Estimates of leaf area index from spectral reflectance of wheat under different cultural practices and solar angle

Estimates of leaf area index from spectral reflectance of wheat under different cultural practices and solar angle

n e M o r e SeNSINC OF eNVIRONMeNt 17.'1-11 1985) Estimates of Leaf Area Index from Spectral Reflectance of Wheat Under Different Cultural Practices ...

615KB Sizes 0 Downloads 25 Views

n e M o r e SeNSINC OF eNVIRONMeNt 17.'1-11 1985)

Estimates of Leaf Area Index from Spectral Reflectance of Wheat Under Different Cultural Practices and Solar Angle*

G. ASRAR, E. T. KANEMASU, AND M. YOSHIDAt Evapotranspiration Laboratory, Kansas State University, Manhattan, KS 66506

Two methods (a regression formula and an indirect technique) were used to assess the influence of several management practices and solar illumination angles on the estimates of leaf area index (LAI) using red and near infrared reflectance measurements from wheat (Triticum aestivum L. and T. durum) experiments conducted. In all three experiments, the management treatments affected the seasonal trend in spectral response of wheat canopies. The effect of each treatment was shown in the estimated LAIs that were based on the measurements of canopy reflectance. Solar illumination angle also affected the spectral properties of the canopies, and hence the estimated LAIs. For both methods, good agreements were obtained between the measured and estimated LAIs over a range of 0.0-6.0 LAI, with major differences only for LAI greater than 6.0.

Introduction

Remote sensing, with its unique synoptic perspective, is a potential means of monitoring terrestrial vegetation. The role of phytomass in the global carbon cycle is extremely important, since it involves the amount of carbon stored in various plant communities. Remote sensing of the vegetation canopy potentially offers a great improvement over conventional destructive techniques, since it allows the researcher to monitor changes in the condition of the same plants with time. In recent years, a considerable amount of ground-based remotely sensed data has been accumulated. These ground based measurements are useful in describing the effect of agronomic management prac-

*Contribution No. 84-319-J from Evapotranspiration Laboratory, Agronomy Department, Agricultural Experiment Station, Kansas State University, Manhattan, KS 66506. This study was supported by NASA Johnson Space Center Contract No. NAS-16457. tSoil physicist, microclimatologist, and CIMMYT Associate Scientist. ©Elsevier Science Publishing Co., Inc., 1985 52 Vanderbilt Ave., New York, NY 10017

tices on spectral response of the vegetation. This information enhances understanding and interpretation of the data acquired by earth observational satellites such as Landsat. Numerous studies have been conducted to assess the influence of atmosphere (Dave, 1980; Slater and Jackson, 1982), management practices (Daughtry et al., 1980; KoUenkark et al., 1982a) and solar illumination angle (Kollenkark et al., 1982b; Pinter et al., 1983) on the quality of reflected radiation from different plant canopies. Hatfield et al. (1984a, b) and Asrar et al. (1984) have shown that both absorbed photosynthetic radiation by plants and green leaf area index can be estimated from measurements of plant canopy spectral reflectance. Goel et al. (1984) in a series of papers described a detailed procedure for estimating agronomic parameters by inversion of plant canopy reflectance models. Their approach, however, requires measurements of plant canopy reflectance at several view azimuth and zenith angles. 0034-4257/85/$3.30

( ; ASRAR ET AL.

Leaf area index (LAI) is an important plant canopy parameter. The magnitude and duration of LAI is related strongly to the canopy's ability to intercept photosynthetically active radiation; therefore, LAI is correlated with canopy photosynthesis and dry matter accumulation in situations where stress (water, disease, pests, etc.) does not predominate. Direct measurements of leaf area are extremely tedious; thus the development of a simple and rapid technique for assessing leaf area would be an important contribution. The main objective of this study was to evaluate the influence of management practices and solar illumination angle on the leaf area index estimated from measurements of wheat canopy reflectance. Materials and Methods Experiment eonditions The data used in this study were obtained from experiments conducted at three different geographical locations. The first experiment was conducted in 1979 1980 at the U.S. Water Conservation Laboratory in Phoenix, AZ (112001 ' W longitude, 33 °26' N latitude). The treatments were five planting dates (1-5) and typically three irrigation levels (A, B, and C) on Produra spring wheat (Triticum

durum Desf.) planted in north-south direction. Only data from planting dates 1 and 5 will be reported here. Planting, emergence, and irrigation dates and quantity of water applied to each treatment are presented in Table 1. The nominal irrigation amount at each date was 10 cm. Six plants were selected randomly from each treatment twice weekly for determining leaf area index. Leaf area was determined by an optical planimeter (LiCor model LI-3100). Reflected radiation from the wheat canopies and a barium sulfate reference panel were measured with a hand-held Exotech Model 100-A radiometer on clear days at solar zenith angle of -= 57 °. The Exotech radiometer is equipped with four wavelength bands corresponding to the Multispectral Scanner (MSS) on board Landsat satellites. Two of the wavelength band are in the visible (550-600 nm and 600-700 nm) and the other two are in the near infrared (700-800 nm and 800-1100 nin) regions of electromagnetic spectrum. The second experiment was conducted during the 1982-1983 season near Manhattan, KS (96 °37' W longitude, 39 °09' N latitude). Newton winter wheat (Triticum aestivum L.) was planted in eastwest and north-south rows at a spacing of 17.8 era. Twenty-five plants were

TABLE 1 Planting, Einergence, Irrigation Dates (Day or Year) and Total Applied Water for the 1979-1980 Experiment on Produra Spring Wheat at Phoenix. AZ TREATMENT

A1 B1 C1 A3 B3 C3 A5 B5 C5

PLANTING DATE

271 271 271 :318 318 318 036 036 036

EMERGENCE DATE

275 275 275 330 330 330 047 047 047

IBRIGATION 1)ATES

271,278, 271, 27S, o72, 278, 319, 1351 319, 079 320, 351 I)39, 100 039, 079, 039, 093,

289, :3:34 290. 313, :345 290, 324

106, 123, 134 114

SPECTRAL ESTIMATES OF LEAF AREA INDEX

harvested randomly from each plot at least once a week, and the leaf area was determined. Canopy spectral reflectance was measured, using a boom-truck assembly equipped with an Exotech Model 100-A Radiometer (Asrar et al., 1984). Reflectance measurements were carried out on days with clear sky conditions at several times from mid-morning to mid-afternoon. Reflected radiation from the barium sulfate panel was measured sequentially. Canopy reflectance factors were determined from a ratio of the canopy to reference panel reflectances. The third experiment was conducted during the 1982-1983 season as Ciudad Obregon, Mexico (109 °59' W longitude, 27 °28' N latitude). Two spring wheat (Triticum aestivum L.) cultivars Ciano and Pavon were planted in north-south rows 17.8 cm apart. The main treatments were four soil moisture regimes. Treatm e n t 1 was well-watered throughout the season (six irrigations). Treatment 2 received only three irrigations during the vegetative growth to promote water stress during the grain-filling period. Treatment 3 was irrigated four times, such that water stress was achieved during stem elongation. Treatment 4 was the severe stress treatment, with only two irrigations during the early part of the growing season. In each irrigation, the basin was flooded and then drained after 12 h. Plant samples were harvested once a week from a 0.34 m e section of each treatment for leaf area determined. Canopy spectral reflectance was measured with a Barnes 12-1000 Modular Multispectral Radiometer (MMR) equipped with seven reflective wavelength bands from visible to middle infrared and thermal infrared wavelength bands. The MMR radiometer spectral bands are the same as for the Landsat

3

Thematic Mapper (TM), except that MMR has an additional near infrared wavelength band. The radiometer was attached to a boom at a height of 4 m above the soil surface. Measurements of reflected radiation from plant canopy and a barium sulfate reference panel were obtained on clear days between the times 1100 and 1500.

Data analysis In the following analysis, only data from the red (MSS = 600-700 nm and MMR = 630-690 nm) and near infrared (MSS = 8 0 0 - 1 1 0 0 nm and M M R = 7 6 0 - 9 0 0 nm) wavelength bands were used. Near infrared (Pn) and red (Pr) canopy reflectance factors from each experiment were used in computing the normalized difference (ND), ND =

(Pn -- Pr)/(Pn "~- Pr),

(1)

and ratio (RI) of near infrared to red reflectance, RI = p./p

(2)

Two approaches were used for estimating LAI from spectral reflectance data. In method 1, regression technique (RGT), LAI was computed from RI and the following empirical equation: LAI = - 0.4222 + 0.279 * RI,

(3)

which was developed by Hatfield et al. (1984b) based upon a data set from an experiment conducted during the 19781979 growing season in Phoenix, AZ. In method 2, the indirect procedure (IND), N D values were corrected for early season effect of soil background and scattering of

4

(;. ASRAR ET AL.

the near infrared radiation by foliage elements, to obtain an estimate of radiation absorbed by plants (Asrar et al., 1984): p = - 0.185+ 1.20*ND

(4)

where p is the estimated fraction of photosynthetic radiation absorbed by the plants. Values of p were used in computing LAI as LAI = - ln(1

- p)/K

(5)

where In(1 - p ) is the arithmetic mean of estimated/9 and K is a mean leaf angular shape coefficient (Fuchs et al., 1984). Assuming a spherical leaf angle distribution, K was computed from K = 0.5/cos 7/,

(6)

where ~/is the solar zenith angle that was computed from the time of reflectance acquisition and geographical coordinates

of each experiment site, using equations given in the Nautical Almanac (Walraven, 1978). Results and Discussion

The date of planting affected development and growth of Produra spring wheat (Figs. 1 and 2). Early planting (treat. B1) resulted in lengthening the growing season (180 days), but reducing the magnitude of peak LAI (Fig. 1). In the later planting (treat. BS) the peak LAI was higher in spite of reduced duration (105 days) of growth (Fig. 2). The difference between these treatments resulted from low mean air temperature (T = 12.7 °C) and insolation ( i = 317.5 Wm 2) due to shorter daylengths for the early planting treatment B1. When planting was delayed in treatment B5, development and growth rate of leaves increased, due to higher air temperature (T = 19.5 °C) and insolation (I = 595.6 W m 2), if soil moisture was adequate.

PHOENIX,AZ 1979 - 8 0 SPRING WHEAT V A R I E T Y PRODURA TREAT. B I

6.0 5.0 X UJ

o

Z

4.0

Ld :5.0

u. 2.0 I,d .J

JOINTING

t.o

• R

O0

/

520

ir



360

35 75 DAY OF THE YEAR

.

.

.

I 15

.

155

FIGURE 1. Temporal profiles of measured (0) and estimated [(O) regre~siom ~x) mdirectj LAI for Produra spring wheat, treatment BI: first planting date and five irrigations for the entire growing season.

SPECTRAL ESTIMATES OF LEAF AREA INDEX

7.{1

5

PHOENIX,AZ 1980 SPRING WHEAT VARIETY PRODURA TREAT. B5

'";!

6.0 x 5.0

IJiJ

E3 Z m

4.0 I,iJ

<

3.0

h

~ ao 1.0 0.0

I

320

I

I

360

I

35 DAY OF

115

75

THE

155

YEAR

FIGURE 2. Temporal profiles oI measured (O) and estimated [(O) regression; ( × ) indirect] LAI for Produra spring wheat, treatment B5: fifth planting date and five irrigations for the entire growing season.

The difference in plant development among treatments was shown in the estimated LAI values that are based on the measurement of canopy spectral reflectance. In general, the LAI values estimated according to IND procedure were in a better agreement with the measured LAI values than those obtained with the RGT method. This is demonstrated visually in Figs. 1-4 and confirmed by a statistical comparison between measured and estimated LAI values. The standard deviations between the mean estimated and measured LAIs for treatments B1 and B5 were 0.20 and 0.74 for the IND procedure and 0.24 and 0.78 for the RGT method, respectively. These differences were not statistically ( a = 0.01) significant, based on unequal variance t-tests. A reduction in soil moisture, due to the reduced number of irrigations in treatments A5 and C5 (Table 1), also resulted in a reduced peak LAI, as compared with

treatment B5 for the same growth period (Figs. 3 and 4). The effect of moisture treatments also was shown in the LAI values estimated by both methods. However, the LAI values obtained [rom RGT method were consistently higher than the measured ones (Figs. 2-4). This indicated that the empirically derived RGT equation did not adequately represent the sparse canopies observed in these treatments. However, better agreements were observed between LAI values from IND procedure and the measured ones. The standard deviation between mean estimated and measured LAYs for treatments A5 and C5 were 0.45 and 0.55 based on the IND and 0.64 and 0.65 based on the RGT methods, respectively. These differences were not statistically (a = 0.01) significant. Some of the seasonal variability depicted in measured LAI values in all treatments was due to nonsymmetric fac-

6

( ; ASRAR ET AL PHOENIX,AZ 1980 SPRING WHEAT

7.0

VARIETY PRODURA TREAT. AS

6.0 x o z

5.0 4.0

bJ e,. 3.0 i1 bJ 2 D ..J 1.0 0.0



i

320

I

A

360

35 DAY

OF

75 THE

115

155

YEAR

FIGURE 3. Temporal profiles of measured (e) and estimated [(Q) regression~ ( x ) indirect] LAI for Produra spring wheat, treatment A5: fifth planting date and two irrigations for the entire growing season.

P H O E N I X , AZ 1980 S P R I N G WHEAT

7.0

V A R I E T Y PRODURA TREAT. C5

6.0 x 5.0 bJ 0 Z -- 4.0 <

b.I =:

<

30

IL. <

"._1 ' 2.0 1.0 0.0

i

520

I

I

560

I

I

35

75

115

155

DAY OF THE YEAR FIGURE 4. Temporal profiles of measured (I) mlct estimated [(©) regressiolL: { ><, indirect] LAI for Produra spring wheat, treatment C5: fifth planting date and threc~ irrigations for entire growing season.

SPECTRAL ESTIMATES OF LEAF AREA INDEX

tors such as differences in physiological development among plants which were either inherent or caused by weatherrelated parameters, a n d / o r spatial variability in soil characteristics. For example, the sharp drop in LAI observed between days 17 and 23 of Fig. 1 coincides with a cool ( T = 6 . 0 ° C ) and rainy (2.8 cm) period that presumably hampered the plant growth. However, the integrated effect of these and other possible sources of variability was incorporated in the estimated LAI values. Table 2 presents a comparison between measured and estimated values of LAI for Newton winter wheat planted in two row orientations (east-west and north-south). The numbers in parentheses are standard deviations of the measured and estimated LAI values. The measured LAI values, for both row orientations were similar on 24 April. The estimated LAI values according to IND and RGT methods were also similar for both row orientations at com-

7

parable sets of solar angles. The small differences between the estimated LAI values at similar solar angles for the two orientations were within the range of their corresponding standard deviations. This indicated that row orientation did not significantly affect the estimated LAI values. Data from 3 and 6 May cannot be used for similar analysis, due to the difference in measured LAI values for the two orientations. The observed variance in both measured and estimated LAI values was due to combined effects of sampling and intrafield variabilities that are inherent in any field experiment. The estimated LAI values, according to both IND and RGR methods for both row orientations, were consistently smaller for the reflectance data acquired around solar noon (Table 2). This was due to the changes in red and near infrared canopy reflectance with the solar illumination angle. The diurnal symmetric variation of the red reflectance was found to be more

TABLE 2 Comparison between Measured and Estimated LAI of Newton Winter Wheat Planted in Two Different Row Orientations in 1983 at Manhattan, KS a

DATE 24 April 1983

3 May 1983

6 May 1983

Row ORIENTATION East-West

MEASUI~D LAI 2.4 (0.20) b

North-South

2.5 (0.25)

East-West

5.1 (0.40)

North-South

3.4 (0.30)

East-West

5.4 (0.40)

North-South

3.4 (0.30)

SOLAR ANCLE (deg) AZIMUTH ZENITH 34.4 - 4.1 - 49.3 28.0 - 9.6 - 52.8 51.2 25.5 48.3 18.3 69.3 56.1 - 55.6 67.2 54.3 - 57.9

27.9 25.4 34.2 29.3 25.6 35.7 31.6 24.5 30.4 23.6 41.0 32.8 33.6 39.3 31.9 33.9

ESTIMATED LAI INDIRECT 2.4 1.8 3.0 2.1 2.0 3.3 6.1 3.0 3.9 2.7 5.0 4.7 3.8 4.4 2.8 2.7

(0.33) (0.22) (0.85) (0.55) (0.57) (0.33) (1.05) (0.92) (0.17) (0.49) (0.68) (0.66) (0.42) (0.65) (0.77) (0.55)

REGRESSION 3.5 2.4 3.6 2.7 2.6 2.8 4.4 3.0 4.2 3.4 4.9 4.1 3.5 4.5 3.7 3.5

(0.70) (0.29) (0.75) (0.68) (0.72) (0.40) (0.73) (0.88) (0.43) (0.40) (0.31) (0.30) (0.45) (0.30) (0.74) (0.50)

aThe LAI estimates are based on measurements of canopy spectral reflectance at different solar angles. b Standard deviation.

S

(;. ASRAR ET AL.

dependent on solar illumination and canopy geometry than the near infrared reflectance (Kollenkark et al., 1982b). Therefore, canopy geometry and solar angle affected both the direct ratio RI [Eq. (2)] and the ND index [Eq. (1)]. Asrar et al. (1984) found that ND is considerably less sensitive to changes in these parameters. However, the leaf angle shape coefficient (K) for the IND procedure is based upon the solar zenith angle [Eq. (6)]; therefore, the LAI values computed from both methods were affected by solar illumination angle. Time of irrigation in relation to stage of plant growth played a significant role in development and production of leaves by Ciano and Pavon spring wheat (Figs. 5 and 6). The upward arrows on the abscissa of each figure correspond to the

OBREGON, MEXICO SPRING WHEAT VARIETY PAVON

9.0

time and number of irrigations. The first irrigation in all treatments was on 28 December 1983 (not shown on the figures). The highest peak LAI for Pavon was observed in treatment 1 with six irrigations (Fig. 6). A decrease in number of irrigations (to three) in treatment 2 did not result in a significant change in duration or peak LAI for Pavon. However, when the number of irrigations was reduced to four, which were applied mainly during the reproductive periods of growth (treat. 3), the peak LAI for Pavon was decreased, considerably (Fig. 6). Similar results also were observed for cultivar Ciano. A comparison between treatments 2 and 3 indicated that the early water stress during the stem elongation period (leaf initiation phase) adversely affected the LAI in treatment 3. However, late

1983

/' '~

TREAT. PI

&O 7.0



t

x

td " 2

6.0 I

5.0 Il t

.

4.0

/

".1 3.0 2.0

1.0 13

t 20

Z4

t. 40

1. 60

8O

I~

120

140

DAY OF THE YEAR FIGURE 5. Seasonal distribution of measured (0) and estimated [ ( 0 ) regression; ( × ) indirect] LAI for Pavon spring wheat, treatment PI: six irrigations for the entire growing season.

SPECTRAL ESTIMATES OF LEAF AREA INDEX

9

OBREGON, MEXICO 1983 SPRING WHEAT VARI ETY PAVON TREAT. P3

6.0 x "'

5.0

<.

4.0

<.

3.0

"'-

~ 2.o I.O I

PO

'

40

60

80

I00

120

|

140

DAY OF THE YEAR

FIGURE 6. Seasonal distribution of measured (e) and estimated [(O) regression; ( × ) indirect] LAI for Pavon spring wheat, treatment P3: irrigated once after planting and three irrigations for the reproductive stages of growth

irrigations (irrigations 3 and 4) delayed senescence of the leaves and prolonged the duration of green LAI in this treatment. In treatment 4, one irrigation during the stem elongation period increased peak LAI, as compared with treatment 3. However, overall decrease in soil moisture in treatment 4 resulted in the shortest duration of green LAI, as compared with treatments 1, 2, and 3. Therefore, in Ciano and Pavon spring wheat, water stress during the vegetative stage resulted in a reduction of peak LAI, while water stress during the reproductive period shortened the duration of green LAI. These changes affected the seasonal trend of spectral reflectance properties of the plant canopy in each treatment. The integrated effect of these changes was incorporated in the estimated LAI for each treatment.

The major differences between the measured and estimated LAI values were observed in early, well-watered treatments 1 and 2. Although the measured LAI in these treatments was > 7.0, the estimated LAI based on either IND or RGT methods did not exceed 6.0. This difference can be attributed to the relatively small changes in canopy reflectance that occur in response to LAI changes at full cover (Asrar et al., 1984). Point-bypoint comparisons of the estimated LAI values, based on the IND and RGT methods, with measured values were not conducted, due to large variance in measured LAI values. However, the standard deviation bars demonstrate the extent of variance associated with each procedure, and the degree of agreement between the methods. As stated earlier, this variance is

10

caused by sampling and intrafield variabilities. Both IND and RGT methods adequately depicted the seasonal trends and the changes caused by management practices that influenced growth and development of plant leaves. Conclusions The influence of several management practices and solar illumination angle on leaf area index (LAI), estimated from measurements of canopy spectral reflectance, was evaluated by two different methods. Date of planting and time of irrigation in relation to the stage of plant growth had significant effects on development of leaves in spring wheat. A reduction in soil moisture adversely affected both the duration and magnitude of the maximum LAI in late planting dates. In general, water stress during vegetative stages resulted in a reduction in maximum LAI, while water stress during reproductive period shortened the duration of green LAI in spring wheat. These management treatments affected the seasonal trend of spectral response of the wheat canopies. The effect of these treatments was shown in the estimated LAI values based on the measurement of canopy reflectance. Canopy geometry and solar angle also affected the spectral properties of the canopies, and hence the estimated LAI. Increase in solar zenith angles resulted in a general increase in estimated LAI obtained from both methods. This was attributed to strong dependence of red canopy reflectance on solar illumination angle, since near infrared reflectance is less sensitive to diurnal changes. The results indicate that the symmetric influence of solar illumination angle on

(;. ASRAR ET AL.

estimated LAI was not as significant as the effects of nonsymmetric factors that were caused by management practices, a n d / o r the intrafield variabilities. However, the effects of both symmetric and nonsymmetric factors were integrated in the LAI estimated from measurements of canopy spectral reflectance. The basic limitation in using the red and near infrared reflectance to estimate LAI is a need for cloudless sky condition for measurements of canopy reflectance.

We thank Drs. R. D. Jackson, P. J. Pinter, R. J. Reginato, and S. B. Idso of the U.S. Water Conservation Laboratory, Phoenix, AZ, and Drs. D. Saunders and ]. Ransom o f CIMMYT, Obregon, Mexico for providing the data sets. Ms. J. M. KiUeen and Ms. S. D. Roepke helped in preparing the manuscript.

References Asrar, G., Fuchs, M., Kanemasu, E. T., and Hat_field, J. L. (1984), Estimating absorbed photosynthetic radiation and lea/ area index from spectral reflectance in wheat, Agron. I. 76:300-306. Daughtry, C. S. T., Bauer, M. E., Crecelius, D. W., and Hixson, M. M. (1980), Effects of management practices on reflectance of spring wheat canopies, Agron. ]. 72:1055-1060. Dave, J. v. (1980), Effect of atmospheric conditions on remote sensing of vegetation parameters, Remote Sens. Environ. 10:87-99. Fuchs, M., Asrar, G., Kanemasu, E. T., and Hipps, L. E. (1984), Lea/ area estimates from measurements of photosynthetically active radiation in wheat canopies, Agric. Forest Meteorol. 31:13-22.

SPECTRAL ESTIMATES OF LEAF AREA INDEX

Hatfield, J. L., Asrar, G., and Kanemasu, E. T. (1984a), Intercepted photosynthetically active radiation estimated by spectral reflectance, Remote Sens. Environ. 14:65-75. Hatfield, J. L., Kanemasu, E. T., Asrar, G., Jackson, R. D., Pinter, R. D., Jr., Reginato, R. J., and Idso, S. B. (1984b), Leaf area estimates from spectral measurements over various planting dates of wheat, Int. J. Remote Sons. 5:(forthcoming). Kollenkark, J. C., Daughtry, C. S. T., Bauer, M. E., and Housely, T. L. (1982a), Influence of cultural practices on the reflectance characteristics of soybean canopies, Agron. J. 74:751-758. Kollenkark, J. C., Vanderbilt, V. C., Daughtry, C. S. T., and Bauer, M. E. (1982b), In-

11

fluence of solar illumination angle on soybean canopy reflection, Appl. Opt. 21:1179-1184. Pinter, P. J., Jr., Jackson, R. D., Idso, S. B., and Reginato, R. J. (1983), Diurnal patterns of wheat spectral reflectance, IEEE Trans. Geosci. Remote Sens. 21:156-163. Slater, P. N., and Jackson, R. D. (1982), Atmospheric effects on radiation reflected from soil and vegetation as measured by orbital sensors using various scanning directions, Appl. Opt. 21:3923-3931. Walraven, R. (1978), Calculating the position of the sun, Solar Energy 20:393-397.

Received 29 February 1984, revised 20 ]une 1984.