a National Research Council postdoctoral fellowship. It was during this period that he first became interested in pattern recognition and remote sensingunder the NASA Large Area Crop Inventory Experiment (LACIE).The author is currently the principal investigatorof a NASA contract investigating multivariate atmospheric variables and their impact upon the NASA space shuttle.