Constrained two layer models for estimating evapotranspiration

Constrained two layer models for estimating evapotranspiration

David L B Jupp
CSIRO Division of Water Resources Canberra,
ACT, 2601, Australia

Abstract
Thermal region remote sensing allows you to map and monitor the distribution of surface temperature. Together with meteorological data, this information is of great value in water resources applications and in environmental remote sensing institutions where the hydrological cycle is a key element. This paper discusses some questions related to the need for complexity in model used to interpret remotely sensed data. In particular, it shows how a class of constrained Two-layer models provide useful improvements on the One-layer, or Penman-Monteith, models while still retaining simplicity and ease of use with the generally limited data available through remote sensing

Introduction
The use of remotely sensed thermal infrared radiation has been increasingly applied to the study of the distribution and flow of water in the earth surface layer. In principle, the application is immediate, since the presence of water in the soil and/or the root zone of plants and trees combined with solar radiation results in evaportranspiration ( ET in the following) and a net cooling of the surface relative to similar areas with less available water as well as retention of heat during the night relative to the drier areas However, in practice, the dynamic exchanges taking place between the atmosphere and surface layer and the nature of the surface layer (trees or grass for example) result in observed temperature distributions which are complex and it is difficult to develop direct relationships between remotely sensed surface temperature and available moisture in the surface layers of the soil.

In this situation, the remotely sensed data must either be used empirically as collateral data in the study of the spatial heterogeneity and distribution of moisture - for which it is ideally suited - or combined with models of the processes involved in mass and energy exchanges in the earth surface layer. The models being used with remotely sensed data have a wide variety in complexity and detail. All are based on the principle of the energy balance:

R_n = lE + H + G---------------------------(1)

Where:
R_n is net radiation in units of Wm^-2;
lE is latent heat flux ( with l as Latent heat of vaporization);
H is sensible heat flux; and
G is flux of heat into the soil or other storages.

The available models disaggregate the energy balance among the components of the surface layer and relate it in various ways to the water balance and the dynamics of the energy and water fluxes in the soil, vegetation and atmosphere. Through the latent heat, or evaportranspiration, term of the energy balance, remote sensing or surface temperature can provide information on the moisture content of the upper soil layers and the resistance of the surface layer to loss of moisture. This paper discusses a range of such models which are simple enough to be supported by the limited data available from remote sensing as well as being adequate to estimate the ET and other fluxes. In particular, it examines some issues raised in the practical demonstration of the methods in Jupp and Kalma ( 1989) and Kalma and Jupp

One Layer Models
> The least complex model which still provides an adequate model for the processes involved is the one-layer Resistance Energy Balance Model ( REBM) or in a linearized form, the Penman-Menteith formula (Monteith, 1975; 1981), This model treats the surface as a single, composite entity, assumes equilibrium of the fluxes and concentrations, is 'external' in that it only includes fluxes of heat and water vapour external to the effective surface and does not include water balance components.

The flux relationship for the one-layer model are governed by the following equations. The sensible heat flux (H) is modeled using a 'resistance' formulation ( monteith, 1975) by:

H = rC_p {(T₀-T_a)/r_a}------------------------(2)

Where T₀ is an effective temperature for the composite surface, T_a is air temperature at a reference height above the surface and r_a is a term describing the resistance to transport of sensible heat between the surface and the reference height. If a model for this resistance is available and computable then the REBM proceeds by indentifying T₀ with the (remotely) measured surface temperature T_s to obtain H and using equation (1) to compute the ET as:

lE = R_n - G - H-------------------------(3)

Which assumes that G and R_n have been adequately modelled.

The ET can also be expressed in resistance form (see Jupp and Kalma, 1989 for greater details in the notation being used here) as:

lE = (rC_p/g){(e_s (T₀) - e_a)/(r_a + r_s)}--------------------------(4)

Where e_s (T₀) is saturated vapour pressure at temperature T₀, e_a is vapour pressure at the reference height and r_s is a composite 'surface' resistance expressing both the intrinsic capacity to extract water through the composite surface and the available water.

The equations above are ideal for combination with remotely sensed data as shown by practical application in Jupp and Kalma (1989). If T₀ is identified with T_s then both lE and rs are computable. Also computable is a term called the 'moisture availability' (m_a) defined as:

m_a = lE / lE_pot ------------------------------(5)

Where lE_pot is the ET that would occur for the same reference meteorological conditions but with no limit on moisture availability through the composite surface. This last condition is assumed equivalent for this work to r_s being zero.

Computing this potential ET, as well as computing ET and T₀ when r_s or m_a are specified rather than t_s, required some care. The equations are nonlinear in that r_s is a function of T₀ and can be numerically difficult in situations of partial cover and unstable and drying meteorological conditions. With an appropriate numerical method, therefore, is possible to derive an effective T₀ corresponding to an input value r_s or m_a and the opportunity this poses for remote sensing of water related effects has been outlined in Jupp et al. (1990). In particular, if the soil wetness (eg the ratio W/W_max) is obtained by mass balance methods and the operating characteristic:

m_a + lE / lE_pot = f(W/W_max)

is known for a cover type, the REBM allows you to combine meteorological data and mass balance data to predict effective surface temperature and therefore bring remote sensing and water balance model into directly comparable form. In Jupp et al. (1990) this temperature Index'.

Two-Layer Rebm Model
As shown by example in Jupp and Kalma ( 1989), the one-layer REBM provided a direct and powerful means for relating meteorological data, remotely sensed surface temperature and evapotranspiration through a single, composite 'surface resistance'. The disadvantage is that this surface resistance is the result of many factors from which one, such as available soil moisture, is difficult to extract. Also, is has been found that in the presence of partial cover and drying conditions the model can perform very poorly. A specific problem was discussed in Kalma and Jupp ( 1990). This takes the form of a systematic discrepancy with occurs between the measured surface temperature and the estimate for To which is obtained when lE and H are obtained by measurement. In the latter case, the value for T₀ which satisfies the observed flux measurements is known as the 'aerodynamic, temperature, T_aer and should be equal to T_s if the assumptions used in the one-layer REBM were correct.

The least complex model that provides a separation of the water loss through transpiration by plants and direct evaporation by soil and provided some explanation of the aerodynamic temperature is the Two-layer REBM. In the resistance formulation, the flux terms take the form:

and the four basic conditions on a solution may be written:

where A _v and A_g are the partition of the total available energy (R_n - G) between vegetation and soil.

These four equations provide constraints for the six unknowns e_o, T₀, T_g, T_v, R_gs and r_vs so that a solution is only possible when two extra conditions are given. These may be from a model for r_gs and r_vs, for example, or from mass balances and models for the relationships between root zone moisture, surface layer moisture and the two 'moisture availability' terms:

m_av = lE_v / lE_{v
pot}

and

m_ag = lE_v / lE_{v
pot}

which provide integration with more realistic water balances and lead to the generation of temperature indices in the same way as with the one-layer REBM.

Of particular significance here is the addition of a measurement on the system through remote sensing of the surface temperature (T_s). This may be modeled by:

T_s = f_v T_v + (1 - f_v)T_g-------------------(10)

Where f_v is the fraction of vegetation cover. In this case, there are five conditions and six unknowns -----resulting in an underdetermined system of equations given only remotely sensed data and reference meteorological data.

Constrained Two-Layer Models
There exist a number of ways in which the Two - layer REBM can be made soluble and provide a model with the convenience of the One-layer REBM of Penman-Monteith formula but with the extra freedom and separation of vegetation and soil water loss that the Two-layer model provides. They each provide an extra condition which effectively reduces the Two-layer model to a one-layer model. The ones being studied currently at CSIRO include:

Observed temperature equal canopy temperature (T_s = T₀)
Canopy temperature equal foliage temperature ( T₀ = T_v)
Foliage temperature equal ground temperature ( T_v = T_g)
Smith et al. (1988) condition (H_g = (1 - a_s) A_g))
Minimum power (P = r_vH_v ² + r_gH_g² + r_aH²)
Canopy Aerodynamic Resistance (L' Homme et al., 1988)
Equal moisture availability (m_av = m_ag)

Only the One-layer REBM (which is equivalent to case 1.) and the minimum Power examples will be compared with data here. A much more complete study is in preparation. Essentially, the 'minimum power' solution is an heuristic which seeks to choose a solution that minimized an energy type condition among possible solutions. The one chosen is an analogue of the variational principle which leads. To Kirchhoff's laws for electrical circuits.

By applying the minimum power principle to the underspecified system of equations a solutions is obtained in which:

where the modified resistance (r'_a) is:

r'_a = r_a + f_v² r_v + (1 = f_v)² r_g------------------(11)

This represents two ways to express the solution - as an expression for To:

T₀ = [(1-r_a/r_a)T_a] + [(r_a/r'_a)/T_s]--------------------(12)

or as a modification of (2) in which T₀ can be indentified with T_s but with a modified resistance term. Either ways, a convenient solution is obtained. Note also that (12) allows T₀ (i.e. T_aer and T_s to be different.

Performance with data
Out of a wide variety of data sets which have been used to examine the models, two somewhat extreme examples will illustrate the findings for this discussion. The data sets are to be found in Choudhury et al. ( 1986) and Choudhury ( 1989) and comprise records of all relevant fluxes over an irrigated wheat field (Phoenix day 657) in the first case and a very hot, dry patchily covered area ( in the Ownes Valley) in the second. Figures 1 and 3 show the results of using the One-layer REBM and Figures 2 and 4 the constrained Two-layer model for the two cases. The solid lines are the predicted fluxes and the points are the measured data. The line with both lines and symbols is an error estimate defined as:

A_d = (lE _means - lE_pred) - (H_meas - H_pred)

And represents a term which explains the discrepancy without altering the energy balance. Its most likely nature is an advective term - hence its notation.

The two situations are very different. Over the irrigated wheat the ET flux is very high, it follows the pattern of net radiation over the day and the sensible heat flux actually becomes negative later in the day. The surface temperature is low and conditions are stable. In the Owens Valley data the sensible heat flux is very high and ET near constant and very small. Surface temperatures are high and the conditions unstable.

The graphs show how constrained Two-layer model performs very well compared with the One-layer REBM on the Owens Valley data with its partial cover and drying conditions. The One-layer model cannot cope with the unstable conditions and produces very large negative ET values! However, the results for the closed canopy wheat data are almost identical in each case and the systematic 'advective' error term is not changed. In common with other examples studied, there is a residual which does not seem to be a function of the disaggregation of the model into layers but is more likely the result of unaccounted advection and capacitance effects. Since these will generally balance over time and space scales the REBM does seem to produce a conservative and (with the constrained model modification discussed here) stable estimates for the ET flux attributable to the net radiation.

Since all relevant fluxes are available for these data, it was possible to derive the vegetation, soil and canopy temperatures (T_v T_g and T₀) using the complete two-layer model. In all cases that have been studied, the areas where the A_d error term is high correspond to areas were the estimates of the component temperatures are unsatisfactory. This shown that while there are significant benefits in using the more flexible model, the remaining residuals are not due to model aggregation into a few land cover components.

Conclusion
In this paper, it has been shown how the performance of the One-layer resistance energy balance model used in Jupp and Kalma (1989) to distribute evaportranspiration spatially in a catchment using airborne thermal data may be improved when the ground is only partially covered by vegetation by the use of a constrained Two-layer model. It has also been shown how the advantage of the two-layer formulation can be retained while still maintaining a level of model simplicity that enables remotely sensed thermal data to provide effective estimates of evaportranspiration and surface resistance over area at catchment scale

The implications of the improved model for soil moisture estimation are that partial covers and drying conditions are much better handled while the simplicity of the One-layer REBM or Penman-Monteith formula is retained. Available moisture is distributed more reasonably between the plant cover and the soil allowing integration with a more sensible water balance model which separates near surface and root zone storages. It is unlikely that remotely sensed thermal data and a reference meteorological data station can support a more complex model without other data. The methods outlined have been incorporated into version 3 of the mocro BRIAN image processing system and provide significantly improved results over areas of partial cover

References

Choudhury, B.J. ( 1989). Estimating evaporation and carbon assimilation using infrared temperature data: vistas in modeling. G. Asrar (Ed). Theory and Applications of Optical Remote Sensing. John Wiley & Sons, NY
Choudhury B.J. Reginato, R.J. and Idso, S.B. ( 1986). An analysis of infrared temperature observations over wheat and calculation of latent heat flux. Agricultural and Forest Meteorology, 37, 75-88
L ' Homme, J-P., Katerji, N., Perrier, A. and Bertolini, J-M ( 1988). Radiative surface temperature and convective flux calculation over crop canopies . Boundary-layer Meteorology , 43, 383-392.
Jupp, D.L.B. and Kalma, J.D. ( 1989). Distributing evaportranspiration in a catchment using remote sensing. Asian-Pacific Remote Sensing Journal, 2, 13-26.
Jupp, D.L.B., Walker, J., Kalma, J.D. and Mc Cicar, T. ( 1990). Using thermal remote sensing to monitor land degradationand salinization in the Murray-Darling Basin of Australia. Proceedings of the 23rd Int Symp on Rem Sens Env, Bangkok, Thailand, April 18-25.
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Smith, R.C.G., Barrs, H.D. and Fischer, R.A. ( 1988) . Inferring stomatal resistance of sparse crops from infrared measurements of foliage temperature. Agricultural and Forest Meteorology, 42 183,-198.