Plant factors influencing irrigation need

Static maximum allowable soil moisture deficits

The key to minimizing irrigation without inhibition of growth is the often repeated finding that soils can lose considerable quantities of water without suppressing crop growth rate (Bailey Reference Bailey1990; Hills et al. Reference Hills, Winter, Henderson, Stewart and Nielsen1990; Krieg & Lascano Reference Krieg, Lascano, Stewart and Nielsen1990; Musick & Porter Reference Musick, Porter, Stewart and Nielsen1990; Bacci et al. Reference Bacci, Battista, Rapi, Sabatini and Checcacci2003; Panda et al. Reference Panda, Behera and Kashyap2003). Much effort has therefore been devoted to quantifying this phenomenon (Denmeade & Shaw Reference Denmeade and Shaw1962; Ritchie Reference Ritchie1973; Meyer & Green Reference Meyer and Green1980; Rosenthal et al. Reference Rosenthal, Arkin, Shouse and Jordan1987; Muchow & Sinclair Reference Muchow and Sinclair1991; Sadras et al. Reference Sadras, Villalobos and Fereres1993; Sadras & Milroy Reference Sadras and Milroy1996; Thompson et al. Reference Thompson, Gallardo, Valdez and Fernandez2007). A summary of this subject is given in the FAO irrigation manual (Allen et al. Reference Allen, Periera, Raes and Smith1998; Savva & Frenken Reference Savva and Frenken2002). It considers that soil water deficit does not inhibit growth unless it also inhibits ET. It is based on the assumption that crop ET remains constant with a decrease in available water until a critical value is reached, after which ET declines linearly with a further decrease in available water until the wilting point is reached, when ET ceases. It summarizes existing knowledge in terms of a maximum allowable soil moisture deficit, expressed as a proportion of the available water to the depth of rooting (MADP):

(1)

where TAW is the total available water in the root zone, i.e. between field capacity and the permanent wilting point. RAW, the readily plant available water, is the proportion of TAW that can be removed before transpiration and growth rates start to decline.

The concept is illustrated in Fig. 1. MADP depends on plant species and the evaporative conditions. Values of MADP standardized to a value of ET=5 mm/day (the reference ET) and referred to as MADP (5), have been provided by Allen et al. (Reference Allen, Periera, Raes and Smith1998) for 92 different crops. Examples are given in Table 1. Standardization to an ET of 5 mm/day was achieved using an equation that enabled values of MADP to be adjusted for differences in ET, over the range 0·1⩽MADP(ET)⩽0·8:

(2)

Fig. 1.

Schematic diagram of the effect of soil water content on crop transpiration. Kt is the transpiration rate (or growth rate) at the given water content expressed as a fraction of the rate when water is not limiting. θ_fc, θ_thr and θ_pwp are the water contents at field capacity, threshold and the permanent wilting point.

Table 1.

Some values of the maximum allowable deficit expressed as a proportion of the available water to the depth of rooting (MADP(5)) and standardized to an ET of 5 mm/day (after Allen et al. Reference Allen, Periera, Raes and Smith1998)

MADP(ET) is the value of MADP for a given value of ET in mm/day. Perhaps the equation should be treated with some caution as there has been a long-standing controversy about the dependence of MADP on ET (e.g. Denmeade & Shaw Reference Denmeade and Shaw1962; Ritchie Reference Ritchie1973). These estimates of the MADPs for different crops can only be approximate, because eqns (1) and (2) are assumed to hold for all soils and environments.

Dynamic maximum allowable soil moisture deficits

Implicit in the FAO tables of MADP(5) (Allen et al. Reference Allen, Periera, Raes and Smith1998) is that MADP(5) for each species is considered to remain constant throughout growth, even though it has long been established that the sensitivity of crop growth to soil water deficit varies during the growing season (Salter & Goode Reference Salter and Goode1967). Examples of the more sensitive stages of growth for a range of crops are given in Table 2 and emphasize that crops are particularly sensitive to water stress at flowering and seed development. Also, the method does not take account of the short-term changes in evaporative demand, plant water stress and growth rate during the growth period. The reason for the omission of these aspects from the FAO tables could be that the bulk of experiments on which they are based only reported the net effects of the average soil water deficit during growth on crop yields and provided little information on the short-term changes during the growing period. Typically in an irrigation experiment (Bailey Reference Bailey1990), treatments consisted of applying irrigation whenever the soil moisture deficit fell to each of different extents during growth. It enabled the minimum soil moisture deficit for maximum final yield to be estimated, which together with the depth of rooting gave the MADP.

Table 2.

Stages of growth that are particularly sensitive to water stress

One possible way of obtaining better information about changes during growth of allowable soil water deficits is to base it on the finding that, although growth increases almost asymptotically with increase in transpiration, growth and transpiration over a wide range of conditions are almost proportional to one another with a gradient that depends on evaporative conditions (Guitjens Reference Guitjens, Stewart and Nielsen1990; Steduto et al. Reference Steduto, Hsiao and Fereres2007). Thus in some lysimeter experiments, once there is complete crop cover of the soil, the rate of loss of water is assumed to be proportional to growth rate. Also the average soil water content can be estimated from the daily loss of water. Non-destructive measurements of transpiration and soil water content, as in some lysimeter experiments, can provide a good indication of changes in the sensitivity of growth to soil water deficit over much of the growing period.

Another approach to the problem is to relate surrogate measures of plant growth rate, such as the rate of leaf expansion, to plant available soil water (PAW) (Sadras & Milroy Reference Sadras and Milroy1996). Essentially, plant measurements are made at intervals over a period during which the soil water content declines in the depth of soil containing most roots. The rate of the surrogate process measured on the plant experiencing soil water deficit relative to that on those with no water deficit is considered to be unaffected by fall in water content until a threshold value is reached when the rate starts to fall. This threshold value and the corresponding apparent maximum allowable deficit, expressed as a fraction of the total available water, are determined. They are analogous to MADP and they are broadly consistent (Sadras & Milroy Reference Sadras and Milroy1996) with values of MADP referred to previously (Allen et al. Reference Allen, Periera, Raes and Smith1998).

The rooting depth problem

All the above methods suffer from the drawback that the amount of crop available water is critically dependent on the depth of rooting, which is often very difficult to estimate. A major problem in generalizing the results at one site to others is that rooting depths are very sensitive to soil conditions (Taylor & Gardner Reference Taylor and Gardner1963; Stalham et al. Reference Stalham, Allen, Rosenfeld and Herry2007). This is illustrated by Fig. 2, which shows that root penetration can vary by four-fold over the normal range of soil resistances induced by compaction and by reduced soil water content. Field experiments and surveys on 602 UK commercial fields of potatoes demonstrated that compaction resulted in shallower rooting than is desirable for efficient use of water and nutrients (Stalham et al. Reference Stalham, Allen, Rosenfeld and Herry2007). Practically useful models are needed to predict the effects of soil conditions on root growth over long periods. A possible way forward is suggested by the finding that penetrometer measurements can give a good measure of the effect of soil conditions on root growth at a point in time (Clark et al. Reference Clark, Gowing, Lark, Leeds-Harrison, Miller, Wells, Whalley and Whitmore2005). It may be possible to develop models for the dependence of penetrometer readings, and thus root penetration, on soil texture, bulk density, organic matter content and soil water content. This would provide a means of estimating the time-course of soil conditions and their effects on root penetration.

Fig. 2.

Effect of soil strength on root penetration of cotton seedling tap roots. Different soil strengths were created by modifying bulk density and soil water content; both variables affected the dependence of root penetration on soil strength similarly. (The figure is derived from Taylor & Gardner Reference Taylor and Gardner1963.)

A new procedure for measuring the maximum allowable deficits without any explicit requirement of rooting depth was introduced by Thompson et al. (Reference Thompson, Gallardo, Valdez and Fernandez2007) for crops in a 400 mm deep soil. Measurements are made of volumetric water content to this depth every 30 min. The loss of water is partitioned into drainage and transpiration losses by considering that only drainage loss occurs at night. It is considered that when transpiration loss is divided by ET starts to fall, growth is inhibited by soil water deficit and thus the MADP can be calculated from the corresponding soil water content. The idea that the net changes in soil water content from drainage (and also from evaporation) can be assessed from measurements during the night might have wider applications. They may be relevant to estimating the depth of rooting from sensor measurements of soil water down the soil profile. The depth of rooting is sometimes taken as the maximum depth at which soil water content declines according to measurements at a given time of day. This could be due to a combination of transpiration, drainage and other soil processes affecting water movement. Ideally what is required is the depth from which transpiration removes water. By taking account of changes in soil water during the night, it should be possible to separate the changes due to soil processes from those due to transpiration and thus obtain a more reliable measure of rooting depth.

Soil factors affecting irrigation need

Required characteristics

Soil water sensors need to measure soil water content or the corresponding soil water potential over the ranges that are found in practice. A recent survey of the MADP(5) of 22 different crop species gave 72 values that ranged from 0·05 to 0·81 with 26 values being greater than 0·5. The majority of MADP(5) were therefore less than 0·5 (D. J. Greenwood, personal communication). A proportion of 0·5 of the maximum available water corresponds to a water potential of approximately −20 kPa for a loamy sand soil and −200 kPa for a clay soil. A different survey gives 33 values of the soil water potentials that have been found to be necessary for good growth of 16 species but only 6 were less than −200 kPa (D. J. Greenwood, personal communication). The results from the two surveys are broadly consistent with one another and suggest that water potential sensors over the range 0 to −200 kPa should cover most requirements. These values are, however, averages, and water is unevenly distributed down the soil profile so the water potential in a specific position, say at the uppermost sensor, could be much lower than −200 kPa. Variations between sensors of the same type in their responsiveness to soil water should be small otherwise the cost of calibration could be considerable. Ideally sensors should need no calibration by the user, and it is notable that according to Decagon Devices (2009) one of their sensors meets this requirement.

There is no advantage of using sensors that measure water content over those that measure water potential because the ability of a given soil to supply water to plant roots is governed by both water potential and water content. Irrigation requirement is dependent on soil water content at field capacity and the threshold value. Thus whichever type of sensor is used the soil moisture release curve is required to calculate the missing parameter and thus the irrigation requirement.

Available sensors

Numerous sensors for monitoring soil water are presently available. There are two main types; those that measure soil water content and those that measure soil water potential. The majority of the former actually measure the dielectric constant of soil, which is largely determined by its water content. One type of dielectric sensor, the frequency domain reflectometer (FDR), also referred to as a capacitance sensor, adjusts the frequency of an oscillating voltage until it identifies the strongest resonating frequency, which is a measure of the dielectric constant and thus the water content of the soil. Another type of dielectric sensor uses the technology ‘time domain reflectometry’ (TDR) to measure the dielectric constant, as the time taken for an electromagnetic pulse travelling down a rod to be reflected along its precise length. Both types of sensor, although expensive, measure a wide range of bulk soil water contents. They can also respond quickly to changes in soil water content (Campbell & Anderson Reference Campbell and Anderson1998; Evett & Parkin Reference Evett and Parkin2005; Nemali & Iersel Reference Nemali and Iersel2006). However, there is evidence that the performance of some sensors may vary with the soil conditions; one commercially available capacitance probe did not give reliable absolute measurements of soil water content although it enabled relative changes in soil water content over time to be estimated (Mwale et al. Reference Mwale, Azam-Ali and Sparkes2005). Although they do not always give reliable measurements of soil water content, dielectric sensors are generally chosen where there is a requirement to measure small but rapid changes in soil water contents needed for precise control of water addition to soil, e.g. in container cropping where volumes of soil are small and it is essential to maintain soil water contents close to a constant value, despite sudden variations in evaporative conditions that in the absence of irrigation would cause rapid changes in soil water content.

Moisture sensors that measure water potential are numerous. However, prominent among sensors that have been used to measure water potential in field soils are granular matrix sensors. Essentially, they consist of two concentric electrodes embedded in a reference matrix material, which is surrounded by a synthetic membrane for protection against deterioration (Chard Reference Chard2005). When a sensor is put into moist soil, the matrix water equilibrates with that in soil. Absorption of water increases the electrical conductivity of the matrix and the water potential of the soil can be calculated from this. Such sensors are comparatively inexpensive, have a range of 0 to −240 kPa, can measure water potentials in small volumes of soil and have functioned satisfactorily in soil for 3·5 years after installation (Qualls et al. Reference Qualls, Scott and Deoreo2001). They are, however, slow to respond to changes in soil water content, especially in drier soils. Granular matrix sensors are usually chosen when soil water changes are gradual, where many sensors are required to monitor soil water to depth over considerable areas and where satisfactory sensor performance over a long period is required.

A novel and probably more accurate sensor for measuring soil water potential consists of porous material that equilibrates with the soil water and a dielectric device that measures the water content of the porous material. A closed-form of hysteresis loop is used to convert the measured water content of the porous material into water potential. The use of two ceramic materials instead of one enables the sensor to measure matric soil water potentials over the approximate range −10 to −200 kPa (Whalley et al. Reference Whalley, Watts, Hilhorst, Bird, Balendonck and Longstaff2001, Reference Whalley, Clark, Take, Bird, Leech, Cope and Watts2007, Reference Whalley, Lock, Jenkins, Peloe, Burek, Balendonck, Takei, Tuzel and Tuzel2009).

Practical experience in the use of sensors

Much experience has been gained in the use of soil water sensors, from which the following practical advice on their use has emerged.

1. (a) Sensors must be calibrated for the given soil before use. Problems can arise, however, because of failure to recognize that soil water potential sensors can only operate over a limited range of water potentials and that equilibration times can be considerable. Manufacturers of some types of sensors provide experimental protocols for deriving calibration equations and others are published in the scientific literature (e.g. Geesing et al. Reference Geesing, Bachmaier and Schmidhalter2004; Groves & Rose Reference Groves and Rose2004). In addition to calibration, it is also important to check the specifications to ensure that the sensor is suitable for the proposed application and soil.

2. (b) Failure to ensure adequate contact between the sensor and surrounding soil is a serious and widespread problem. Protocols for embedding sensors in soil (Allen et al. Reference Allen, Periera, Raes and Smith1998; Savva & Frenken Reference Savva and Frenken2002) are especially important for small sensors as it is easy to damage the ceramics (Bacci et al. Reference Bacci, Battista, Rapi, Sabatini and Checcacci2003), which results in poor contact with the soil. Dielectric sensors must be maintained in close contact with soil otherwise measurements can be largely dominated by air gaps between sensor and soil.

3. (c) Failure to include a means of detecting sensor or electronic malfunction and an associated routine for automatically cutting off the irrigation can result in considerable waste of water (Qualls et al. Reference Qualls, Scott and Deoreo2001).

4. (d) The presence of roots around sensors can, if the sensors are small, result in their outputs losing sensitivity to changes in soil water, possibly because the sensors become coated with material that is impermeable to water (Bacci et al. Reference Bacci, Battista, Rapi, Sabatini and Checcacci2003).

Shallow-rooted crops

Typical examples are container crops, lawns and urban landscapes, as the rooting depths are often no more than about 200 mm, often because they overlay a very compact soil layer, or an extremely stony horizon that is a barrier to root penetration. The water contents in such soils change very quickly and water needs to be added frequently so as to ensure that the plant is never restricted by a temporary soil water deficit and yet ensure there is only minimal drainage. Soil water sensors at only one depth are required for these cropping systems.

Horticultural plants, such as ornamentals for the retail market, are grown in containers under evaporative conditions that are often high and vary during the day and from day to day. Efficient irrigation practice has included taking readings at a specific time of day and adjusting irrigation practice for the evaporative conditions (Klein Reference Klein2004). The amount of water that can be held in the rooting medium for container crops is small so the water content can change quickly. What is required is a way of maintaining the soil water at a constant value irrespective of changes in weather conditions, an objective that has been achieved by Nemali & Iersel (Reference Nemali and Iersel2006) who devised a system in which a controller uses dielectric substrate moisture sensors interfaced with a data logger and solenoid valves that supply irrigation. Measurements of substrate water were made every 20 min, which enabled the substrate water content to be maintained within 2–3% of the target value over 40 days. The key to success was the use of a very rapidly responding dielectric sensor.

Sensor-driven irrigation systems for urban landscapes in China and the USA are described in a number of papers (e.g. Qualls et al. Reference Qualls, Scott and Deoreo2001; Guo et al. Reference Guo, Zheng, Zhao, Li and Wang2005). Scheduled irrigations have been based on applying sufficient water to compensate for estimates of water loss from evaporative demand. However, in one system (Qualls et al. Reference Qualls, Scott and Deoreo2001) granular matrix sensors (Watermark) were embedded in soil at different points within the area and transmitted their readings by wire to an electronic module that either allows or prevents a scheduled irrigation cycle depending on the soil moisture condition. The system was adopted on 22 sites of average area 2177 m², and resulted in an average reduction of 27% of water applied and savings of $US 331/yr per sensor.

Irrigation of lawns has, as for urban landscapes, been based on supplying sufficient water to compensate for estimated evaporative demand. It can result in excessive drainage losses. To prevent such losses, sensors that detect free water are embedded in soil at a suitable depth and when they indicate the presence of free water they send a signal that cuts off the irrigation supply (Stirzaker & Hutchinson Reference Stirzaker and Hutchinson2005). A rather more sophisticated system has been reported, using a simulation model to estimate water requirement together with a subsurface time domain sensor to monitor soil water content (Blonquist et al. Reference Blonquist, Jones and Robinson2006). Its use resulted in 53% less irrigation than the currently recommended fixed rate of 50 mm water/week and also resulted in no detectable drainage below 300 mm from the surface.

Few papers in the literature are on sensor-controlled irrigation of field crops that penetrate to only a shallow depth. One such paper (Kang & Wan Reference Kang and Wan2005) related the growth and quality of radish (Raphanus sativus L.) to sensor-measured water potentials at 200 mm from the surface. It reported that although maintaining water potentials at each of different values over the range −15 to −65 kPa had no effect on yield, they did affect root cracking, which is an important quality attribute. A literature review of irrigation scheduling controlled by soil water sensors at a depth of approximately 250 mm also indicated that good growth of a variety of crops required a water potential greater than −65 kPa (Boote & Ketring Reference Boote, Ketring, Stewart and Nielsen1990; Stanley & Maynard Reference Stanley, Maynard, Stewart and Nielsen1990; Wright & Stark Reference Wright, Stark, Stewart and Nielsen1990; Munoz-Carpena et al. Reference Munoz-Carpena, Dukes, Li and Klassen2005; Wang et al. Reference Wang, Kang, Liu and Hou2007). It is possible that at least some of these crops are deep-rooted.

As mentioned earlier, sub-surface drip irrigation (Banedjschafie et al. Reference Banedjschafie, Bastani, Widmoser and Mengel2008) can improve water use efficiency by minimizing evaporation from the soil surface. It is notable that water sensors at only 50 mm depth have been effectively used to control decisions by irrigation from drip tapes installed 250–300 mm from the soil surface (Nogueira et al. Reference Nogueira, Dukes, Haman, Scholberg and Cornejo2003). Sensors at shallow depths within the soil profile can therefore provide useful information for irrigation scheduling.

Deep-rooted agricultural crops

Limited support for the view that sensor determinations at about 250 mm depth may provide a useful indication of irrigation need of some deeper rooted crops is provided by Stieber & Shock (Reference Stieber and Shock1995). These authors concluded that irrigation of potatoes could best be determined by maintaining soil water potential above −59 kPa with sensors at a single depth of between 0·1 and 0·2 m, offset from the centre of the ridge.

Crops with roots that penetrate to a depth of 1–2 m can, if the soil conditions are satisfactory, extract water to that depth. Many such soils are at near field capacity immediately before planting, often as a result of winter rainfall or monsoon rains. Considerable soil water deficits can occur on such soils without growth being inhibited. Irrigating according to the distribution of soil water down the profile so as to make maximum use of the stored water could therefore enable substantial saving of irrigation water. As irrigation would be infrequent, the surface of the soil would be dry for long periods which would reduce evaporation from it. For these reasons, some growers have been using sensors to monitor soil water at different depths down the profile and using the information obtained to schedule irrigation as described by free advice from extension services (Thomson & Ross Reference Thomson and Ross1996; Werner Reference Werner2002).

Estimation of when and how much irrigation is required for deep rooted crops can be based on MADP using the information and sequence of decisions summarized in Fig. 3. Determinations of the rooting depth and volumetric water content to that depth require further explanation. Thomson & Ross (Reference Thomson and Ross1996) inferred the rooting depth from soil water distribution measured by sensors. Other workers deduced the time course of the depth of rooting from mean daily temperatures (Pedersen et al. Reference Pedersen, Zhang, Thorup-Kristensen and Jensen2009) or an algorithm that enables the rooting depth to be calculated from plant dry weight (excluding fibrous roots), depth of rooting at final harvest and an equation that defines the increase in plant dry weight with time (e.g. Greenwood et al. Reference Greenwood, Cleaver, Loquens and Niendorf1977, Reference Greenwood, Gerwitz, Stone and Barnes1982; Zhang et al. Reference Zhang, Greenwood, White and Burns2007). The average volumetric water content to the depth of rooting can be calculated from the measured soil water potentials down the profile and the soil water release curves. These calculations can be aided by models that are subsequently described.

Fig. 3.

Flow chart of irrigation scheme based on sensor readings and model predictions.

Simulation models for soil water and its effect on crop growth

Principles about water dynamics in the soil–crop system, such as those described above, have been encapsulated into simulation models that calculate changes in soil water and plant growth over time. The models are of varying complexities and include SWATRE (Belmans et al. Reference Belmans, Wesseling and Feddes1983), CROPWAT (Clarke Reference Clarke1998), IRSIS (Raes et al. Reference Raes, Lemmens, Van Aelst, Vanden Bulcke and Smith1988) and SWAP (Kroes et al. Reference Kroes, Van Dam, Groenendijk, Hendriks and Jacobs2008). They aim to provide a widely applicable means of estimating water distributions down the soil profile and their effects on plant growth. SWAP is one of the most sophisticated models of its kind. The model simulates transport of water, solutes and heat in the vadose zone interactively with the development of vegetation. The governing equation for soil water flow is solved using an implicit finite difference method. The model has been widely tested and has given promising results. However, such a complex model requires many data inputs, which could cause difficulties for any user who does not have an excellent knowledge of soil and plant sciences. Furthermore, the adopted chosen numerical scheme is associated with instability. The most recent model, AquaCrop, developed by FAO (Hsiao et al. Reference Hsiao, Heng, Steduto, Rojas-Lara, Raes and Fereres2009; Raes et al. Reference Raes, Steduto, Hsiao and Fereres2009a, Reference Raes, Steduto, Hsiao and Fereresb; Steduto et al. Reference Steduto, Hsiao, Raes and Fereres2009), simulates the effects of the aerial environment and soil water on plant processes. The model updates a range of variables for each day. It calculates canopy cover, root distribution, stomatal opening, the roots' ability to meet transpiration demand and transpiration. Each of these variables depends to different extents on thermal time, potential ET and the fraction of available soil water to the depth of rooting. Plant biomass is calculated from transpiration and is modified for atmospheric CO₂ and soil fertility. Water evaporates from the soil surface that is not covered by crop canopy. Water movement throughout the body of soil is calculated by a cascade method using semi-empirical algorithms. The model has been calibrated and tested against field experimental data for some crops with promising results (e.g. Farahani et al. Reference Farahani, Izzi and Oweis2009; Hsiao et al. Reference Hsiao, Heng, Steduto, Rojas-Lara, Raes and Fereres2009). In the model, soil water is central to controlling crop growth, but the treatment of soil movement using a cascade method appears to be questionable for some circumstances such as where there is a relatively high groundwater table and upward capillary flow makes an important contribution towards meeting ET. Further, the model requires a large number of inputs which are often difficult to obtain, such as those associated with soil water movement. Cascade methods, though easy to implement, are associated with unsatisfactory simulations of capillary flow and poor predictions of daily soil water changes (Gandolfi et al. Reference Gandolfi, Facchi and Maggi2006; Cannavo et al. Reference Cannavo, Recous, Parnaudeau and Reau2008; Yang et al. Reference Yang, Zhang, Zhang, Greenwood, Hammond and White2009). If a way could be found of replacing the cascade method in the AquaCrop model with ones based on the fundamental theory of soil water movement, then the resulting model might be more widely applicable than the current version.

Application of classical theory of soil water movement

Application of this theory, besides improving existing models for water dynamics in the soil–crop system, is also important for interpreting soil water sensor data. Although there are many situations where soil water content increases with depth and the total soil moisture deficit can be readily obtained from the integral of a fitted empirical equation between sensor measured soil water content and depth, there are other situations where the patterns are complex and the fitting procedure is unsatisfactory, especially when soil sensors are few. Also it is always difficult, from soil sensor measurements, to distinguish between evaporative and drainage losses and crop transpiration.

The key to solving these problems probably lies in applying classical theory of soil water transport, despite complications associated with the dependence of the soil water release curve on changes in bulk density after cultivation and on whether the soil water content is rising or falling (Warrick Reference Warrick, Stewart and Nielsen1990). Classical theory is encapsulated in the Richards' equation (Bastiaanssen et al. Reference Bastiaanssen, Allen, Droogers, D'urso and Steduto2007; Yang et al. Reference Yang, Zhang, Zhang, Greenwood, Hammond and White2009). The equation is differential and highly non-linear and until recently complex procedures were required to solve it. Often their use requires specialized expertise that many potential users lack, which could explain why the cascade method for soil water movement is still favoured in many crop models used to solve practical problems.

The extremely rapid rate of computing by PCs provides an alternative and easy-to-use means of solving the Richards' equation (Lee & Abriola Reference Lee and Abriola1999). The essential idea behind the advance can best be understood by simulating water movement down a column consisting of sequential soil layers with uniform thickness. It is assumed that at any instant although the average water contents in each layer may differ, the water within each individual layer is uniformly distributed. Water flows between adjacent layers are according to the flow equation. The calculations are repeated for a very small time step, in the order of 0·001 days, and give estimates of water distribution that are similar to those obtained by solving the Richards equation using the finite-element method.

A new model, by extending the work by Lee & Abriola (Reference Lee and Abriola1999), has recently been constructed for simulating water dynamics in the soil–crop system (Yang et al. Reference Yang, Zhang, Zhang, Greenwood, Hammond and White2009). The model treats infiltration of water into the surface layer and evaporation from it; it also includes algorithms for root growth and the associated transpiration. Potential evaporation and transpiration are estimated from Allen et al. (Reference Allen, Periera, Raes and Smith1998). Soil hydraulic functions are those defined by Mualem (Reference Mualem1976) and Van Genuchten (Reference Van Genuchten1980). Simulations with the model of the distributions of water down the soil profile in different cropped soils at time intervals during growth were in excellent agreement with measurements (Yang et al. Reference Yang, Zhang, Zhang, Greenwood, Hammond and White2009). The model as such could be used to predict irrigation requirements as shown in Fig. 3.

In the sensor-based irrigation system, a model of this kind can be calibrated against the sensor data and then used to calculate the daily distributions of water down the profile and the evaporative and drainage losses. To calibrate the model, inverse modelling techniques, based on optimization theory to obtain the best fit between simulation and measurement, could be used to estimate uncertain parameter values. One possible set of parameters for such estimation are those defining soil hydraulic properties which are often determined by PTFs in terms of percentages of clay, silt, and soil organic matter and bulk densities as proposed by Wösten et al. (Reference Wösten, Lilly, Nemes and Le Bas1999) and Cresswell et al. (Reference Cresswell, Coquet, Bruand and McKenzie2006). Estimating soil hydraulic properties using PTFs is widely applied, but has not proved to be accurate enough on many occasions. Also, inverse modelling techniques can be employed to deduce root development and root distribution for the given soil.