Remote sensing of row crop structure and component temperatures using directional radiometric temperatures and inversion techniques

REMOTE SENSING OF ENVIRONMENT 13:33-55 (1983)

33

Remote Sensing of Row Crop Structure and Component Temperatures Using Directional Radiometric Temperatures and Inversion Techniques

D. S. KIMES Earth Resources Branch, National Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt, Maryland 20771

A physically based sensor response model of a row crop was used as the mathematical framework from which several inversion strategies were tested for extracting row structure information and component temperatures using a series of sensor view angles. The technique was evaluated on ground-based radiometric thermal infrared data of a cotton row crop that covered 48% of the ground in the vertical projection. The results showed that the accuracies of the predicted row heights and widths, vegetation temperatures, and soil temperatures of the cotton row crop were on the order of 5 cm ( _ 10% of mean values), 1°, and 2°C, respectively. The inversion techniques can be applied to directional sensor data from aircraft platforms and even space platforms if the effects of atmospheric absorption and emission can be corrected. In theory, such inversion techniques can be applied to a wide variety of vegetation types and thus can have significant implications for remote sensing research and applications in disciplines that deal with incomplete vegetation canopies.

Introduction Most remote sensing applications in natural resource disciplines deal with vegetation or with substrate materials covered by vegetation. A mixture of vegetation and substrate components within the field of view of a sensor complicates accurate scene interpretation. Wide variability in mixtures of natural scenes occurs, causing wide variability in spectral respouse and difficulty in making accurate inferences, Agricultural row crops are an extreme example of composite scenes. Incomplete row crop canopies are important remote sensing targets because of the need to identify crops, acreage, growth stage, and crop conditions early in the growing season, to schedule irrigations, and to predict final crop yields (Jackson et al., 1979). There have been a number of studies ©Elsevier Science Publishing Co., Inc., 1983 52 Vanderbilt Ave., New York, NY 10017

concerning the use of effective radiant temperatures of components and strucrural information of the crop along with other variables to infer these crop parameters and conditions (Byrne et al., 1979; Jackson et al., 1979). Remote sensing techniques which uniquely infer component temperatures and scene structure from composite scenes need to be developed. A new technique for obtaining additional knowledge about the components and structure of composite scenes utilizes multiple view angles in conjunction with physically based, sensor response models as demonstrated by Kimes (1981) on incomplete wheat canopies. In this study a geometric projection model for a row crop was used as the mathematical model from which several inversion strategies were tested on field data for extracting component temperatures (sunlit and shaded 00344257/83/010033-23503.00

34

vegetation and soil) and row structure (mean height, width, and spacing) from a series of sensor view angles. The field data were collected on an irrigated cotton row crop in Phoenix, Arizona. This technique has significant implications for natural resource disciplines which must deal with composite scenes composed of vegetation and substrate. The technique may be applied to sensor data from aircraft or satellite sensors with scanning or off-nadir capabilities, Theory The remote sensing problem is to infer scene attributes, given measured response(s) of the scene. The classic quantitative approach to this problem is to develop a physically based, deterministic, mathematical model which relates the sensor response to underlying scene attributes. Inversion techniques can then be applied to this system to infer underlying scene attributes using remote measurements. In many cases the classic approach is not directly applicable because too many unknown variables are involved, or unacceptable solutions are obtained due to experimental errors, or simply a readily invertible model cannot be conceived, The row crop model of Jackson et al. (1979) was modified, validated, and verified by Kimes and Kirchner (1982) and was used in this study. The model abstracts the rows as extended rectangular solids which have no gaps, and calculates the proportions of projected surface area of four surfaces (sunlit and shaded vegetation and soil) which are in direct line of sight of a particular view direction (Fig. 1). The projected proportions are a function of view angle, row spacing, row

D.S. KIMES

height, row width, row orientation, and solar zenith and azimuth angles. These four proportions (the sum of which equal 1.0) are then weighted by the respecrive component temperatures in degrees Kelvin to the fourth power. The fourth root of the sum of these four terms is equal to the sensor response (effective radiant temperature) in degrees Kelvin. The modification is a simplification of Jackson's model in that angular effects within the field of view were not considered. This simplification is advantageous in that the sensor response is dependent on the fewest number of variables. Thus, direct inversion techniques which infer unique component temperatures a n d / o r row structure with a minimum of a priori knowledge or assumptions will be most feasible in remote sensing missions. In addition, the model was expanded to treat any sensor orientation rather than only sensor angles in the plane normal to the rows. The algorithm used to calculate the proportions of sunlit and shaded vegetation and soil is presented in the Appendix. Inversion schemes for inferring component temperatures and row structure parameters were as follows. First, a system of equations was developed for inferring component temperatures with a priori knowledge of row structure. The system of linear equations used was similar to that used by Kimes (1981) on wheat canopies: A[ = g +

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ROW CROP STRUCTURES

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FIGURE I. Abstraction o[ row crop model where rows are extended rectangular solids (side view shown). H is the row height, W is the row width, S is the row spacing, ab and bc are the sunlit vegetation surfaces, cd is the shaded vegetation surface, de is the shaded soil surface, efis the sunlit soil surface and 0 is termed the solar row angle. Any solar zenith and azimuth orientation can occur. Three view directions are shown with dashed lines representing the projection o[ these surfaces in each view direction. The projected surfaces 1, 2, 3, and 4 are sunlit vegetation, shaded vegetation, sunlit soil, and shaded soil, respectively. The proportions o[ the [our projected surfaces for each view direction are normalized so that they sum to 1.0. View directions 1, 2, and 3 are denoted as left, nadir, and right o[ the rows, respectively.

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where s,r~4 is the effective radiant temperature in degrees Kelvin to the fourth power of a sensor viewing the row crop in direction i (i = 1, n where n is the num-

ber of sensor view directions) with zenith and azimuth view angles a and Y, respectively, t~ is the mean effective radiant temperature in degrees Kelvin to the fourth power of the jth component of the canopy ( j = 1, m where m is the number of canopy components, e.g., for m = 4 the four components may be sunlit and shaded soft and vegetation), and ~ is the error vector. The elements of the A matrix (pq) characterize the row crop structure and each element is the relative proportion of canopy component (j) that presents itself to direct view to a sensor with view angle aYi (Fig. 1). These proportions are further discussed by Kimes and Kirchner (1982)and are calculated as a fimction of row height and width, row orientation, solar azimuth and zenith an-

36

D.S. KIMES

gle, and sensor azimuth and zenith view angles as presented in the Appendix. Two linear inversion schemes were used to solve for /. Exact systems were used where n is equal to m (n sensor view angles and m canopy components). The solution ignoring the unknown error term is i = A-lg. In addition, overdetermined systems were used where n is greater than m (more equations than unknowns) by using a least-squares approach. A form of the solution is t = (ArA)-IArg. Finally, a system of equations (F) was developed for inferring both row structure parameters and component temperatures simultaneously. A system of n nonlinear equations with n unknowns was constructed for row crops with a specific azimuthal row orientation. Each equation represents a particular view angle and the unknowns are the row structure parameters and component temperatures (total unknowns equal n). The system is

F = ,

P ~I(H, W, S ) t4 + p 12(H, W, S ) t4 H, W, S )t 4 - s~14 P21(H, W, S)t 4 + p ~ ( H , W, S )t 4 • " " Plm(

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where pil(H,W, S) is the relative proportion of canopy component j that has a direct view to a sensor with view direction aT~ and is a nonlinear flmction of row height (H), width (W), and spacing (S) as described in the Appendix.

Experiment The experiment was conducted at Phoenix, Arizona, during the period of 22-25 June 1981. The observations reported here were made on a homogeneous plot of irrigated cotton with eastwest row orientation. Cloudless weather prevailed during all measurement periods. Other supporting environmental data and the structural characteristics of the canopy were reported in detail by Kimes and Kirchner (1982). The mean measurements pertinent to this study are row height 44.0 cm, row width 48.1 cm, row spacing 100 cm, and 48% plant cover. The effective radiant temperatures of the canopy were obtained through the use of a thermal infrared radiometer which has a 15 ° field of view, a stated accuracy of +0.5°C, a resolution of +0.1°C, and a spectral band width of 8-14/~m. Radiometric measurements from above the canopy were made on a 5.2-m platform. Five zenith view angles were sampled (0 °, 20 °, 40 °, 60 °, and 80 °) within the plane normal to the rows and with a south-looking orientation. The samples for each view angle were taken systematically by advancing the sensor's horizontal position approximately 10 cm within the plane normal to the rows for each sample. The four component temperahtres were sampled on the ground. Sunlit vegetation was sampled by orienting the view direction normal to and at the midpoint of both surfaces a---band b---c(Fig. 1). Shaded vegetation was measured with the view direction normal to and at the midpoint of surface cd. Sunlit and shaded soil were sampled to include in the field of view only sunlit and shaded surfaces of ef and de, respectively. Measurements were taken at hourly intervals beginning and ending at 0800-

ROW CROP STRUCTURES

37

1700 MST on 23 June 1981, and 0600-1300 MST on 25 June 1981, for a total of 18 measurement periods. The radiometric and environmental measurements for each period were recorded within a 20-min span. The sample size, mean, and variance of all radiometric data were reported by Kimes and Kirchner (1982). The above data set of the 18 measurement periods served as validation data for testing various inversion schemes. The first class of inversion schemes tested assumes a priori knowledge of the structure of the row crop. Several cases were solved, Ten cases of an exact 2 × 2 system (n view angles and m canopy component temperatures) using all possible pairs of zenith view angles, e.g., 00/20 ° , 00/40 ° ..... 200/40 ° , etc., were tested. In addition, an exact 3 x 3 and 4 × 4 system using zenith view angles of 00/200/40 ° and 0 0 / 2 0 0 / 4 0 0 / 6 0 °, respectively, were tested. Finally, several overdetermined systems were tested: 3 × 2, 4 x 2, and 5 × 2 systems using zenith view angles of 0 0 / 2 0 0 / 3 0 °, 0 0 / 2 0 0 / 4 0 0 / 6 0 ° and 0 0 / 2 0 0 / 4 0 0 / 6 0 0 / 8 0 °, respectively; 4 × 3 and 5 × 3 systems using zenith view angles of 0 0 / 2 0 0 / 4 0 0 / 6 0 ° and 0 0 / 2 0 0 / 4 0 0 / 6 0 0 / 8 0 ° , respectively; and a 5 × 4 system using zenith view angles of 00/200/400/60o/80

°.

The solutions to the exact and overdetermined systems were obtained by the respective LEQT2F and LLSQF numerical algorithms from the International Mathematical and Statistical Library (1979). The solution of these systems were compared to the measured component radiant temperatures as follows. For the systems with two canopy components these predicted component temperatures were compared with the measured radiant temperatures of sunlit vegetation and

mean soil. The mean soil temperature was calculated as the mean of sunlit and shaded soft temperatures weighted by their respective ground surface areas, since these latter temperatures are most important to the crop status and they are viewed by a nadir-looking sensor. For the systems with three canopy components, the predicted component temperatures were compared to the measured radiant temperatures of sunlit vegetation and sunlit and shaded soil. Finally, the predicted component temperatures of the four component systems were compared to the radiant temperatures of sunlit and shaded vegetation and soil. In all cases the statistic used for accuracy of prediction was the root-mean-square (rms)of deviations between the predicted and measured radiant temperatures for all 18 measurement periods. The second class of inversion schemes assumed no knowledge about the geometric struchlre of the row crop. Only the simple 3 × 3 systems it) this class of inversion schemes were found to be stable. A 3 × 3 system was constructed using zenith view angles of 00/200/40 ° and three unknowns: vegetation and soil temperahtres, and a relative row width-height parameter. This parameter was obtained by setting relative width equal to relative height. The relative width and height were treated as W / S and H / S , respectively, where S is the row spacing and is set equal to 1.0 unit. These relative structural parameters are adequate for most applications. In real-time applications this simple equality can be replaced by more realistic growth equations which relate row width to row height. Other 3 × 3 systems were tested using zenith view angles of 0 0 / 2 0 0 / 6 0 ° , 00/400/60 ° , and 0 0 / 4 0 0 / 8 0 ° . These systems were applied to those measurement periods where the

38

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ZENITH V I E W ANGLE F I G U R E 2. The root-mean-square (rms) of deviations between the measured sensor response from above the canopy and the measured sunlit vegetation temperature for all 18 measurement periods, as a hmction of the sensor's zenith view angle. Also shown in a similar fashion is the rms of deviations between the measured sensor response from above the canopy and the measured soil temperature.

sensor was to the right of the rows and sun (Fig. 1). The solution to this nonlinear system of equations was obtained by numerical algorithm ZSYSTM from the International Mathematical and Statistical Library (1979). The predicted vegetation and soft tempera~res, and the row width-height parameter were compared to the measured sunlit vegetation, and weighted mean soil temperature, and the mean of the measured row height and width, respectively. The rms deviation between the predicted and measured values of all measurement periods was cornputed for each system,

Results and Discussion The measured structural characteristics of the canopy, the environmental characteristics, the mean and standard error of all radiometric measurements, and model validation and verification were presented by Kimes and Kirchner (1982). Figure 2 presents the root-mean-square (rms) of deviations between the measured sensor response from above the canopy and the measured stmlit vegetation temperature for all 18 measurement periods, as a hmction of the sensor's zenith view angle. Also shown in a similar fashion is

R O W CROP STRUCTURES

the rms of deviations between the measured sensor response from above the canopy and the measured soft temperature. The soil temperature was calculated as the mean of the sunlit and shaded soil temperatures weighted b y their respective ground surface areas. It is clear from Fig. 2 that the response of a nadir sensor is a composite measurement and cannot be used to infer mean vegetation or soil temperatures directly except during special environmental conditions. As the zenith view angle increases, however, the sensor's response converges toward the vegetation temperature and diverges away from the soil temperature. By employing inversion strategies and multiple •

39

TABLE 1 Accuracyof the PredictedSunlit Vegetation Temperatures and Mean Soil Temperatures for the Exact 2 × 2 Systems a

V~w A~G~s OFSYs=M (t) 00/20° (2) 00/40° (3) 0 0 / 6 0 ° (4) 00/80°

rlns DEVIATIONS(°C)

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SO~ 3.1 2.2 2.5 1.9

(5) 200/40° (6) 2 0 o / 6 0 ° (7) 200/80°

1.5 1.1 0.9

(8) 40°/60°

1.1 9.7 0.9 4.9 --No solution--

(9) 4 0 0 / 8 0 ° (10) 6 0 0 / 8 0 °

2.5 3.9 2.5

aThe root-mean-square (rms) of the deviations between the predicted and measured

view angles these errors can be greatly reduced, The first class of inversion schemes tested assumed a priori knowledge of the

sunlit vegetation temperatures and between

structure of the row crop. Table l presents the rms deviations between the measured and predicted sunlit vegetation temperatures, and between the measured and predicted mean soft radiant temperatures for the exact 2 × 2 systems. All of these systems (except #10) greatly improved the accuracy of inferring the radiant temperature of vegetation a n d / o r soil as compared to the single sensor measurements as shown in Fig. 2. Theoretitally, the sensor views only vegetation (ground is obscured from the sensor) at zenith view angles of 60 ° and 80 °. Consequently, the rms for vegetation is relatively low for any system which utilizes one of these angles. System # 1 0 has no solution because, theoretically, the ground is not visible at either of these angles 6 0 0 / 8 0 °. In general, the soft and vegetation rms are improved as the difference between the two view angles increases, These results suggest that several of these

systems may be applied to data from scanning radiometers on aircraft platforms. For example, System # 2 ( 0 ° / 4 0 °) seems to be a practical case to apply to aircraft data. Before System # 2 can be declared practical under field conditions, however, we need to know how the accuracies of the predictions from Systems # 2 change with errors in the sensor and row structure measurements for various row crops. This question was answered b y the following sensitivity analysis. Figure 3 shows the sensitivity of the 0 ° and 40 ° sensor radiant temperatures on the predicted vegetation and soft radiant temperatures. Figure 4 shows the sensitivity of the row height and width measurements on the predicted vegetation and soft radiant temperatures. To make the results in Figure 4 as general as possible, relative units were used to define the row height and w i d t h - - t h a t is one unit is equal to

the predicted and measured mean soft temperatures for all 18 measurement periods are

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A R A D I A N T TEMPERATURE OF 0 ° SENSOR FIGURE 3. Sensitivity of the 0 ° and 40 ° sensor radiant temperatures on the predicted vegetation and soil radiant temperatures using System ~-2 (Table 1). Initial vegetation and soil radiant temperatures were 29 ° and 54°C, respectively. Five row crop canopies with different percent cover of vegetation were simulated as explained in the text.

the row spacing. Thus, for a 1-m row spacing the relative units of row height and width are expressed in meters. The initial vegetation and soft radiant temperatures were 29 ° and 54°C, respectively, Five row crop canopies were simulated where the percent ground cover varied from 10% to 90%. In all cases the row height was equal to row width and row spacing was 1 unit. For the canopies with 10%, 30%, 50%, 70%, and 90% cover the row width was 0.1, 0.3, 0.5, 0.7, and 0.9 units, respectively,

Figures 3 and 4 show that one can expect relatively accurate inferences of vegetation temperatures for canopies having intermediate and dense vegetation densities (e.g., percent cover >_ 30%). For example, a + 0 . 8 ° C error and a + 0 . 5 ° C error in the 0 ° and 40 ° sensor responses, respectively, resulted in < 1.5°C error in vegetation temperature. A +__0.04 and +0.05 unit error of height and width, respectively, resulted in < 1.5°C error in vegetation temperature. The converse is true for inferring soil temperatures--ac-

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curate inferences of soft temperature can be expected for canopies having sparse and intermediate vegetation densities (e.g., percent cover < 70%). For example, a _+0.6°C error and _+0.8°C error in the 0 ° and 40 ° sensor responses, respectively, resulted in a < 1.g°c error in soil ternperature, and a + 0.04 and + 0.02 unit error of height and width, respectively, resulted in < 1.5°C error in soil temperature. The errors in sensor response and row structure measurements which yield aeenrate predictions ( < 1.5°C) of vegetation and soil temperature are obtainable. Noise equivalent temperatures of thermal infrared sensors are commonly less than

0.5°K. Height and width measurements can be measured with errors (standard error of estimate) of less than 0.02 units (Kimes and Kirehner 1982). Furthermore, it is stressed that the sensitivity analysis was conducted on relatively extreme initial eonditions. The above analyses show that System #'2 would be practical under field conditions for a wide range of percent canopy covers. In the above inversion schemes, a priori knowledge of row structure is assumed. However, research has documented that information on canopy structure speeifieally percent cover of vegetation (equivalent to row w i d t h / r o w spacing), may be obtained remotely using shortwave spee-

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tral bands as reviewed by Kimes (1981). Further research is needed to show if complete characterization of row structure may be inferred from such remote spectral measurements, Table 2 presents the rms deviations between the measured and predicted component temperatures for the remaining systems which assume a priori knowledge of row structure. The overdetermined System #11 does not improve the accuracy of inferences as compared to the simpler System ¢;~2. Systems #12 and #13 show some improvement over their corresponding simpler systems in Table 1, i.e., #1, ;~2, #3, #5, #6, #8, and # 1 - 9 , respectively. Only three measurement periods were included in System #14

because the other periods had algorithmically singular matrices. System #15 showed accurate inferences of the sunlit vegetation and sunlit and shaded soil, and these accuracies were improved with System #16. All matrices of System #17 were algorithmically singular. Finally, System #18 showed that even four components could be accurately separated in some cases. In general, the overdetermined systems improved the stability and accuracy of the systems. Several of these systems may be applied to data from scanning or pointable radiometers from aircraft platforms. The second class of inversion schemes assumed no a priori knowledge of the structure of the row crop. These nonlin-

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ear 3 × 3 systems are advantageous when no knowledge of row crop structure can be obtained practically. However, their major disadvantage is that only two component temperatures can be predicted as compared to three or four component temperatures for the linear systems. Table 3 presents the rms deviations between the relative row width-height parameter, and sunlit vegetation and mean soil temperatures {or the nonlinear 3 × 3 systems. All o{ these systems ( # 1 9 - # 2 2 ) demonstrate that geometric structure, together with vegetation and soil temperatures can be in{erred accurately using remotely sensed data. In real-time applications the simple equality between row height and width can be replaced by more realistic growth

equation which relate row width to row height. The solution of these nonlinear systems requires an initial guess for the root. In this study an initial guess was used which could be readily used in real time remote sensing applications. For example, the initial vegetation and soft temperatures were defined as the local air temperature (°C), and air temperature plus 20°C, respectively. (Note: an initial soil temperature defined as air temperature plus 10°C gave equivalent results.) In general, using this information, the system converged successfldly {or all ten measurement periods using an initial guess o{ the row width-height parameter within _ 16% of the true value, and for > 5 measurement

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& ROW HEIGHT (RELATIVE UNITS) FIGURE 4. Sensitivity of the height and width measurement on the predicted vegetation and soil radiant temperatures using System ~¢2 (Table 1). Initial vegetation and soil radiant temperatures were 29 ° and 54°C, respectively. Five row crop canopies with different percent cover of vegetation were simulated as explained in the text.

periods using an initial guess within + 27% of the true value. Percent cover measurements obtained by inferences from shortwave spectral measurements as discussed earlier can be used as an initial guess of the relative row width-height parameter. However, in many cases this parameter can simply be varied until the system successhdly converges, In the above analyses the azimuth view angle was always in the plane normal to the row structure. But what may one expect for other azimuths? Using the model of Kimes and Kirchner (1982) the

sensor response for two different row crop structures were simulated as a fimction of zenith and azimuth view angles (Fig. 5). It is clear that the same sensor response can occur at many different view directions. Yet, for a given azimuth plane the trajectory of response as a function of zenith view angle is unique for a given row crop structure and solar row angle as is shown in Fig. 6. Exceptions to this are when azimuth angles are parallel to the rows and when component temperatures are equal. As the azimuth approaches an angle parallel to the rows the sensor re-

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sponse as a hmction of zenith angles approach a constant value as explained by Jackson et al. (1979). However, the implication of this in terms of the above inversion schemes is that the system to be inverted becomes singular at some point as the azimuth approaches 90 ° a n d / o r 270 °. In general, the optimum set of view angles for most remote sensing missions has an azimuth angle normal to the row direction and zenith angles positioned so that only one angle occurs in the fiat position of the trajectory as seen in Fig. 6. This is done to avoid redundancy and to assure that the maximum differences of

sensor responses occur between successive zenith view angle positions. Specifically, the optimum azimuth orientation depends on the row structure, the range, position, and number of view angles that can be collected, and the nature of the inferences being made. The row crop model can be used to choose the optim u m number and position of view angles for a given mission by analyses of sensitivity of various inversion schemes. In some remote sensing missions the azimuthal orientation of rows may not be known. However, in many instances this knowledge may not be necessary. For example, many applications simply re-

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quire separation of vegetation and soil temperatures and need no knowledge of the absolute structural measurements (e.g., width as measured normal to the rows). Thus, the nonlinear schemes as demonstrated above can be applied. The solution of these inversion schemes yields the vegetation and soil temperatures, the relative height, and the relative width of the rows. The relative width and height are relative to row spacing in the partieular azimuth direction. Thus, in essence, the structure (width and spacing) is elongated as the azimuth deviates from being normal to the rows. As mentioned above, however, this approach will fail at some point as the azimuth direction ap-

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proaches an angle parallel to the rows and matrices become singular. Conclusions and I m p l i c a t i o n s A physically based sensor response model of a row crop was used as the mathematical framework from which several inversion strategies were tested for extracting row structure information and component temperatures using a series of sensor view angles. The results show that the accuracy of the predicted vegetation and soil component temperatures of the cotton row crop are on the order of 1 ° and 2°C, respectively, assuming a priori knowledge of the row structure. The ac-

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I--

o ~

--1-

I---

(

~_ ~

, _

-3~-

~

_ -4

--.10

__1

-- .08

--.06

--.04

-- .02

.00

.02

.04

.06

.08

.10

A ROW WIDTH (RELATIVE UNITS) FIGURE 4d.

curacy of the predicted row structure parameters, and vegetation and soft temperatures are on the order of 5 cm ( ___10%), I°C, and 2°C, respectively, assuming no a priori knowledge of row structure. The inversion techniques demonstrated in this paper can be applied to directional sensor data from well-planned aircraft missions that use thermal IR scanners or the more desirable pointable pushbroom systems which use mtdtilinear sensor arrays that are now being developed, These techniques can extract the unique component temperatures from a composite scene which are superior information for evapotranspiration, yield, and

irrigation studies. In addition, the techniques can extract information on crop structure which is valuable for determining the type, growth stage, condition, and final yield of the crop. The same potentials may exist from satellite systems with pointable sensors, but with more restrictions and complications. Research needs to address the problems of atmospheric absorption and emission for pointable, thermal infrared sensors. Correction techniques for these effects would be required before these inversion schemes could be applied to satellite data. In theory, such inversion techniques can be applied to a wide variety of vege-

48

D . S . KIMES TABLE 2

Accuracy of the Predicted C o m p o n e n t Temperatures for Various Systems ~'

rms DEVIATION(0(3) SUNLIT VEGETATION

SYSTEM (11)

3 × 2 system using 0 0 / 2 0 0 / 4 0 °

SHADED VEGETATION

MEAN SOIL

SUNLIT SOIL

SHADED SOIL

1.9 ( n = 3)

3.9 (n = 3)

3.7 (n = 3)

1.1 ( n = 18)

2.4 i n = 17)

2.7 b ( n = 5)

0.5 (n = 18)

2.2 (n = 17)

2.5 b ( n = 5)

2.4': ( n = 7)

2.5 ~ (n = 5)

1.7 ( n = 18)

2.2 ( n = 18)

0.9 (n = 18)

2.5 ( n = 18)

0.6 ( n = 18)

2.2 (n = 18)

view angles (12)

(13)

4 × 2 system using 0 0 / 2 0 0 / 4 0 0 / 6 0 ° view angles 5 x 2 system using 0 ° / 2 0 ° / 4 0 ° / 6 0 ° / 8 0

°

view angles (14)

(15)

3 × 3 system using 0 ° / 2 0 ° / 4 0 ° view angles 4 X3 system using 0 0 / 2 0 0 / 4 0 ° / 6 0

°

view angles (16)

(17)

5 × 3 system using 0 0 / 2 0 0 / 4 0 0 / 6 0 ° view angles 4 × 4 system using 0 ° / 2 0 ° / 4 0 ° / 6 0 °

All matrices were algorithmically singular

view angles (18)

5 X 4 system using 0 0 / 2 0 0 / 4 0 0 / 6 0 0 / 8 0 view angles

°

0.5" ( n = 18)

0.3': (n = 1)

aThe root-mean-square (nns) of the deviations b e t w e e n the predicted and measured c o m p o n e n t t e m p e r a t u r e s are presented for each system. The n u m b e r of m e a s u r e m e n t periods included in each tans value is indicated in parenthesis. If n was less than 18, the missing points were w h e n the c o m p o n e n t did not exist or w h e n the matrix was algorithmically singular. Exceptions are noted. bAny period with a solar row angle (O) _< 0.8 ° was not included in n n s value since the proportion of projected shaded soil area in the nadir direction was small ( _< 0.076) and u n a c c e p t a b l e solutions connnonly occurred. CAny period with a solar angle of 0.0 < 8 _< 9.8 ° was not included in the rms value since the system yielded unacceptable solutions for all components.

tation types as corroborated by Kimes (1981). These findings have significant implications for remote sensing research and applications in agriculture, forestry, geology, hydrology, and other earth resource disciplines where the scene of interest is the vegetation or the underlying substrate rather than the composite scene,

The author acknowledges the SoilPlant-Atmosphere Systems Research Group at the USDA, U.S. Water Conservation Laboratory, Phoenix, Arizona, for providing the field experiment site and

instrumentation. He thanks J. Kirchner for her part in collecting the field data.

Appendix The algorithm for calculating the relative proportions of projected surface area (with direct view to the sensor) of the four scene components is presented. Only rows with an east-west orientation are treated here. Three cases are shown. The variables are shown graphically in Figs. 1 and A.1.

ROW CROP STRUCTURES

49

TABLE 3 Accuracy of the Predicted Relative Row Width-Height Parameter, and Sunlit Vegetation and Mean Soil Temperatures for the Nonlinear 3 × 3 Systems a

rms DEVIATION SYSTEM (19)

3 × 3 nonlinear system using 00/200/40 ° view angles b 3 × 3 nonlinear system using 00/200/60 ° view angles 3 × 3 nonlinear system using 00/40o/60 ° view angles 3 × 3 nonlinear system using

(20)

(21)

(22)

Row WIDTH-HEIGHT PA~aAMm~R(cm)

SUNLrr VECETATION (°C)

MEaN SoIL (°C)

10.4

2.2

2.5

6.7

.85

2.9

4.9

1.1

2.1

1.6

1.1

2.1

00/400/80 ° view

angles aThe root-mean-square (rms) of the deviations between the predicted and measured values are presented for each system. bTwo measurement periods resulted in algorithmicaUy singular matrices.

Algorithm

If x > xs

F o r all cases: tan 8 - cos ~ tan z tan v = cos 7 t a n a

xs=W+

p(1) =

W/S

p(2)=

1-x/S

p ( 3 ) = (x - x s ) / S

p(4)=(xs-W)/S.

n(tan0)

If x < xs a n d x > W C a s e (1) for nadir sensor: p(1) =

p(1) =

W/S

W/S

p(2) = 1 -

p(2)--0

v(3)=o

p(3) = 1 - xs/S

p(4)

p ( 4 ) = ( x s - W)/S.

x/S

= (x - W)/S.

Ifx
v(1) = w / s

C a s e (2) for s e n s o r to right of r o w s ( o p p o s i t e sun, Fig. A1):

p(2) = 1p(3) = 0

x = S - n ( t a n v)

p ( 4 ) = 0.

W/S

50

D.S. KIMES NORTH 0o

ROW WIDTH 80o

3o°

=

15 c m

ROW HEIGHT =

15 c m

SOLAR ROW ANGLE =

5°

60 ° ua -J

"--'45.4-,

Z <

~

46

/ !;

NADIR

,~51.4

I I

- -

/

< --

~ so .~

/

20 °

90°

Z

~',

I--

40 °

60 °

8

0

°

~

150 °

180 °

FIGURE 5. Simulated sensor radiant temperatures as a function of zenith and azimuth view angles for two row crops having different row height and width measurements. Contour lines of equal radiant temperatures in I ° C intervals are plotted in polar coordinates where the azimuth view angle is defined such that a sensor with a 180 ° azimuth view angle is looking toward the north. The rows of the crops were oriented east-west and the solar azimuth angle was 180 °. Only half of the total exitance hemisphere is shown in these plots because the plots are symmetric about the north-south line. The sunlit and shaded vegetation temperatures, and sunlit and shaded soil temperatures were defined as 28.5 °, 29.5 °, 55.3 °, and 31.1°C, respectively.

ROW CROP STRUCTURES

51

NORTH

oo

80 ° - _ ~

.~'~°°~de/'~. "-..

/

/.

6°°---

>

= 48 cm

HEIGHT

=

44 c m

ROW SPACING

=

100 c m

SOLAR ROW ANGLE

=

5°

""

. ./

<

ROW WIDTH ROW

7".,

~-

40 ° ~

~

32.3 , ~ , . . , , . , ~

~

~

~',

NADIR

90 °

/I

40 °

80 °

-- -- ~

150 °

180 °

FIGURE 5'0.

<~

52

D.S. KIMES

ROW W I D T H = 15cm ROW HEIGHT = 15cm ROW SPACING = 100cm SOLAR R O W ANGLE

= 5°

52 - -

',\

-

.,

44

--

<

_

W

~z~<

40

-A Z I M U T H VIEW ANGLE

F,

o°~--

<

Ik%

30 °

~:

o

03

~ \

60 ° .

36

80 ° ~

Z

.

.

__l ~

.

m

90 °

k,l.I 03

I

32 - -

28

I

1

0

I

20

I

J

1

40

ZENITH

VIEW

I

l

60

80

ANGLE

F I G U R E 6. S i m u l a t e d sensor r a d i a n t t e m p e r a t u r e s as a f u n c t i o n of zenith v i e w a n g l e for selected a z i m u t h view angles. T w o r o w c r o p s h a v i n g different r o w h e i g h t a n d w i d t h m e a s u r e m e n t s are s h o w n as in Fig. 5.

C a s e (3) for sensor to left of rows ( s a m e side as sun, Fig. A1): x = W+

H(tanv).

If x >__S a n d xs < x

p(3) = 0 p(4) = 0 If x > S a n d xs > x p(1) = [W+(Sp(2) = 1- p(1)

p(1) = 1

p(3) = 0

p(2) = 0

p ( 4 ) = 0.

W)tanp/tanS]/S

ROW CROP STRUCTURES

53

R O W W I D T H = 48 c m R O W HEIGHT = 44 cm R O W SPACING = 100 cm S O L A R R O W A N G L E = 5°

44 -

•. k 40--

. •

\,

,,

=

\

I--

"'

•

%

I-

%

z

•=:,

-

\,

AZIMUTH VIEW ANGLE

-"

oo

Z

\

~

\~\

30 ° . . . . . .

k 80 ° - - _

\

~,

w 32 --

~i,

\

~

- -

I

900 "

'\ '

\

~

~k

\,

/

,

\ i

I

28 0

I

I

I

20

I

40

ZENITH

VIEW

I I

~

60

ANGLE

FIGURE 6b.

If x < S and x s > S p(1)= [W + (S-

If x < S and x s > x and xs < S

W)tan~,/tanO]/S

p(1) =

x/S

p(2)=x/S-p(1)

p(2)=0

p(3) = 0

p(3)= 1 - xs/S

p(4) = 1 -

x/S.

p(4) = (xs -

I 80

x)/S.

54

D.S. KIMES

SENSORLEFT

SENSORRIGHT 0 F R/WS

OiiOWS

I

/

/

/ / (o.o1/

x"

/

x

/ FIGURE A/. A horizontal projection of a row crop ]ooking paraHe] to the rows, Several variables are shown as defined in the Appendix, where xs and x are measured on the X axis as shown. The intersection point x is shown for a sensor located to the right (x) and to the left (x') of the rows.

If x < S and xs < x p(1) = x/S p(2) = 0 p(3) = 1 p(4) = 0.

x/S

Notation P,)i = 1 , 2 , 3 , 4 = T h e proportion of sunlit and shaded vegetation, and sunlit and shaded soil respectively, in direct view of the sensor, H = Row height, W = Row width, S = Row spacing, z = Solar zenith angle,

~ = Solar azimuth angle measured positive to the east of true north, /9 = Solar row angle, a = Zenith view angle, 3' = A z i m u t h view angle measured positive to the east of true north, 1,=Sensor row angle (Fig.

A1),

x = R e l a t i v e horizontal dist a n c e of i n t e r s e c t i o n point of sensor vector with ground as shown in Fig. A1, xs = Relative horizontal dist a n c e of i n t e r s e c t i o n point of solar vector with ground as shown in Fig. A1.

ROW CROPSTRUCTURES References Byrne, G. F., Begg, J. E., Fleming, P. M., and Dunin, F. X. (1979), Remotely sensed land cover temperature and soft water status--A brief review, Remote Sens. Environ., 8:291. IMSL Library (1979), edition 7 (FORTRAN IV) IBM 370-360 Series, International Mathematical and Statistical Libraries, Houston, Texas. Jackson, R. D., Reginato, R. J., Pinter, P.j., Jr., and Idso, S. B. (1979), Plant canopy information extraction from composite

55 scene reflectance of row crops, Appl. Opt., 18:3775. Kimes, D. S. (1981), Remote sensing of temperature profiles in vegetation canopies using multiple view angles and inversion techniques, IEEE Trans. Geosci. Remote Sens., GS-19:85. Kimes, D. S., and Kirchner, J. A. (1982), Directional radiometric measurements of row crop temperatures, Int. 1 Remote Sens., in press. Received 10 February 1982; revised25 luly 1982.