Abstracts
Empirical models are mathematical equations that can be fitted to experimental results. The use of these models aims to evaluate or predict observed phenomena or experimental data with the objective of helping the development of adequate soil management practices. Based on these considerations, eight mathematical models described in the literature are compared in the present work, using as experimental data the mineral N accumulated during 32 weeks of incubation in Southern Brazilian soils. To obtain mineralization values experimentally, an incubationwashing procedure with 0.01mol L1 CaCl2 was used. Mineral N was determined at the beginning of the incubation and in the 2nd, 4th, 8th, 16th and 32nd weeks. Among the models, the best fit was obtained with the simple exponential model to describe the mineralization of organic N in the soils. The double exponential models showed quite good fit, but may be superparametrized. In addition, the hypothesis on which these models are based, i.e., the presence of two forms of organic N susceptible to mineralization, cannot be sustained in this study.
potentially mineralizable N; mathematical models; simple and double exponential; N pools
Modelos empíricos são equações matemáticas que podem ser ajustadas a resultados experimentais. Esses modelos podem ser utilizados para avaliar ou predizer fenômenos observados ou dados experimentais e auxiliar no desenvolvimento de práticas adequadas de manejo do solo. Desse modo, o presente trabalho teve por objetivo comparar oito modelos matemáticos descritos na literatura, utilizando como dados experimentais o N mineralizado de dez solos do Rio Grande do Sul, acumulado durante 32 semanas de incubação. O N mineralizado foi obtido experimentalmente em um experimento de incubação, seguido de lixiviação com CaCl2 0,01mol L1. O N mineral foi determinado no começo do período de incubação e ao final da 2ª, 4ª, 8ª, 16ª e 32ª semanas. Entre os modelos testados, o melhor ajuste do N mineralizado foi obtido com os modelos exponenciais simples, ao passo que a obtenção desses ajustes nos modelos exponenciais duplos esteve condicionado ao aumento de parâmetros na equação. Em função dos resultados observados e das condições experimentais, concluise que a hipótese em que os modelos exponenciais duplos estão baseados, isto é, na presença de dois compartimentos de nitrogênio suscetíveis à mineralização, foi rejeitada.
N potencialmente mineralizável; modelos matemáticos simples e duplo exponencial; compartimentos de N
EMPIRICAL MODELS TO PREDICT SOIL NITROGEN MINERALIZATION
MODELOS EMPÍRICOS PARA A PREDIÇÃO DA MINERALIZAÇÃO DO NITROGÊNIO DO SOLO
Flávio Anastácio de Oliveira Camargo^{1}1Professor do Departamento de Solos, Universidade Federal do Rio Grande do Sul (UFRGS), CP 776, 90001970, Porto Alegre, RS. Email: Autor para correspondência.Professor do Departamento de Solos, Universidade Federal do Rio Grande do Sul (UFRGS), CP 776, 90001970, Porto Alegre, RS. Email: Autor para correspondência. Clesio Gianello^{1}1Professor do Departamento de Solos, Universidade Federal do Rio Grande do Sul (UFRGS), CP 776, 90001970, Porto Alegre, RS. Email: Autor para correspondência.Professor do Departamento de Solos, Universidade Federal do Rio Grande do Sul (UFRGS), CP 776, 90001970, Porto Alegre, RS. Email: Autor para correspondência. Marino José Tedesco^{1}1Professor do Departamento de Solos, Universidade Federal do Rio Grande do Sul (UFRGS), CP 776, 90001970, Porto Alegre, RS. Email: Autor para correspondência.Professor do Departamento de Solos, Universidade Federal do Rio Grande do Sul (UFRGS), CP 776, 90001970, Porto Alegre, RS. Email: Autor para correspondência. João Riboldi^{2}1Professor do Departamento de Solos, Universidade Federal do Rio Grande do Sul (UFRGS), CP 776, 90001970, Porto Alegre, RS. Email: Autor para correspondência.Professor do Departamento de Solos, Universidade Federal do Rio Grande do Sul (UFRGS), CP 776, 90001970, Porto Alegre, RS. Email: Autor para correspondência. Egon José Meurer^{1}1Professor do Departamento de Solos, Universidade Federal do Rio Grande do Sul (UFRGS), CP 776, 90001970, Porto Alegre, RS. Email: Autor para correspondência.Professor do Departamento de Solos, Universidade Federal do Rio Grande do Sul (UFRGS), CP 776, 90001970, Porto Alegre, RS. Email: Autor para correspondência. Carlos Alberto Bissani^{1}1Professor do Departamento de Solos, Universidade Federal do Rio Grande do Sul (UFRGS), CP 776, 90001970, Porto Alegre, RS. Email: Autor para correspondência.Professor do Departamento de Solos, Universidade Federal do Rio Grande do Sul (UFRGS), CP 776, 90001970, Porto Alegre, RS. Email: Autor para correspondência.
SUMMARY
Empirical models are mathematical equations that can be fitted to experimental results. The use of these models aims to evaluate or predict observed phenomena or experimental data with the objective of helping the development of adequate soil management practices. Based on these considerations, eight mathematical models described in the literature are compared in the present work, using as experimental data the mineral N accumulated during 32 weeks of incubation in Southern Brazilian soils. To obtain mineralization values experimentally, an incubationwashing procedure with 0.01mol L^{1} CaCl_{2} was used. Mineral N was determined at the beginning of the incubation and in the 2^{nd}, 4^{th}, 8^{th}, 16^{th} and 32^{nd} weeks. Among the models, the best fit was obtained with the simple exponential model to describe the mineralization of organic N in the soils. The double exponential models showed quite good fit, but may be superparametrized. In addition, the hypothesis on which these models are based, i.e., the presence of two forms of organic N susceptible to mineralization, cannot be sustained in this study.
Key words : potentially mineralizable N, mathematical models, simple and double exponential, N pools.
RESUMO
Modelos empíricos são equações matemáticas que podem ser ajustadas a resultados experimentais. Esses modelos podem ser utilizados para avaliar ou predizer fenômenos observados ou dados experimentais e auxiliar no desenvolvimento de práticas adequadas de manejo do solo. Desse modo, o presente trabalho teve por objetivo comparar oito modelos matemáticos descritos na literatura, utilizando como dados experimentais o N mineralizado de dez solos do Rio Grande do Sul, acumulado durante 32 semanas de incubação. O N mineralizado foi obtido experimentalmente em um experimento de incubação, seguido de lixiviação com CaCl_{2} 0,01mol L^{1}. O N mineral foi determinado no começo do período de incubação e ao final da 2ª, 4ª, 8ª, 16ª e 32ª semanas. Entre os modelos testados, o melhor ajuste do N mineralizado foi obtido com os modelos exponenciais simples, ao passo que a obtenção desses ajustes nos modelos exponenciais duplos esteve condicionado ao aumento de parâmetros na equação. Em função dos resultados observados e das condições experimentais, concluise que a hipótese em que os modelos exponenciais duplos estão baseados, isto é, na presença de dois compartimentos de nitrogênio suscetíveis à mineralização, foi rejeitada.
Palavraschave : N potencialmente mineralizável, modelos matemáticos simples e duplo exponencial, compartimentos de N.
INTRODUCTION
Models represent attempts of mathematical description of a natural event. In the case of Nmineralization, the main purpose of modeling is to obtain quantitative data to recommend nitrogen addition (TANJI, 1982). The degree of complexity of a model is determined by the understanding of the system to be modeled, by the availability of data and by the way the model will be applied. The main limitation of the models that include the mineralization of organic N is that the organic N pool is very large when compared to the inorganic N pool (CAMARGO et al., 1997a), causing large errors when estimating the quantity of organic N available to plants (TANJI et al., 1979).
Initially, the mathematical fittings obtained for Nmineralization as a function of time were described by simple exponential models, considering the existence of only one form of potentially mineralizable N (N_{o}), decomposing at a rate proportional to its concentration (STANFORD & SMITH, 1972). When this concept was introduced, several mathematical attempts to estimate N_{o} were proposed, using the accumulated values of mineralized N. The majority of the new models separate the organic N pool, trying to reduce the errors in the estimates of N_{o}. These models consider the existence of two or more fractions occurring in the process, a more stable and a less stable form, also referred to as readily and less readily mineralizable N (MOLINA et al., 1980; INOBUSHI et al., 1985).
Comparative studies of simple and double exponential models and other functions show that the best fit to Nmineralization in incubation studies is obtained with the double exponential function, especially for short incubation times (MOLINA et al., 1980; DEANS et al., 1986; CABRERA & KISSEL, 1988). However, only a simple fit for comparison is generally shown, not considering the biological response of the observed data (CAMARGO et al., 1997b). The present work aims to compare mathematical models described in the literature, using experimental organicN mineralization data from soils of the Rio Grande do Sul State, Brazil.
MATERIAL AND METHODS
Theoretical background: STANFORD & SMITH (1972) proposed a biological method to study Nmineralization from organic matter incubated at constant temperature and moisture. The mineralization potential for soil nitrogen (N_{o}) was defined as the fraction of the N pool considered susceptible to mineralization, as long as the process follows a first order kinetic. They suggested that the mineralization potential, N_{o}, and the mineralization constant, k, could be used to predict the soil nitrogen availability to plants in a time period, considering that mineralized N is a linear function of the square root of time. This exponential model is based on the assumption that there is only one form of potentially mineralizable nitrogen, decomposed at a rate proportional to its concentration (Table 1). This form is assumed to be of a discrete size (N_{o}) and its magnitude approaches asymptotically that of mineralized N (N_{m}) accumulated through time.
Ever since STANFORD & SMITH (1972) introduced the concept of potentially mineralizable N (N_{o}), different mathematical approaches have been used to estimate its value using mineralized N data. From the simple exponential model (Table 1) a physical meaning for k (specific decomposition rate) is obtained, and k is not linear with time (DEANS et al. , 1986). When it is difficult to obtain a nonlinear regression, N_{o} can be obtained by the hyperbolic equation described by JUMA et al. (1984). According to MOLINA et al. (1980), neither the forms obtained by a nonlinear nor those by linear methods give satisfying results for N_{o} or k.
A quite popular model is the double exponential described by MOLINA et al. (1980) and INOBUSHI et al. (1985). Both authors consider the existence of two fractions within the process, one being a less stable form, and the other one more resistant. These models are mathematically and conceptually identical, but with some differences in interpretation of their parameters. MOLINA et al. (1980) described this behavior by the equation in table 1, where S and (1S) represent, respectively, the less stable and the resistant fractions of organicN, decomposing at specific rates h and k. The model described by INOBUSHI et al. (1985) was used to fit Nmineralization data for submerged soils. These authors proposed a double exponential equation (Table 1), where N_{oq} and N_{os} are described by first order kinetic reactions, with readily and lessreadily mineralizable N, decomposed at specific rates of kq and ks, respectively. According to DEANS et al. (1986) the simple exponential equations exhibit a systematic subestimation of N_{o} and a superestimation of the mineralization rate k. These authors observed that the double exponential models gave better estimates of k with lower mean squares, which supports the hypothesis that at least two organic N pools contribute to mineralized N.
Notwithstanding the relevance of these studies, TALPAZ et al. (1981) proposed a nonlinear fitting, considering only one mineralizable Npool, whereas SIERRA (1990) considered the simple and double exponential models only as approaches to the mineralization that happens within the soil organic matter. This author showed that the mineralization rate diminishes continuously, and that it is impossible to recognize mineralized N pools as proposed for exponential models, due to the non significance of the pool size (mineralized N). From a practical point of view, the number of pools to be included in the model depends on the fit and the desired precision.
Experimental data: Eight mathematical models (Table 1) were compared using experimental data from the average of ten representative soils from Rio Grande do Sul, Brazil, comprising a wide range of organic matter (17  56g kg^{1}), nitrogen (0.9  3.6g kg^{1}), pH (4.0  6.4) and clay content (180  550g dm^{3}), described by CAMARGO et al. (1997b). Initially, these soils were incubated for four years in microplots (30L) with drainage and exposed at field conditions. This procedure was used to minimize sampling and handling effects at the beginning of incubation. One sample was colleted to determine soil moisture and another one with natural moisture to be used, immediately, in the procedures of incubation described below. This method of incubation described by STANFORD & SMITH (1972) was used. The method consist of using 25g of each soil mixed with 25g of washed sand in leaching tubes of 100mL. These tubes were washed first with 0.01mol L^{1} CaCl_{2}, and then a solution without N (0.002mol L^{1} CaSO_{4}.2H_{2}O, 0.002 mol L^{1} MgSO_{4}, 0.005mol L^{1} Ca(H_{2}PO_{4})_{2}. H_{2}0 and 0.00025mol ^{1} K_{2}SO_{4}) was added and the water excess removed by suction at 80 kPa. The tubes were closed to maintain aerobic conditions and incubated at 35^{o}C. At the beginning, and after 2, 4, 8, 16 and 32 weeks, the soils were washed using the same procedure, according to STANFORD & SMITH (1972). NH_{4}^{+} (distillation with MgO) and NO_{3}^{} + NO_{2}^{} (distillation with devarda alloy) were determined in the wash water.
The fitting of the mathematical models was done using SASProcNLIN (SAS Institute, 1990), an iterative method using MARQUARDT (1963) algorithm, obtaining convergence when the decrease in residual squares in relation to that of the previous iteration become negligible. The method depends on initial values for the parameters, important to reduce the number of iterations and to avoid nonconvergence problems.
The initial estimates were based on the behavior of the results in relation to those observed in the literature and afterwards determined by the least number of iterations for convergence, least standard error and coefficient of variation, by the variation inflation factor (VIF) and by the least dependence between parameters (1/(1VIF)), given by the SIGMASTAT (1994) software, using the MARQUARDT (1963) algorithm for nonlinear models. To identify the most convenient model, the mathematical properties of the functions were considered (from a biological point of view) as well as the determination coefficient (R^{2}), the adjusted coefficient of determination (adjusted R^{2}), the residual mean of squares (RMS), the residual standard deviation (RSD), the distribution of residues and the asymptotic correlation matrix for the parameter estimates.
RESULTS AND DISCUSSION
Due to the similarity of the R^{2} values (Table 2) calculated from the 8 models, it was not possible to determine the best model, using this index. Eliminating the effect of the number of parameters by the adjusted R^{2}, the fitted models showed differences, with a slight superiority of the simple exponential ones. This behavior could be expected, given the similarity of the R^{2} and the higher number of parameters of the double exponential models. Among the models studied, that of JONES (1984) is outstanding by its fitting capacity and description of observed data. With one parameter less than the models of MOLINA et al. (1980) and INOBUSHI et al. (1985), this model showed highest adjusted R^{2} value, showing that the use of superparametrized models, like the double exponential, is not essential for obtaining an adequate fit.
Comparing the models by residual mean squares and residual standard deviation, the superiority of JONES' (1984) model is clearly seen (Table 2). The fit of the double exponential models (MOLINA et al., 1980; INOBUSHI et al., 1985) was of lower quality, being superior only to the model described by BROADBENT (1986), that, in the context of the present study, was the most inadequate. As far as the distribution of residues is concerned, the models were alike; however, JONES (1984) model showed a distribution of residues that suggests better fit. Examining the asymptotic correlation matrix of the parameter estimates, high positive and negative correlations are observed for the double exponential models (MOLINA et al., 1980; INOBUSHI et al., 1985), indicating that they are superparametrized for the used data set. Models with less parameters show low correlations, especially that of JONES (1984).
Considering the mathematical properties of the functions, based on the graphical representation of the fitted models, using the simple exponential as reference (Figure 1), it can be seen that, as they turn into typical zero order kinetic reactions, a slight subestimation of mineralized N occurs for the last point of measurement (32 weeks). JONES (1984) model fits this point better (Figure 1), including the concept of initial flux of mineralization and active nitrogen fraction.
The parameter added to the model of STANFORD & SMITH (1972) by MARION et al. (1981), a time exponent, improved the fitting (Figure 1). STANFORD & SMITH (1972) observed that the model, made up of two parameters, does not describe adequately the initial nitrogen mineralization rate. These authors corrected their estimates adding N mineralized during the first two weeks of incubation to the potentially mineralizable N, obtained by the simple exponential function (containing two parameters), on the mineralized N after 30 weeks of incubation. The parameter (time exponent) added by MARION et al. (1981) to the basic mineralization equation described by STANFORD & SMITH (1972) has the same objective and is a simpler way. Recently, CABRERA (1993) used a simple exponential model, including the mineralization rate of the most stable nitrogen pool. Therefore, the application of this model is based on the existence of two pools of organic N being mineralized.
STANFORD & SMITH (1972) verified for 39 soils that the variation of Nmineralization rate was very small (k=0.054 ± 0.009. week^{1}), suggesting that the main source of mineralized N was the same for all soils. A reevaluation of the experimental data of these authors by MOLINA et al. (1980) revealed that a better fit of the simple exponential model depends on the presence of various organic N pools, each one with its own mineralization rate. Following this principle, INOBUSHI et al. (1985) assumed the existence of two nitrogen pools in the mineralization process in 30 flooded soils from Japan. The application of the double exponential model described by these authors (MOLINA et al. 1980; INOBUSHI et al. 1985) to the results of the present paper showed good fit to the data observed after the second week (Figure 1).
The first data point was not well fitted by the models, with a deviation of 11.55mg N kg^{1} of soil. The only model that estimated zero time well was the simple exponential model described by JONES (1984), representing an important factor for the mathematical estimation of residual nitrogen present at the beginning of mineralization. This result was expected because the other authors did not generally use time zero in their adjustments because at this time the mineralization had not begun. The hyperbolic function (JUMA et al., 1984) did not describe the data as well as the double exponential models, mainly for data at week eight (Figure 1).
STANFORD & SMITH (1972) observed that most of their results fitted well to the parabolic equation (Table 1). However, they did not give any more attention to this model, used only as a mean of preestimating N_{o} for the simple exponential model. It is a fact that the parabolic model fitted the data better than the simple exponential model for some soils, not for all of them. BROADBENT (1986), based on the hypothesis that a better estimate of mineralized N could be obtained using the parabolic equation, used values for the time exponent different from 0.5. A comparison between the parabolic model and the exponential model described by STANFORD & SMITH (1972) (Figure 1) showed a better fit with the late model. In their original paper, STANFORD & SMITH (1972) estimated k and N_{o} by a nonlinear fitting of mineralized N as a function of incubation time. This fit is worse than the nonlinear model when estimating the mineralization parameters and for the fitting of exponential data (SMITH et al., 1980). As in the present paper, the treatment given to the exponential model is nonlinear; hence the greate difference between the models can be justified.
Some other differences among the parameters estimated by the eight models can be observed (Figure 1). As far as the potentially mineralizable N (N_{o} = 145.92 mg kg^{1}) in the model of STANFORD & SMITH (1972) is concerned, some models subestimated its value, such as those described by JONES (92.2% of N_{o}), CABRERA (88.1% of N_{o}) and BROADBENT (47.2% of N_{o}), while other superestimated it, like those of MARION et al. (103.6% of N_{o}), JUMA et al. (113.9% of N_{o}), INOBUSHI et al. (118.1% do N_{o}), and MOLINA et al. (118.1% of N_{o}).
Nitrogen mineralization rates obtained in this work were higher than those cited in the literature. This may be due to the nature of the process, with intense mineralization in the first weeks followed by a relatively long period of stabilization when compared to the mineralization periods. The occurrence of these high initial rates may be associated with an increase of microbial activity, induced by the handling and preparing of the samples. This is a recognized fact, and some authors even neglect the zero time mineralized N for this reason (STANFORD & SMITH, 1972), while other (MATAR et al., 1991) start their calculations of accumulated mineralized N from the 4^{th} week. Applying these methods to the data of the present paper, mineralization rates decrease from an average of 0.3196 week^{1}, to 0.2530 week^{1}, if considered from the 2^{nd} week, and to 0.129 week^{1}, if considered from the 4^{th} week (CAMARGO et al., 1997b).
The comparison of the models reveals that the simple exponential model described well the mineralization of organic N in soils from Southern Brazil. The active fraction (N_{2}) and the initial flux (N_{1}) parameters that compose the model of JONES (1984) can represent the residual mineral N (N_{1}) that, together with the potentially mineralizable fraction (N_{2}) and could be used as a simplification for N fertilizer recommendation. Considering that the models studied in this paper are linked to that one of STANFORD & SMITH (1972), but including a higher number of parameters, the only model that presented a significant increase 10% of the explanation variability for mineralized N (F value; Table 2), in relation to the model of STANFORD & SMITH (1972) was the model of JONES (1984).
The double exponential models fit the data satisfactorily, however, only a little improvement was observed when compared to the simple exponential ones, indicating that the superparametrization of these models is not justified when few points are used. In this case, the use of models with more than tree parameters did not fit very well because the low number of degrees of freedom. Nevertheless, it is possible to conclude that during the growing period, or during the incubation period studied (32 weeks), there was no contribution from the recalcitrant fraction to the mineralized organic N balance. This fact is confirmed by data obtained from 26 soils by KEENEY & BREMNER (1966), who divided organic N into that found before and after a year of incubation. The differences between the divided organic forms in this period were not significant, as well as the recovery of exchangeable ammonium, showing that such a situation does not require a higher number of parameters for a better fit.
The unstable or easily mineralizable fraction is mineralized at rates up to 10% per year (HEBERT, 1982). In the present study, a rate of about 7% was observed, characterizing the absence of nitrogen originating from the recalcitrant fraction. The double exponential models give a better fit than the simple exponential models, which is expected given the elevated number of parameters. However, those models present indications of superparametrization and the hypothesis that these models characterize the presence of two forms of organic N susceptible to mineralization can not be sustained or is sometimes impossible (DENDOOVEN et al., 1997). To test the hypothesis that there are two organic N pools a 12 yearperiod of incubation is need. In this case, it is possible that the double exponential models would show some advantages over the single exponential models. Results from DOU et al. (1996) study show that the pool size and mineralization rate parameters in the different models are merely mathematically defined quantities and do not represent any rigorouslydefined pool size of potentiallymineralizable N and their mineralization rate constants in the soils.
CONCLUSIONS
Based on the results obtained in this study and on the numerical and graphical analysis of the models, is possible to conclude that the simple exponential model proposed by JONES (1984) is the one that best describes the mineralization process and the potential of organic N to be released to the environment and to become available for plants.
Ciência Rural, v.32, n. 3, 2002.
^{2}Professor, Departamento de Estatística da UFRGS, Av. Bento Gonçalves, 9500, 91540000, Porto Alegre, RS.
^{Recebido para publicação em 02.05.01. Aprovado em 08.08.01}
 BROADBENT, F.E. Empirical modeling of soil nitrogen mineralization. Soil Science, Baltimore, v.141, p.208213, 1986.
 CABRERA, M.L. Modeling the flush of nitrogen mineralization caused by drying and rewetting soils. Soil Science Society of America Journal, Madison, v.57, p.6366, 1993.
 CABRERA, M.L., KISSEL, D.E. Potentially mineralizable nitrogen in disturbed soil samples. Soil Science Society of America Journal, Madison, v.52, p.10101015, 1988.
 CAMARGO, F.A. de O., GIANELLO, C., VIDOR, C. Comparative study of five hydrolytic methods in the determination of soil organic nitrogen compounds. Communications in Soil Science and Plant Analysis, New York, v.28, p.13031309, 1997a.
 CAMARGO, F.A. de O., GIANELLO, C., VIDOR, C. Potencial de mineralização do nitrogênio em solos do Rio Grande do Sul. Revista Brasileira de Ciência do Solo, Campinas, v.21, p.575580, 1997b.
 DEANS, J.R., MOLINA, J.A.E., CLAPP, C.E. Models for predicting potentially mineralizable nitrogen and decomposition rate constants. Soil Science Society of America Journal, Madison, v.50, p.323326, 1986.
 DENDOVEN, L., MERCKX, R., VERTRAETEN, L.M.J. et al. Failure of an iterative curvefitting procedure to successfully estimate two organic N pools. Plant and Soil, The Hague, v.195, p.121128, 1997.
 DOU, Z., TOTH, J.D., FOX, R.H., et al. Soil nitrogen mineralization during laboratory incubation dynamics and model fitting. Soil Biology and Biochemistry, Oxford, v.28, p.625632, 1996.
 HEBERT, J. Nitrogen. In: BONEAU, M., SOUCHER, B. Constituents and properties of soil New York : Academic, 1982. p.435442.
 JONES, A. Estimation of an active fraction of soil nitrogen. Communications in Soil Science and Plant Analysis, New York, v.15, p.2332, 1984.
 JUMA, N.G., PAUL, E.A., MARY, B. Kinetic analysis of net mineralization in soil. Soil Science Society of America Journal, Madison, v.48, p.465472, 1984.
 INOBUSHI, K., WADA, H., TAKAI, Y. Easily decomposable organic matter in paddy soil. VI. Kinetics of nitrogen mineralization in submerged soils. Journal of Soil Science and Plant Nutrition, Tokyo, v.34, p.563572, 1985.
 KEENEY, D.R., BREMNER, J.M. Characterization of mineralizable nitrogen in soils. Soil Science Society of America Journal, Madison, v.30, p.14718, 1966.
 MARION, G.M., KUMMEROW, J., MILLER, P.C. Predicting nitrogen mineralization in chaparral soils. Soil Science Society of America Journal, Madison, v.45, p.956961, 1981.
 MARQUARDT, D.W. An algorithm for least squares estimation of nonlinear parameters. Journal of Society of Applied Mathematic, Washington, v.11, p.431441, 1963.
 MATAR, A.E., BECK, D.P., PALA, M., et al. Nitrogen mineralization potentials of selected Mediterranean soils. Communications in Soil Science and Plant Analysis, New York, v.22, p.2336, 1991.
 MOLINA, J.A.E., CLAPP, C.E., LARSON, W.E. Potentially mineralizable nitrogen in soil: the simple exponential model does not apply for the first 12 weeks of incubation. Soil Science Society of America Journal, Madison, v.44, p.442443, 1980.
 SAS Institute. SAS User's Guide 1989 Edition. New York : SAS Institute, 1990. 651p.
 SIERRA, J. Analysis of soil nitrogen mineralization as estimated by exponential models. Soil Biology and Biochemistry, Oxford, v.22, p.11511153, 1990.
 SIGMASTAT, Scientific graphing sofware. Transform & curve fitting San Rafael : Jandel Scientific Revision SPW 2.0, 1994. 455p.
 SMITH, J.L., SCHNABEL, R.R., McNEAL, B.L., et al. Potential errors in the firstorder model for estimating nitrogen mineralization potentials. Soil Science Society of America Journal, Madison, v.44, p.9961000, 1980.
 STANFORD, G., SMITH, S.J. Nitrogen mineralization potentials of soils. Soil Science Society of America Journal, Madison, v.36, p.465472, 1972.
 TALPAZ, H.P., FINE, P., BARYOSEF, B. On the estimation of Nmineralization parameters from incubation experiments. Soil Science Society of America Journal, Madison, v.45, p.993996, 1981.
 TANJI, K.K. Economic implications of controls on nitrogen fertilizer use. In: STEVENSON, F.J. Nitrogen in agricultural soils Madison : ASA/SSSA, 1982. p.721772.
 TANJI, K.K., BROADBENT, F.E., MEHRAN, M., et al. An extended version of a conceptual model for evaluating annual nitrogen leaching losses from croplands. Journal of Environmental Quality, Madison, v.8, p114120,1979.
Publication Dates

Publication in this collection
03 Nov 2003 
Date of issue
June 2002
History

Accepted
08 Aug 2001 
Received
02 May 2001