- Email: [email protected]

289

Optimal Spatial Sampling Techniques for Ground Truth Data in Microwave Remote Sensing of Soil Moisture R. G. S. RAO AND F. T. ULABY University of Kansas Center for Research, Inc., Remote Sensing Laboratory, Lawrence, Kansas 60045 Microwave remote sensing of soil moisture is currently being explored by a series of both active and passive experiments with the sensor output then related to soil moisture laboratory measurements made on field-collected samples taken at the time of microwave data acquisition. In addition to diurnal variation, soil moisture varies widely with surface location and depth; furthermore, the cost of sample extraction increases markedly with depth. Therefore it is desirable to identify sampling techniques which give acceptable statistical validity while minimizing the effort involved in sample extraction. Data from an extensive soil sample collection program carried out in April, 1976 near Perry-Topeka, Kansas were used as input to five statistical sampling tests based on both simple random sampling and stratified sampling. In addition to the relation between desired sample size and depth, the tests were applied to various field cell sizes corresponding to the resolution cell of the microwave remote sensor; field cells ranging from 2.5 to 40 acres in size were considered, and depths to 45 cm were included. If the total number of samples taken during a ground truth mission can be prespecified, then stratified sampling based on optimal allocation is to be preferred; otherwise, simple random sampling should be used. As an example, in the top 0-1 cm layer of a 20-acre field, 35 soil samples would be required using simple random sampling whereas only 19 samples would be required using optimal allocation stratified sampling. This reduction in the number of samples is a consequence of the higher, weighting assigned to the surface layers which exhibit the greatest soil moisture variability with spatial position.

1. Introduction It has been demonstrated in the past several years that microwave remote sensors can be used to measure soil moisture content in the top layers o f soil. Both active approaches (Ulaby and Batlivala, 1976) and passive techniques (Schmugge et al., 1974) are useful in the 3-21 cm wavelength bands, b u t in all cases the sensor o u t p u t in any given experiment must be related to an adequate data base of ground truth. The most important ground truth measurement for such experiments is that of soil moisture content over the resolution cell o f the sensor and for various soil depths.

In a recent paper, Jackson et al. (1976) have examined the temporal variability of soil moisture in Avondale loam and have found that even though the diurnal variation depends strongly on depth, irrigation times and other factors, the 24-hour average volumetric water content can be accurately found b y sampling a b o u t one hour before solar noon. This finding provides a considerable increase in sampling e c o n o m y insofar as temporal variations are concerned. There is also wide variability in spatial distribution of soil moisture, b o t h with surficial location and with depth. In this paper we discuss optimal sampling techniques for obtaining accurate spatial averages of soil moisture with the mini©Elsevier North-Holland, Inc., 1977

290 mum number of samples. Since the time and effort required to obtain a soil sample increases markedly with depth, we wish to know how many samples are required for various sensor resolution cells and for various depths.

2. Experimental In April, 1976 the Joint Soil Moisture Experiment (JSME) 1 group conducted a carefully coordinated soil moisture remote sensing experiment involving airborne and ground-based multifrequency scatterometers over a test site in the vicinity of Perry-Topeka, Kansas. An extensive collection of soil samples was acquired from twenty nine 40-acre fields of which twenty four were bare and five were planted in wheat. The soil samples were acquired from 19 predetermined locations in each field, at coordinates shown in Fig. 1. The number of sampling location was based on the criterion of acquiring the soil samples within a time window centered on the aircraft overflight time. Gravimetric moisture content was first measured by weighing a sample before and after drying it in a microwave oven for all samples from the eight sampling depths: 0-1 cm, 2-5 cm, 5-9 cm, 9-15 cm, 0-15 cm, 15-30 cm and 3 0 4 5 cm. To convert gravimetric moisture content to volumetric moisture content, bulk density measurements were made from four 1JSME is a NASA sponsored research team whose membership includes researchers from NASA/JSC, NASA/GSFC, University of Arkansas, Universityof Kansas and Texas A & M University.

R.G.S. RAO AND FAWWAZT. ULABY locations and seven depths as shown in Fig. 1. The soil types involved were silt clay loam, clay loam, clay and silt loam with predominantly silty texture in most of the fields. The variability in soil moisture may be conveniently expressed in terms of the coefficient of variation (}/defined as the ratio of the standard deviation to the mean value, as computed from all the samples within a depth / of a given field i. The frequency distribution of Ci] is plotted in Fig. 2 for three soil depths: 0-1 cm, 5-9 cm and 9-15 cm. It is evident that the spatial soil moisture variability decreases with depth.

3. Statistical Analysis Several statistical analysis techniques have been used with the data base in order to arrive at an optimal sampling stratagem which could be used for future ground truth missions. These can be divided into two basic procedures: random sampling and stratified sampling. Random sampling considers each soil depth individually and assumes that the soil moisture values are normally distributed within that depth. For the samples from the April 1976 mission, measured data indicated that the mean soil moisture within a given depth of a specified field could fluctuate +10% from the observed mean. Both u-test and t-test graphical procedures and analytical techniques have been used for estimating the number of samples N~ required to detect a + 10% change in the observed mean moisture value at depth ] in field i.

SOIL MOISTURE SAMPLING TECHNIQUES

291

19 Location Sampling Grid 1320'

t1

"1

- 260, -I--200,-t- ~ , + ~ " 1 - 20°'-I- 260, I

I I

210'

-F

150'

I¢O'

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3

6

0

0

2

5

"~ 1

o

~

o

112

16

19

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150'

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~

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15

18

V --1--

i50'

-I-, 150'

4

v

-~^ Is

o

14

t,I

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I I

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Seven soil moisture sample locations; five samples each (0-1 cm, 1-2 cm, 2-5 cm, 5-9 cm, 9-15 cm). o

Twelvesoil moisture sample locations; eight samples each (0-1 cm, 1-2 cm, 2-5 cm, 5-9 cm, 9-15 cm, 0-15 cm, 15-30 cm, 30-45 cm).

X ._ Four bulk density sample locations; seven samples each , (0-1 cm, 1-2 cm, 2-5 cm, 5-9 cm, 9-15 cm, 15-30 cm, 30-45 cm). ----

Aircraft ground track

FIG. 1. JSME Lawrence-Topeka site, 19 location sampling grid.

T h e n u m b e r o f samples for the ] t h d e p t h and the ith field can be calculated using the u-test (Hald, 1952) and is given by:

NiT= u ;

- ~ / 2 sij'/L~ij - .~ol ~

where U~ _ ~/2 = the value o f standard normal distribution at a probability 1 - a/2,

Si/

standard deviation o f the observed soil m o i s t u r e values

292

R.G.S. RAO AND FAWWAZ T. ULABY

!lt " 0.35 I-'1 I'-I O-lcm Depth I-"1 I I'~ [ r--1 Volumetric Data

~, 0.2 =

IHIIII

I I,(a)I I 1-71

~0.0

'

I t - 0.15 ,-~'0"3

5-9cm Depth Volumetric Data ~- 0.15

o.31

~.

o,o

1"71/ (b)

0.3

,

t

J

- - 0.12

¢,3 C

9-15cm Depth Volumetric Data E- 0.12

gO. 1 -~ 0.

n , 20.4 I

0.1 0.2 0.3 0.4 0.5 0.6 Standard Deviation to Mean Ratio (c) FIG. 2. Frequency distribution of standard deviation to mean ratio plotted for soil moisture samples from (a) top 1 cm, (b) 5-9 cm, and (c) 9-15 cm. O"/).0

in the field, =

Xo

jth

d e p t h o f the ith

m e a n value o f the observed soil m o i s t u r e values in the / t h d e p t h o f the ith field,

= hypothesized

mean

value.

T w o o f the i m p o r t a n t analytical procedures discussed in this p a p e r are based o n the u-test. The first o f these considers o n l y the samples within a given field i and gives the n u m b e r o f samples N ~ which will, with a p r o b a b i l i t y o f 0.95, allow a d e t e c t i o n o f + 10% change in the

SOIL MOISTURE SAMPLING TECHNIQUES

293

observed mean moisture__value. Thus with ot = 0.05, -~o = 1.1 Xij or 0.9 Xij, and U I _ ~/2 = 1.645, the above equation reduces to:

1

NF

NF

i=1

NF

N~ = 270.7

Cij2 ,

(Test No. 1)

(1)

where

Nj=

Z i=1

N~j,

Nij = number of observed soil moisture

Cij = the coefficient o f variation o f soil moisture values in t h e / t h depth of the ith field

values in the jth depth o f the ith field.

N F = number of fields.

= sq / Yq. The number of samples N/* for the j t h depth can now be calculated as the average o f Ni/over all fields i and is given b y

Nj*=

1 NF Z N/~. N F i=1

Thus the number o f samples for each depth can be calculated using the coefficient of variation Cij (Test No. 1) or the average coefficient of variation C! (Test No. 2). A third random sampling approach uses a skewness coefficient (Cochran, 1963) so that N~ = 25(Gq) 2,

In the second analytical u-test approach, the soil moisture value from the jth depth of all fields is considered so that N~ = 270.7

C/2 ,

(Test No. 2)

(2)

(3)

where the skewness coefficient given by

Gi] is

Nil ~:

where

(x o -

xq) 3

GO = i=l

Cj = average coefficient of variation of soil moisture values in the j t h depth

= s i / x i, NF

and

(Test No. 3)

Sj 2 = ~I/(Nj-NF)~ iZ__l (Nq- 1)Sq 2, ~ J

(N o -

(4) 1)(So) 3

Stratified sampling procedures, unlike simple random sampling, recognize the dependency o f soil moisture at one layer on the soil moisture in adjacent layers, and that since moisture variability is greatest in the surface layers a greater weighting should be assigned to those

294

R.G.S. RAO AND FAWWAZT. ULABY

layers. Each layer is assigned a weighting factor Wij normalized such that

N/) j=l

for all i

Wo=l,

(5)

where ND is the number of depths. We now consider any given field i to be represented by a single population whose sample means and variances are given by

Since the soil moisture variability decreases with depth ] (when averaged over all fields), this allocation of weights has the effect of increasing the relative importance of those surface layers where the variability is greatest. Within this stratified sampling framework, there are two convenient ways of estimating the required number of samples N~'. The first of these makes use of proportional sampling, i.e.,

N~ = WqNi,

AiD Xi = ~, Wi! Xii, j=l

vat(X0

=

(6)

Nv (S;j)2 /~l= (w;J)2 Nij ,

(7)

a~

(8)

N~ aO 1=1 where

1 aij = ~

No ~"

X o j=l

Xij'"

(9)

(10)

where N i is the total number of samples for the ith field, all depths considered. The second technique uses optimal allocation (Hald, 1952) according to tP qjV ¼ Wq S q N i ~.

* = Nq ND

where NiT is the sample size for depth / and field i. Therefore the variance of the sample mean depends on the variation within each stratum while the variation between strata is eliminated. The weights were chosen for the problem at hand as follows:

(Test No. 4)

,

(Test No. 5)

j=l (11) where Pij is the relative cost involved in extracting the samples from 'the jth stratum of the ith field. Equation (1 1) gives an optimal sample size based not only on the standard deviation of the /th stratum, but also its statistical weight (related to soil moisture variability) as well as the economic and time cost of sample extraction. It is noted that if the extraction cost is independent of depth and all strata have the same variance, (11) reduces to (10). In all of the tests previously discussed, the estimated sample size is greatly in-

SOIL MOISTURE SAMPLING TECHNIQUES fluenced by the standard deviation Si/; therefore, errors in a single sample which would influence the true value of Si] need to be identified and eliminated. In the analysis procedure applied to the April, 1976 data base, these outliers were identified by three statistical tests, removed and N~ was recalculated. The N~ values were then multiplied b y a finite sample correction factor to obtain a more reliable estimate o f N~ (Cochran, 1963).

295

tion of optimal allocation sampling give adequate estimation of sample size while the results from the test involving skewness coefficient give a lower limit; those from porportional sampling give an upper limit on the sample size. These results also indicate that the Test No. 1 involving the average coefficient of variation over all fields (Test No. 2). This further dem. onstrates the importance of soil moisture variability in the top strata on a field-to field basis. The sample size required in correlating soil moisture to the o u t p u t of the remote 4. Results and Discussion microwave sensor clearly depends on the resolution cell size, or antenna footprint Five analytical procedures for estimat- on the surface. Enough samples must be ing N~/. have been given, three in connec- taken within the resolution cell to give tion with simple random sampling and some reliability in the statistics. This two with stratified sampling. Table 1 situation was simulated by subdividing shows the average number o f samples the 40-acre fields into smaller sized N~ (average of N~ values over all i) fields of 30, 20, 13, 10, 5, and 2.5 acres for a forty-acre field c o m p u t e d using with various combinations and geometanalytical Tests 1-5 (Eqs. 1, 2, 3, 10 and rical arrangements considered. For ex11) as well as estimates based on graphi- ample, four different combinations of cal u- and t-test procedures. The value of 30-acre cells and their corresponding Pi/ in Test No. 5 was assumed to be soil moisture data sets were considered equal to unity for all depths and for all as four different fields. N o w using these fields. It is evident from Table 1 that the four different "fields" in each o f the n u m b e r of required samples N~ obtained twenty nine original fields, the average from Tests No. 1, 2 and 5 are close to number of samples N ~ for each 30-acre those obtained by averaging the t- and cell was calculated for all depths j. A u-test results. Test No. 3 results in smaller similar exercise was carried out for other estimates of N~ while larger estimates of sub-cell sizes, with the results shown in N : are obtained from Test No. 4, partic- Figs. 3-6. 1 ularly for the top strata. However the It is seen from these graphs that the results from Tests No. 3 and 4 can be re- results from Tests No. 1 and 2 are similar garded as the respective lower and upper and that the results from Tests No. 4 and limits on the sample size requirements. 5 are similar. There is considerable reducThese results indicate that sample size tion in sample size from Tests No. 4 and estimated by using coefficient of varia- 5 for all depths as the resolution cell size

296

R.G.S. R A O A N D F A W W A Z T. U L A B Y

~

'~

'~

'~ ~ o .~ 6

~.~

~

c0

< 0

o U~

<

~o ~ ~,~ ~ -

o"

~.g

~

em

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E

~o~O~ ~ m

~

SOIL MOISTURE SAMPLING TECHNIQUES

40,

297

•

40 Acres 30 Acres • 20 Acres 13 Acres o 10, 5, 2 1/2 Acres •

25 o'Z

20

v-,

15

10

0 k

0-I

I-2

~

2-5

m

5-9 9-15 Depth (cm)

l

15-30 30-45

0-15

FIG. 3. Sample size requirements for various depths and for various resolution ceils using the agerage coefficient of variation (Test No. 1). is decreased, but a very significant reduction with cell size is obtained for the top strata. For greater depths all the tests give approximately the same sample size. Therefore stratified sampling procedures (Tests No. 4 and 5) allocate the sample size in a better way for different depths and for different resolution cell sizes than does the simple random sampling procedures (Tests No. 1 and 2). We have n o w related the sample size to depth and cell size in a way which not only is help-

ful in the collection of ground truth but also in the statistical analysis of the remote sensor response (e.g., brightness temperature of scattering cross-section) to the ground truth data. Stratified sampling procedures have been shown to give a better sample allocation than simple random Sampling procedures, when estimating sample size variations with resolution celt size. Stratified sampling procedures require a knowledge of the total number of

298

R.G.S. RAO AND FAWWAZ T. ULABY

• • •

40 Acres 30 Acres 20 Acres 13 Acres o 10 Acres m 5 Acres

ill

"

2 ll2Acres

~oa2 0 E

Y' 15

10

0 L

0-1

I

I

i

1-2

2-5

5-9

I

9-15

I

I

15-30 30-45

l

0-15

Depth (cm) FIG. 4. S a m p l e size r e q u i r e m e n t s for various d e p t h s and for various r e s o l u t i o n cells using the average coefficient of v a r i a t i o n

(Test No. 2).

samples (Ni) to be taken in a field for all depths. This can usually be estimated from a knowledge o f the available time and number of people involved in extracting soil samples in a given ground truth mission. If it is not possible to prespecify the total num be r o f samples (N i) then simple random sampling should be used.

5. Final R e c o m m e n d a t i o n s The final recom m endat i ons on sample size requirements are based on the results previously discussed and are summarized in Table 2. For each depth and for each cell size two sample sizes are recomm ended; the one without parentheses was selected from simple random samp-

SOIL MOISTURE SAMPLING TECHNIQUES

299

,5f

• • •

4O

o o "

40 Acres 30 Acres 20 Acres 13 Acres 1O Acres 5 Acres 2 1/2 Acres

3O Z tt~

a, 25 E

20

10

0 I

0-1

I

I

I

I

1-2

2-5

5-9

9-15

I

I

15-30 30-45

0-15

Depth (cm)

FIG. 5. Sample size requirements for various depths and for various resolution cells using stratified sampling (proportional sampling).

ling procedures corresponding to optimal allocation (Test No. 5). It can be seen from Table 2 that the number of samples is larger in random sampling as compared with stratified random sampling, especially for the top strata; furthermore, the number of samples required using random sampling does not significantly decrease with smaller sized resolution cells.

6. Conclusions In this paper we have considered the sample size required to accurately predict the soil moisture at various depths and for cell sizes ranging from 2½ acres to 40 acres. Both simple random sampling and stratified sampling procedures have been used to arrive at a set ofrecom-

300

R.G.S. RAO AND FAWWAZT. ULABY

4 0-[

•

• •

35

~t 13 Acres o o

30

z" ' -

®

40 Acres 30 Acres 20 Acres

10 Acres 5 Acres cres

25

20'

E

L'

5 0 0-I

I-2

2-5

5-9 9-15 Depth (cm)

15-30 30-45

0-15

FIG. 6. Sample size requirements for various depths and for various resolution cells using stratified sampling (optimal allocation).

mended sample sizes for each depth and for each cell size. On the basis of the results presented, the following conclusions can be made: 1. The number of samples required decreases with increasing depth. 2. If the total number of samples (Ni) cannot be prespecified or the moisture in only one single layer is of interest, then a simple random sampling procedure (Test No. 1) should be used which

is based on the observed mean and standard deviation for data from a single field. If the total number of samples can be prespecified (as, for example, by knowing the time and number o f people available to carry out the ground truth collection)and the objective is to measure the soil moisture profile with depth, then stratified random sampling based on optimal allocation (Test No. 5) should be used.

SOIL MOISTURE SAMPLING TECHNIQUES

301

TABLE 2 Final Recommendations on the Sample Size Requirements for Various Depths and Various Resolution Cell Sizes Depth (cm)

40 acre

30 acre

20 acre

13 acre

10 acre

0-1 1-2 2-5 5-9 9-15 0-15 15-30 30-45

36 24 13 18 6 8 6 6

36 24 14 11 9 10 7 7

35 23 12 8 8 9 5 5

32 22 12 8 6 8 5 5

32 22 11 6 4 6 3 2

(31) (23) (12) (12) (10) (12) (8) (8)

(27) (22) (11) (11) (9) (10) (7) (7)

(19) (18) (10) (8) (8) (9) (5) (5)

This is based on both statistical weighting factors (derived from a knowledge of soil moisture variability in each layer) and economic weighting factors (derived from a knowledge of the effort involved in extracting samples from various depths). 3. Decreasing the resolution cell size results in fairly large decreases in sample sizes when using stratified sampling procedures whereas only a modest decrease is obtained in simple random sampling procedures. Research was supported by N A S A Johnson Space Center contract N A S 9-14052. The authors wish to thank Dr. Keith Carver f o r reviewing this paper.

(19) (15) (10) (8) (6) (8) (5) (4)

(11) (9) (5) (4) (3) (4) (2) (2)

5 acre 32 22 11 6 4 5 2 2

(10) (8) (5) (4) (3) (4) (2) (2)

2½ acre 32 22 11 6 3 4 2 2

(7) (6) (4) (3) (2) (3) (2) (2)

References

Cochran, W. G., 1963, Sampling Techniques, John Wiley & Sons, Inc., New York Hald, A., 1952, Statistical Theory with Engineering Applications, John Wiley & Sons, Inc., New York. Jackson, R. D., Reginato, R. J., and Idso, S . B . (1976), Timing of ground truth acquisition during remote assessment of soil-water content, Re. Sens. Environ., 4,249-255. Schmugge, T., Gloersen, P., Wilheit, T., and Geiger, F. (1974), Remote sensing of soil moisture with microwave radiometers, J. Geophys. Res. 79, 317-323. Ulaby, F. T., and Batlivala, P. P. (1976), Optimum radar parameters for mapping soil moisture, 1EEE Trans. Geosci. Electronics GE-14, 81-93. Received 6 December 1976; revised 21 March 1977.