Calibration and validation of a general infiltration model

Calibration and validation of a generalin®ltration model

Surendra Kumar Mishra,1 Shashi Ranjan Kumar1 and Vijay P. Singh2,*1National Institute of Hydrology, Roorkee-247 667, U.P., India

2Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803-6405, USA

Abstract:A general in®ltrationmodel proposed by Singh andYu (1990) was calibrated and validated using a split samplingapproach for 191 sets of in®ltration data observed in the states ofMinnesota andGeorgia in the USA. Of the ®ve

model parameters, fc (the ®nal in®ltration rate),So (the available storage space) and exponent `n' were found to bemore predictable than the other two parameters: m (exponent) and a (proportionality factor). A criticalexamination of the general model revealed that it is related to the Soil Conservation Service (1956) curve number

(SCS-CN)method and its parameterSo is equivalent to the potential maximum retention of the SCS-CNmethodand is, in turn, found to be a function of soil sorptivity and hydraulic conductivity. The general model was foundto describe in®ltration rate with time varying curve number. Copyright # 1999 John Wiley & Sons, Ltd.

KEY WORDS in®ltration model; calibration; validation; split sampling; curve number; SCS-CN method;sorptivity; hydraulic conductivity

INTRODUCTION

In®ltration is an integral part of the rainfall±runo� process whose modelling is required for planning anddesign of water resource systems. A number of in®ltration models have been developed over the years. Thesemodels can be classi®ed as physically based models which are based on soil physics, conceptual models basedon systems theory and empirical models based on ®eld or laboratory experiments. Examples of physicallybased models are those of Philip (1957, 1969), Mein and Larson (1971), Morel-Seytoux and Khanji (1974),Parlange (1974), and Smith and Parlange (1978). The models of Green and Ampt (1911), Kostiakov (1932),Horton (1938), Holtan (1961) and Overton (1964), Dooge (1973), Smith (1972), Collis-Groege (1977) andSingh and Yu (1990) are conceptual. The Soil Conservation Service (1956) curve number (SCS-CN)-basedin®ltration model (Mishra and Singh, 1998), the Hydrologic Engineering Center (HEC, 1981) loss model, thehydrograph model (Dunin, 1969) and the Snyder (1971) model are examples of empirical models. Conceptualmodels have been derived using simple hypotheses and veri®ed with the results of ®eld or laboratoryexperimentation. These models have gained much popularity in recent years (Singh and Yu, 1990).

The usefulness of a model in ®eld applications depends on how precisely the parameters of the model canbe predicted from the information on physically measurable soil characteristics and ®eld conditions. It is,therefore, useful to calibrate and validate a model which is more general in form and broadly represents awide variety of available in®ltration models. The Singh and Yu (1990) conceptual model, derived using asystems approach, is one such general in®ltration model, for it specializes in the models of Horton,Kostiakov, Overton, Green and Ampt and Philip, among others. The objective of this paper is to calibrateand validate this model for various soil types.

CCC 0885±6087/99/111691±28$17.50 Received 12 June 1998Copyright # 1999 John Wiley & Sons, Ltd. Revised 29 October 1998

Accepted 25 November 1998

HYDROLOGICAL PROCESSESHydrol. Process. 13, 1691±1718 (1999)

*Correspondence to: Professor Singh, Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge,LA 70803-6405, USA.

FEATURES OF THE GENERAL SINGH±YU MODEL

The general in®ltration model is based on three fundamental hypotheses. (1) The excess in®ltration ( f ÿ fc)at any time is directly proportional to an mth (m 6� 0) power of the available storage space in the soilcolumn. (2) The excess in®ltration at any time is inversely proportional to an nth (n 6� 0) power of thecumulative in®ltration up to that time. (3) After the soil is ®lled up, i.e. S � 0 (Figure 1), in®ltration is thesame as the ®nal in®ltration. These hypotheses, relating the rate of in®ltration to the available and non-available storage spaces, are analogous to tra�c ¯ow movement on a highway. Here the vehicle speedcorresponds to the rate of in®ltration, the distance to be covered for reaching the destination corresponds tothe available storage space, the distance already covered corresponds to the cumulative in®ltration and thespeed of the vehicle within the approach of the destination city corresponds to the ®nal rate of in®ltration.

Thus, the ®rst hypothesis can be explained as follows. If the distance to be covered is large, the driver willhave a tendency to drive the vehicle as fast as possible so as to reach the destination at the earliest possibletime, and vice versa. This is true in many situations and holds true with the rate of in®ltration. Experimentaldata for in®ltration exhibit such a phenomenon. The second hypothesis is actually a complement to the ®rsthypothesis. As the distance covered by the vehicle, or the magnitude of in®ltrated water, is large, theremaining distance to be covered, or the available storage space, will be less and, therefore, the secondhypothesis will hold. The third hypothesis corresponds to the situation when the driver is within theproximity of the destination city where the speed of the vehicle will be at a bare minimum.

It should be made clear that the distance used in the analogy of tra�c ¯ow is related to the depth of the soilhorizon in in®ltration. Although the depth of the soil horizon does not appear explicitly in the generalin®ltration model, it is considered through parameter S, which is the potential storage space available formoisture retention. Singh and Yu (1990) found the initial value of S;So as 0

.25 m4So4 0.45 m. Given thevalue of soil porosity, the depth of the soil horizon is obtained. For a porosity of, say, 0.35, the depth of soilin the case of Singh and Yu will be 0.71±1.29 m, which is consistent with the soil horizon for agriculturalcrops. This type of explanation was also given by Overton (1964). Another way to look at the meaning ofdepth through S is by recalling the SCS-CN parameter S de®ned in inches as S � �1000=CN� ÿ 10, whereCN is the curve number depending on the soil type, land use, hydrological conditions and antecedent soilmoisture conditions. For practical cases CN varies from 50 to 100, the value of S will then vary from 0 to10 inches. This is also consistent with what was obtained earlier.

Figure 1. Components of the general in®ltration loss model

Copyright # 1999 John Wiley & Sons, Ltd. HYDROLOGICAL PROCESSES, VOL. 13, 1691±1718 (1999)

1692 S. K. MISHRA, S. R. KUMAR AND V. P. SINGH

The general model calibrated here is a conceptual model for macroscopic treatment of in®ltration. Inpractical situations, the scale of heterogenieties in both space and time is large and it is therefore reasonedthat a systemic approach capable of capturing the salient features of the in®ltration hydrology withoutinclusion of microscopic details either of soil water dynamics or soil systems or both is desirable. Thisapproach has been a mainstay in systems hydrology, especially in the area of rainfall±runo� modelling andconstitutes the rationale for the general in®ltration model developed by Singh and Yu (1990).

Mathematically, the general model of Singh and Yu (1990) is expressed as

f�t� � fc �a�S�t��m�So ÿ S�t��n �1�

where f(t) is the in®ltration rate (LTÿ1) at time t; fc is the ®nal in®ltration rate (at t! 1 ) (LTÿ1); S(t) is theavailable storage for water retention in a soil column at time t (L); So is the potential storage space availablefor moisture retention in the soil column (L); a, m and n are, respectively, the coe�cient and exponents ofthe variables S(t) and �So ÿ S�t��. Thus, the general model consists of ®ve parameters, namely, fc;So; a;mand n.

The model parameters have physical meaning as described here. Parameter `m' is the slope of the lineplotting log� f ÿ fc� against log(S), and re¯ects the degree of non-linearity of the relationship between � f ÿ fc�and the available space in the soil. Parameter `a' is the value of the intercept of the line. Exponent `n' is theslope of the line plotting log � f ÿ fc� against log�So ÿ S� or log(F) (accumulated in®ltration) and is anindication of the degree of non-linearity of the relationship between � f ÿ fc� and �So ÿ S�. If two soils are tohave similar in®ltration patterns, they will have the same values of `m' and `n'. However, soil heterogeneityseldom allows two soils to be exactly identical and, therefore, leads to variation in the parameter values fromone soil to another. The range of variation of these parameters with varying soil type and their physicalsigni®cance have not yet been reported in literature.

Singh and Yu (1990) hypothesized that the two parameters fc and So could be determined from theavailable in®ltration data and soil properties, including soil porosity and initial soil moisture condition. Thestrategy used for calibration of the remaining three parameters of their conceptual model, `a', `m' and `n', wasbased on the least-squares approach. An interpretation of potential storage space permits us to relate thegeneral model with the popular Soil Conservation Service (SCS) curve number (CN) model (SCS, 1956).This relationship is elaborated below.

Relationship with the SCS±CN model

Parameter So is shown to be equivalent to the potential maximum retention of the SCS±CN model(Mishra and Singh, 1998). Mathematically, it is expressed as

So �fo ÿ fc

k�2�

where fo is the initial rate of in®ltration (LTÿ1) and k is the in®ltration decay factor (Tÿ1). Parameters fo andk are analogous to those of the Horton (1938) model. Collis-George (1977) showed that the right-hand sideof Equation (2) was equal to �p=4��S2

s=K�, where Ss is sorptivity (Philip, 1957) of the soil and K is hydraulicconductivity of the soil. Therefore, Equation (2) becomes

So �fo ÿ fc

k� p

4

S2s

K�3�


INFILTRATION MODEL CALIBRATION AND VALIDATION 1693

Equation (3) implies that the potential maximum retention of the SCS (1956) curve number model, So , is aquantity that can be derived from the physical properties of a soil. Furthermore, the potential maximumretention, So , is related to CN by

So �1000

CNo

ÿ 10 �4�

where CNo represents the initial curve number. From Figure 1 it is evident that So is the maximumstorage space available at a time equal to zero. As time increases, S decreases. It implies that CN increaseswith time. Indirectly, the general in®ltration model relates in®ltration to CN. If the general equation iswritten in terms of CN, the resulting equation will be dimensionally homogeneous, as CN is a dimensionlessnumber and parameter `a' will take on the dimensions of Tÿ1. This can be demonstrated as follows

Referring to Figure 1, one can express

S�t� � So ÿ F�t� �5�

or

F�t� � So ÿ S�t� �6�

The general form of Equation (4) can be written as (Ponce, 1989; Misra and Singh, 1998)

S�t� � Sabs

100

CNt

ÿ 1

� ��7�

where CNt is the time-varying curve number. Substitution of Equations (4) and (7) into Equation (6) yields

F�t� � 100 *Sabs

1

CNo

ÿ 1

CNt

� ��8�

where Sabs is de®ned as the maximum potential retention (L). Equation (8) is the relationship betweencumulative in®ltration and time-varying CN (or CNt). Equations (4) and (7) can be substituted inEquation (1) of the general model to yield in®ltration rate

f � fc �aSmÿn

abs

102n

� � 100

CNt

ÿ 1

� �m

1

CNo

ÿ 1

CNt

� �n �9�

To make Equation (9) dimensionally homogeneous it is necessary that the exponent of Sabs , m ÿ n, be equalto 1 so as to let `a' assume the dimensions of (Tÿ1). Therefore.

m � n � 1 �10�

Substitution of Equation (10) into Equation (9) leads to

f � fc �aSabs

102n

� � 100

CNt

ÿ 1

� �n�1

1

CNo

ÿ 1

CNt

� �n �11�



or

f � fc � aSabs

CNo

100

� �n �100 ÿ CNt�n�1CNt�CNt ÿ CNo�n

�11a�

which is another form of the general equation. To show the variation of CN (or S) with time, Singh and Yu(1990) visualized several relationships which are shown in Table I. Its variation can also be ascertained fromEquation (8) since the variation of F(t) is known from ®eld data.

CALIBRATION OF MODEL PARAMETERS

In®ltration data

In®ltration tests conducted in Minnesota and Georgia in the USA were utilized in this study. In all, 191in®ltration data sets were used. Of these, 147 data sets were used for calibration and 44 data sets forvalidation. Table II shows pertinent characteristics of the in®ltration data that were generated from severallaboratory tests used by Mein and Larson (1971). For the ®rst three soils in the table, the data weredesorption data, but because in®ltration is an absorption process it is important to use the absorption orwetting curve of the soil characteristic curves. A reasonable estimate of the wetting curve can be obtained bydividing the drying curve by 1.6 (Mein and Larson, 1971). However, the data in their primitive form wereused in this study. The in®ltration data represented a range of soils. For example, Columbia Sandy Loamand Ida Silt Loam represent two completely di�erent soil types.

The hydrological soil groups for the in®ltration data observed in Georgia are shown in Table III. Thesedata are from the Tifton Upland physiographic region of the Georgia Coastal Plain. These tests were

Table I. Special cases of the general model [Source: Singh and Yu (1990)]

Case Parameters Resulting model

m n fc

0 m n fc General model1 0 1 fc Philip two-term or Green±Ampt model

2 0 4ÿ1 0 Kostiakov model3 0 4ÿ1 fc Modi®ed Kostiakov model

4 1 0 fc Horton model5 2 0 fc Overton Model6 40 0 fc Holton model

7 1 1 fc

Table II. In®ltration data for Minnesota, USA

Sl. no. Soil Reference Ks (cm/s) Porosity No. of tests

1 Plain®eld sand (PFS)(disturbed sample)

Black et al. (1969) 3.44� 10ÿ3 0.477 12

2 Columbia sandy loam (CSL)(undisturbed sample)

Laliberte et al. (1966) 1.39� 10ÿ3 0.518 36

3 Guelph loam (GL)(air dried, sieved)

Elrick and Bowman(1964)

3.67� 10ÿ4 0.523 12

4 Ida silt loam (ISL)(undisturbed sample)

Green (1962) 2.92� 10ÿ5 0.530 42

5 Yolo light clay (YLC)(disturbed sample)

Moore (1939) 1.23� 10ÿ5 0.499 12



conducted near Tifton in south-central Georgia and in the vicinity of the Little River ExperimentalWatershed (Figure 2). The surface soils of the watershed exhibit distinct A and B horizons, with textureranging from a loamy sand to a sandy clay loam in the top 91 to 122 cm, except for Kershaw, which is acoarse sand. At the 91±122 cm depth there is usually a semi-permeable clay layer. Nearly 90% of the area hasa slope of 5% or less, but along the sides of some valleys, the slope ranges from 5 to 15%. The Little RiverExperimental Watershed comprises 20 soil series and in®ltration tests were conducted on 11 of these series,accounting for 95% of the watershed area and representing a wide range of conditions. Natural conditionswere preserved at each in®ltration site, except where vegetation was tall; it was then cut to 15 to 20 cm height.Vegetation provided between 50 and 80% of the ground cover. The in®ltration data were published by theUSDA ARS (1976)

Data sets of each group for Minnesota and Georgia (Tables II and III) were split so that one group of datawas used for calibration and the other group for validation. For the calibration data sets, the parameterswere computed using a constrained least-squares approach based on the e�ciency as a goodness-of-®tcriterion.

Goodness-of-®t

The criterion for goodness-of-®t was evaluated by determining the e�ciency as:

Efficiency � �1 ÿ D1=Do� � 100 �12�

where D1 is the sum of squares of deviations between computed and observed data and Do is the modelvariance, which is the sum of the squares of deviations of the observed data about the observed mean.Expressed mathematically

D1 � S�Yo ÿ Y�2 �13�

Do � S�Yo ÿ �Y�2 �14�

where Yo are the observed data, Y, with `^' and `�' stand for computed data and the mean of the observeddata, respectively. The e�ciency varies on a scale of 0 to 100. It can also assume a negative value if D1 4Do,implying that the variance in the observed and computed in®ltration values is greater than the modelvariance. In such a case, the mean of the observed data ®ts better than the applied model. An e�ciency of100 implies that the computed values are the same as the observed ones, which is the perfect ®t.

Table III. In®ltration data for Georgia, USA

Sl. no. Soil Reference Hydrological soil group No. of tests

1 Alapaha loamy sand (ALS) ARS (1976) D 42 Carnegie sandy loam (CASL) ARS (1976) C 43 Cowarts loam sand (COWLS) ARS (1976) C 194 Dothan loamy sand (DLS) ARS (1976) B 45 Fuquay loamy sand (FLS) ARS (1976) B 46 Fuquay pebbly loamy sand (FPLS) ARS (1976) B 67 Kershaw coarse sand (KCS) ARS (1976) A 38 Lee®eld loamy sand (LLS) ARS (1976) C 49 Robertsdale loamy sand (RSL) ARS (1976) C 410 Stilson loamy sand (SLS) ARS (1976) B 411 Tifton loamy sand (TLS) ARS (1976) B 912 Troup sand (TS) ARS (1976) A 12



Initial parameter estimates

The parameters of the general model were estimated using the non-linear Marquardt algorithm of theleast-squares procedure of the Statistical Analysis System (SAS, 1988). The initial guesses for the threeparameters `a', `m' and `n' were arbitrarily chosen as zero in all cases. The other two parameters, fc and So ,were taken to be near but greater than zero, i.e. of the order of 0.0001. The range was, however, ®xed by trialand error, and the lower and upper limits were described by ®rst making a few runs. If the parameterassumed a value equal to its upper or lower limit, the limits were extended accordingly on either side for the

Figure 2. Location of Georgia in®ltration sites



next run, with the general restriction that fc 5 0 and So 5 0. If the subsequent runs produced estimateswithin the prescribed range, the parameter estimates were assumed to be globally optimal. Approximately,three to ®ve runs were needed for computing the ®nal estimates.

Model calibration

In all, 147 data sets were used for model calibration. Each test on a soil sample re¯ects soil characteristics,including heterogeneity and anisotropy. This set of calibrated parameter values of the model for a given soilis a unique combination re¯ecting these soil properties. This means that representative values of theparameters for each soil type de®ned on the basis of the usual nomenclature of sand, silt and clay and theirmixes need to be determined. Figure 3 describes various types of soils and the standard proportions of thesethree basic ingredients. The hydrological soil classi®cation is, however, based on the hydraulic conductivityof soils. Table IV shows various soil types and their in®ltration rate values. Apparently, as the in®ltrationrate increases, the soil type shifts from clay to silt and silt to sand.

Since the performance of the general model varies from one test to the other, because of the varying levelsof soil heterogeneity, it is instructive to evaluate the model performance, using the aforementioned e�ciencycriterion, before the parameters are calibrated. The performance of the model on Minnesota and Georgia

Figure 3. Soil textural classi®cation (triangular) as per US Public Road Administration

Table IV. Soil group classi®cation [Source: Singh (1991)]

Group Soil characteristics Minimum in®ltration rate (in/h)

A Deep sand, deep loess and aggregated silts 0.30±0.45B Shallow loess and sandy loam 0.15±0.30C Clay loams, shallow sandy loam, soils in organic

content and soils usually high in clay0.05±0.15

D Soils that swell upon wetting, heavy plastic claysand certain saline soils

0±0.05



soils is shown in Figures 4 and 5, respectively. The evaluation helps describe the general workability of themodel to a type of soil on one hand and re¯ects abnormality in the behaviour of a soil to an in®ltration teston the other. For example, the general model did not satisfactorily model in®ltration for some particular setsof test data collected on Columbia sandy loam (CSL), Alapha loamy sand (ALS), Cowarts loamy sand(COWLS), Lee®eld loamy sand (LLS) and Troup sand (TS). Figure 6 shows such abnormal behaviour to aparticular test for Columbia loamy sand. However, in most cases, the model performed satisfactorily (withe�ciency greater than or equal to 95%). For Minnesota data, the above-sampled example (Figure 6) is theonly exception. In general, the model performed satisfactorily for both Minnesota and Georgia data. Sincethe available number of tests for Minnesota for a soil type is much larger than that for Georgia, theMinnesota data were used to show the variation of the computed values of the parameters with soil type.

The variation of the optimized parameters on Minnesota soils is shown in Figures 7±11. In these ®gures,the test number on the x-axis refers to the serial number by which the tests are documented by Mein andLarson (1971). In a similar manner, the tests on Georgia soils were numbered. The calibrated parametervalues are the optimized parameters for all calibration tests and are given in Table V. The calibration of the®ve parameters of the general model is discussed below.

Calibration of fc . Parameter fc in Figure 7 shows a generally decreasing trend with the type of soil. As thesoil type changes from PFS (test numbers 1±12) to CSL (test numbers 13±48) or CSL to GL (test numbers49±60) or GL to ISL (test numbers 61±102) or ISL to YLC (test numbers 103±114), fc goes on decreasingand is, therefore, fairly predictable. For PFS it is of the order of 17 cm/h, for CSL 9 cm/h, for GL 3 cm/h, forISL 0.2 cm/h and for YLC 0.1 cm/h.

Figure 4. Performance of the general model with Minnesota data



Figure 5. Performance of the general model with Georgia data

Figure 6. Modelling of an example test (no. 22) showing abnormal in®ltration behaviour



Calibration of So . The variation of parameter So is depicted in Figure 8. It shows two distinct clusterings inthe combined PFS and CSL range (tests numbers 1±48), and in the combined ISL and YLC soil range (testsnumbers 61±114). Except for the lower cluster in the PFS and CSL range and the upper one in the ISL andYLC range, parameter So also shows a fairly consistent decreasing trend with varying soil type in theaforementioned order. Here, GL acts like a transition soil. Parameter So is of the order of 35 cm for PFS andCSL, 5 cm for ISL and YLC and varies between 35 and 5 cm for GL.

Table V. Number of tests taken for calibration and validation

Soil type Calibration Validation Soil type Calibration Validation

PFS 9 3 FLS 3 1CSL 26 10 FBLS 5 1GL 9 3 KCS 2 1ISL 32 10 LLS 3 1YLC 9 3 RLS 3 1ALS 3 1 SLS 3 1CASL 3 1 TS 8 1COWLS 17 2 TLS 9 3DLS 3 1

Figure 7. Variation of fc with varying soil types

Figure 8. Variation of So with varying soil types



Calibration of à'. Parameter à' ranges from 9 to 0.005 (Figure 9). On the whole, it has a narrow range andhovers about 0.001 irrespective of the soil type.

Calibration of `m'. Parameter `m', as shown in Figure 10, is fairly constant. It assumes a value of 3.0 forPFS and CSL, and is of the order of 4.0 for GL. However, it shows a greater scatter in the ISL and YLCrange than in other soils. The values, however, range between 1.0 and 2.0.

Calibration of `n'. Parameter `n', shown in Figure 11, exhibits a more distinct and de®nitive trend. It clearlytakes on a value of 1.0 for PFS and CSL and 2.0 for ISL and YLC. A wider variation is found, however, inthe GL range (49±60).

Representative parameter values

To derive representative values of the model parameters the distribution of each of the parameters wasinvestigated. It is worth mentioning that the parameter values of the data used for validation are excludedfrom the investigated distributions. These distributions for di�erent soil groups are shown in Figures 12±19.Referring to these ®gures, the values of a, m, n, fc and So were obtained and these values are shown inTable VI. These values are considered representative values and were the ones that occurred for most of thetests. In other words, their frequency of occurrence was the highest. These will be referred to as the model

Figure 9. Variation of à' with varying soil types

Figure 10. Variation of `m' with varying soil types



values. The distributions shown in Figures 12±19 help not only to eliminate those values that are far apartfrom the modal values (or in other words, these graphs help eliminate outliers), but also to provide aconsistent way of obtaining a representative value for a parameter. It should be noted that the calibration ofthe parameters of the general model on Georgia soils are based on a very limited number of tests, as indicatedin Table V. Therefore, a judicious guess was employed for ®nally assigning a representative value to aparameter.

MODEL VALIDATION

Using the parameter values shown in Table VI, the remaining samples were simulated for validation. Itshould be emphasized that the samples considered here were not included in calibration. The validationresults are shown in Table VII and the selected simulations are depicted in Figures 20±24. In the table and

Figure 11. Variation of `n' with varying soil types

Table VI. Calibrated parameters for Minnesota and Georgia soils

Soil type Calibrated parameters of th general model

fc So a m n

PFS 17.0 18.6 0.0009 3.7 1.0CSL 9.1 36.7 0.0009 3.05 1.17GL 3.2 12.0 0.001 3.6 1.39ISL 0.18 12.1 0.008 0.8 2.0YLC 0.1 8.5 0.0015 1.75 2.0ALS 1.35 46.0 0.0009 2.0 2.5CASL 4.0 60.0 0.0016 1.9 1.01COWLS 6.5 30.0 0.032 1.0 1.5DLS 6.9 25.0 0.04 1.8 0.30FLS 2.8 35.0 0.04 1.5 0.5FBLS 6.75 50.0 0.01 1.7 0.5KCS 16.0 65.0 0.01 0.7 1.0LLS 4.0 50.0 0.001 2.5 3.0RLS 2.5 17.0 0.002 3.0 1.0SLS 4.7 25.0 0.001 2.6 1.0TS 5.0 24.0 0.0025 3.2 3.0TLS 3.1 21.5 0.002 2.7 1.0



®gures, test numbers are pre®xed by M and G which correspond, respectively, to Minnesota and Georgia;the number following the pre®x is, however, the same as the aforementioned test number.

As shown in Table VII, encouraging results were obtained for the PFS, GL and YLC soils. The modele�ciencies ranged between 97.98 and 99.34% for PFS, between 95.67 and 98.59% for GL, and between90.60 and 94.15% for YLC. Sample simulation results are shown in Figures 20, 22 and 24 for PFS, GL andYLC soils, respectively, and the ®t was found to be quite satisfactory. However, the model did not performsatisfactorily on three out of 10 simulated tests on CSL and six out of nine simulated tests on ISL. On CSLthe model generally performed satisfactorily with e�ciencies ranging between 75.68 and 99.60%. Figure 21corresponds to CSL and Figure 23 to ISL.

The model performance on the Georgia soils was generally less than satisfactory, although the modelperformed reasonably well on RLS (Robertsdale loamy sand) (test no.G52) and SLS (Stilson loamy sand)(test no.G56) for which the model e�ciencies were 79.80% and 88.15%, respectively. The model e�cienciesfor CALS (test no.G8), GLS (Fuquay loamy sand) (test no.G35) and one test on TS (Troup sand) (testno.G75) were 67.34, 64.62 and 66.45%, respectively. On all other soils, the model performed poorly as thee�ciencies are either negative or low.

On the Georgia soils, the model, in general, did not perform well for several of the soils. It is not preciselyclear as to why the model did not do well for these soils and this is a subject for further study. Nevertheless,some plausible explanations are advanced here.

(1) Becuase of a limited number of tests available for a soil type for calibration, the representative valuesmight not be optimal. This argument can be supported by the satisfactory performance of the model onthe Minnesota soils where the number of tests for each soil group is quite large (Table VII).

(2) It is important to emphasize again that an in®ltration test represents an independent identity re¯ecting allthe properties, including soil heterogeneity and its initial conditions, whereas the simulated resultscorrespond to the average conditions. The satisfactory model performance on the Minnesota soils lendscredence to the value of the general model and its potential for future work in this direction.

(3) Soil systems are, in general, complex: highly heterogeneous, isotropic, existence of crusts and densestructures, crop residues, etc. The e�ect of these complexities is considered in an overall sense. Forparticular cases, the e�ect of individual complexities may be so large as to invalidate the model.

(4) In many cases it was observed that after some time, in®ltration started increasing rather than continuingto decrease as normally expected. This sudden increase in in®ltration rate is due to the escape ofentrapped air. The general model does not accommodate this phenomenon of air release. Thus, it is notsurprising that in these cases the model performance was less than satisfactory.

Table VII. Validation results

Test No. E�ciency(%)




M10 99.18 M47 Negative M99 Negative G31 NegativeM11 99.34 M48 Negative M100 Negative G35 64.62M12 97.98 M58 96.89 M101 Negative G41 NegativeM39 87.48 M59 98.59 M102 Negative G44 NegativeM40 99.42 M60 95.67 M112 93.27 G48 NegativeM41 99.25 M93 29.63 M113 94.15 G52 79.8M42 99.6 M94 98.66 M114 90.6 G56 88.15M43 75.68 M95 98.01 G4 3.94 G65 NegativeM44 78.85 M96 97.87 G8 67.34 G75 66.45M45 82.83 M97 Negative G26 Negative G76 NegativeM46 Negative M98 Negative G27 Negative G77 Negative



Figure 12. Histograms showing variation of parameters of the general model for Plain®eld sand



Figure 13. Histograms showing variation of parameters of the general model for Columbia sandy loam



Figure 14. Histograms showing variation of parameters of the general model for Guelph loam



Figure 15. Histograms showing variation of parameters of the general model for Ida silt loam



Figure 16. Histograms showing variation of parameters of the general model for Yolo light clay



Figure 17. Histograms showing variation of parameters of the general model for Cowarts loamy sand



Figure 18. Histograms showing variation of parameters of the general model for Tifton loamy sand



Figure 19. Histograms showing variation of parameters of the general model for Troup sand



Figure 20. Validation on in®ltration data set M10 (Plain®eld sand)

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Figure 21. Validation on in®ltration data set M40 (Columbia sandy loam)

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Figure 22. Validation on in®ltration data set M59 (Guelph loam)

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Figure 23. Validation on in®ltration data set M94 (Ida silt loam)

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Figure 24. Validation on in®ltration data set M113 (Yolo light clay)

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CONCLUSIONS

The parameters of the ®ve-parameter general model (Singh and Yu, 1990) were calibrated for 191 sets ofin®ltration data for soils in Minnesota and Georgia. All parameters of the model have physical meaning.Parameters fc and So and `n' of the general model are found to be fairly predictable. The model performedwell for practically all test soils in Minnesota, but the same was not true for the test soils in Georgia.A critical examination of the general model reveals that it describes the in®ltration phenomenon in terms ofthe time-varying curve number.

REFERENCES

ARS 1976. `Calibration of selected in®ltration equations for the Georgia Coastal Plain'. ARS-S-113. Agricultural Research Service,US Dept of Agriculture, Washington, D.C.

Black, T. A., Gardner, W. R., and Thurtell, G. W. 1969. `The prediction of evaporation, drainage, and soil water storage for a bare soil',Soil Sci. Soc. Am. Proc., 33, 665±660.

Collis-George, N. 1977. Ìn®ltration equations for simple soil systems', Wat. Resour. Res., 13, 395±403.Dooge, J. C. I. 1973. `Linear theory of hydrologic systems'. Tech. Bull. No. 1468. US Dept of Agric. Service, Washington, D.C.Dunin, F. X. 1969. `The in®ltration component of a pastoral experimental catchment', J. Hydrol., 7, 121±131.Elrick, D. E. and Bowman, D. H. 1964. `Note on an improved apparatus for soil moisture ¯ow measurements', Soil Sci. Am. Proc., 28,

450±453.Green, R. E. 1962. Ìn®ltration of water into soils as in¯uenced by antecedent moisture'. PhD Thesis. Iowa State Univ., Iowa.Green, W. H. and Ampt, C. A. 1911. `Studies on soil physics, I. Flow of air and water through soils', J. Agric. Sci., 4, 1±24.Holtan, H. N. 1961. À concept of in®ltration estimates in watershed engineering', URS41-41. US Dept of Agric. Service, Washington,

D.C.Horton, R. I. 1938. `The interpretation and application of runo� plot experiments with reference to soil erosion problems', Proc. Soil

Sci. Soc. Am., 3, 340±349.HEC (Hydrologic Engineering Center) 1981. HEC-1, Flood Hydrograph Package Ð Users Manual. US Army Corps of Engineers,

Davis, California.Kostiakov, A. N. 1932. Òn the dynamics of the coe�cients of water percolation in soils'. Sixth Commission, Int. Soc. Soil Science,

Part A, pp. 15±21.Laliberte, G. E., Corey, A. T., and Brooks, R. H. 1966. `Properties of unsaturated porous media'. Hydrol. Paper 17, Colorado State

Uni., Fort Collins, Colorado.Mein, R. G. and Larson, C. L. 1971. `Modeling the in®ltration component of the rainfall±runo� process'. WRCC Bull. 43. Water

Resources Research Center, University of Minnesota, Minneapolis.Mishra, S. K. and Singh, V. P. 1999. `The SGS-CN method revisited', Wat. Resour. Res., under review.Moore, R. E. 1939. `Water conduction from shallow water tables', Hilgardia, 12, 383±426.Morel-Seytoux, H. J. and Khanji, J. 1974. `Derivation of an equation of in®ltration', Wat. Resour. Res., 10, 794±800.Overton, D. E. 1964. `Mathematical re®nement of an in®ltration equation for watershed engineering'. ARS 41-99. US Dept of Agric.

Service, Washington, D.C.Parlange, Y. 1974. `Water movement in soils', Geophys. Surv., 1, 357±387.Philip, J. R. 1957. `Theory of in®ltration: 4. Sorptivity and algebraic equations', Soil Sci., 84, 257±265.Philip, J. R. 1969. `Theory of in®ltration', Adv. Hydrosci., 5, 215±296.Ponce, V. M. 1989. Hydrology: Theory and Practice. Prentice Hall, Englewood Cli�s.SAS 1988. SAS/STAT User' Guide, Release 6.03 Edition. Statistical Analysis System Institute Inc., Cary.Singh, V. P. 1991. Elementary Hydrology. Prentice Hall, Englewood Cli�s.Singh, V. P. and Yu, F. X. 1990. `Derivation of in®ltration equation using systems approach', J. Irrig. Drainage Engng, ASCE, 116,

837±857.Smith, R. E. 1972. `The in®ltration envelope: results from a theoretical in®ltrometer', J. Hydrol., 17, 1±21.Smith, R. E. and Parlange, J.-Y. 1978. À parameter-e�cient hydrologic in®ltration model', Wat. Resour. Res., 14, 533±538.Synder, W. M. 1971. À proposed watershed retention function', J. Irrig. Drainage Div., ASCE, 97, 193±201.SCS (Soil Conservation Service) 1956. `Hydrology', in National Engineering Handbook, Supplement A. Soil Conservation Service,

USDA, Washington, D.C. Section 4, Chap. 10.



Calibration and validation of a general infiltration model

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