A METHOD FOR ESTIMATING THE TOTAL ORGANIC CARBON TRANSPORT IN WORLD RIVERS

Chuan Liang¹ and Yongming Shen²

¹College of Hydraulic & Hydroelectric Engineering, Sichuan University

Chengdu, Sichuan, 610065, China

Tel: +86-028-5402887, Fax: 028-5405604 email: LChester@scu.edu.cn

²Dalian University of Technology, Dalian, 116024, China

Tel: +86-0411-4708514-8303, E-mail: ymshen@dlut.edu.cn

Abstract: Based on the two-dimensional probability combination theory, an estimation method of hydrology for the total organic carbon transport in world rivers is discussed. For various ecosystems and drainage basins the depth of runoff and the TOC export can be taken as independent and random variables. Applying frequency analysis used commonly in hydrology it is quite feasible to estimate the TOC export by world rivers to the oceans making use of the data at hand. According to this method, global TOC export from forest areas and from total continental area was calculated as 0.37 and 0.57×10¹⁵g TOC /a, respectively. These estimates are 40% higher than the results obtained by SCHLESINGER & MELACK (1981), who used a subjective assignment of carbon losses from various ecosystems. These findings illustrate that the choice of the estimation method is an important factor when discussing total TOC export to the oceans.

Keywords: total organic carbon, probability, estimating method

1    INTRODUCTION

SCHLESINGER & MELACK (1981) reviewed past estimates of global organic (dissolved and particulate) carbon transport by rivers. These estimates range from 0.03 to 1.0×10¹⁵g C/a Although the TOC (total organic carbon) transport by rivers constitutes only a small flux within the global carbon cycle, it nevertheless plays a significant role for continental and marine ecosystems and for the formulation of balanced models of global biochemical cycles. Estimating global TOC transport by rivers is, consequently, a historical concern of ecologists, environmentalists, geologists, hydrologists and other geophysicists worldwide.

The large difference between estimates of global TOC transport given by various authors has been attributed to the lack and inaccuracy of the available data. However, more important for obtaining estimates is to find a reasonable, objective and rational method, which can be applied to the limited data sets already available.

In this respect, Schlesinger & Melack (1981) provided us with a valuable precedent. Their first and second approximates of the world rivers′ TOC transport amounted to 0.37 and 0.41×10¹⁵g C/a, respectively, narrowing down the wide range mentioned above Unfortunately, their first approximation was still based upon the extrapolation of an empirical function derived from data which were far beyond the world scope. Their second approximation, perhaps to be taken as a mere recommendation, was derived using subjective mean annual denudation rates of organic carbon in different ecosystems multiplied by the global areas of these ecosystems. Obviously, these approaches did not show stringent rationale and methodology and were not based upon a certain kind of objective sampling theory.

It is the main purpose of this paper to improve the estimation method making use of the available data as fully as possible by using probability theory objectively and reproducibly than before.

2    RATIONALE

No matter how complicated the physical, chemical, and biological processes affecting carbon transported in rivers may be, the key point to control the amount of carbon transported is the downstream cross section of the river. Therefore, the river’s TOC transport over time (0,T) can be expressed as follows:

Where V_T and v(t) represent the total TOC transport in a time interval (0,T) and the instantaneous carbon transport rate in g/s, as a function of time (t) in seconds, respectively. Similarly, the total amount of time(t) in seconds, respectively. Similarly, the total amount of water (W_T) passing the cross section is expressed as:

Where Q (t) is the instantaneous discharge of water in m³/s as a function of time variable (t) in seconds.

The average carbon concentration (C_T) in the same period of (0,T) will be:

                                (3)

It is quite evident that this concept of average carbon concentration is neither the same as the instantaneous concentration:

Nor as the time averaged carbon concentration, which can be expressed as follows:

Although (3)-the discharge weighted concentration-should be preferred for transport calculations it may be necessary to use (5) because it is much easier obtained from the original measurements. The estimate of the total carbon transport then becomes:

                             (6)

Next, let us assume that the catchment area of the basin above the cross section of a river’s outlet is A with the dimension of m². After dividing both sides of (6) by A, we have:

Where E_T represents the specific rate of carbon loss from the upstream part of the watershed in g/m² for the time interval (0,T) and Y_T is the average rate of runoff in cm for the same area and time period. If the period considered is one year as usual, then:

Reflects the relation between the annual carbon loss from the watershed (E), the annual runoff (Y) and the annual average concentration of carbon (C) in the river.

It is well known that (Y) and (C) are two variables correlated only weakly or not at all with each other. Through frequency analysis both variables may show their estimated probability distribution functions and corresponding parameters, such as long-term mean values, i.e. expected values (Y₀, C₀), coefficients of variance (CV_Y, CV_C) and skewnesses (CS_Y, CS_C). Also, the two- dimensional distribution may be estimated according to the correlation characteristics of the two hydrological variables, that is the correlation coefficient (r_{Y, C}) and the coefficients of regression (r_{Y. C}Cv_YY₀/Cv_CC₀ or r_C.Y Cv_CC₀/Cv_YY₀), etc.

As a result, on the basis of (8), by using the probability combination theory and the technique of two-dimensional random variables, we may find out the probability distribution function of average carbon loss form watershed above the gauging station:

(P_E represents the probability of the event that random variable (X) is equal to or greater than a certain carbon loss rate (E)). We also may find corresponding parameters, one of which is the expected value of (E), namely:

                             (10)

The long-term average carbon loss rate wanted. After multiplying this rate by the catchment area (A, m²), we have the theoretically more correct value of the river’s total carbon transport.

                            (11)

With respect to a certain ecosystem type, for example tropical forest, the total amount of the carbon transported is sum of all the carbon transported in all rivers from that ecosystem:

Where, k is the number of rivers in that ecosystem.

Usually it is vary difficult to get detailed data from all the rivers in a certain ecosystem Therefore, a suitable method should be developed to make use of limited data in order to reasonable estimate total carbon transport of that ecosystem.

Let us suppose that the long-term annual mean runoff rate Y_0j and the corresponding carbon concentration C_0j in a certain watershed j, (j=1,2,k) represent the samples of two independent and random variables (C₀.and Y₀) with certain types of probability distribution functions typical for a certain ecological system. If we then take the limited river data available as a representative sample from an unknown but objectively existing population distribution, we obtain following (8):

Using the combination procedure on two-dimensional random variables and under the condition of (13), the expected value of carbon transport rate (E₀) in a certain ecosystem can

be found:

This combination result would be fully correct under the two mentioned preconditions, which are (I) the independence of Y₀ and C₀ and (II) the representative of the sample set from the population. Thus in field situations, where Y₀ and C₀ correlated only weakly with each other and where the sample size is large enough the result will not lead to a large mistake.

For the purpose of estimating the total TOC transport by world rivers, we should only use the weighted average method:

Where V_0s--the global TOC transport by rivers from all ecosystems in g/a; E_0l--the average carbon loss rate in each ecosystem (l) calculated from (14) in g/m²; A_l--the total area of each ecosystem in m².

In summary, a reasonable estimate of global river TOC transport may be obtained by passing through two steps of probability combination analysis, one for the different rivers in the same ecosystem, the other for the different ecosystems in the world. The final result can be obtained by summing up areally weighted carbon concentration averages. In doing so subjective arbitrariness will be avoided.

3    COMPUTATION

Using the limited date base available from 38 rivers in the forest areas of the world (see Table 1), That is, the average runoff rates (Y_0j) and the average TOC concentrations (C_0j), (j=1,2,38), the total amount of TOC export from the global forest area and the corresponding total world rivers’ TOC transport were calculated according to the rationale described above. However, the first step of the probability combination procedure had to be left out, that is the step which would involve using the individual data of each of the rivers in the forest ecosystem simply because they were not available from the literature.

Table 1    Data sources used for organic transport estimation in forestry areas in the world

Thus, the computation proceeds as follows:

(1) A correlation analysis between Y_0j and C_0j, (j=1,2,38) yields a correlation coefficient r=0.3 and a linear regression C₀= –0.0195Y₀+7.76 as graphed in Figure 1.

Fig.1    Scatter plot and linear regression of depth of runoff versus average TOC concentration for 38 watersheds from forest ecosystems (data from SCHLESINGER & MELACK, 1981)

(2) The runoff probability distribution function for the forest ecosystem is obtained by a frequency analysis of the Y_0j set, (j=1,2,38). Here the Gumbel’s distribution function is chosen. The statistical parameters are: Y₀=90.35cm; Cv=1.10; Cs=5.0; α=0.011; μ=43.02. The mathematical expression of the distribution is:

                   (16)

As shown in Figure 2.

Fig.2    The probability distribution for TOC concentration for 38 watersheds from forest ecosystems (data from SCHLESINGER & MELACK, 1981)

(3) By the same technique, we obtain the Gumbel’s probability distribution function for the average TOC concentration of rivers of the forest ecosystem area with its parameters: C₀=7.00g/m³; Cv=1.10; Cs= 5.00; α =0.148; μ=0.335 and its mathematical expression:

                  (17)

As shown in Figure 3.

Fig.3    The probability distribution for depth of runoff 38 watersheds from Forest ecosystems (data from SCHLESINGER & MELACK, 1981)

(4) According to (13) and by use of the probability combination technique (which is not shown in this paper for the sake of simplicity), the average TOC loss from the forest area was thus obtained as: E₀=6.50 g/m².

(5) The total amount of rivers’ TOC transport from the global forest area (A=57×10¹²m²) was found by multiplying the E₀ by A, thus V₀=6.50×57×10¹²=0.37×10¹⁵g TOC/a.

(6) Assuming that the TOC transport from forest area can be linearly extrapolated to global TOC transport, and using the Schlesinger & Melack (1981) percentage of forest to global export of 64.4%, our global estimate becomes: V_0S=0.57×10¹⁵g TOC/a.

4    CONCLUSION

As was shown in this paper, the accuracy in estimating TOC transport by world rivers does not necessarily depend on an exhaustive data collection. If only a set of representative samples is available, which is the usual situation in such a study, then the accuracy of the transport estimate depends to a large degree on the estimation technique applied. Because of the random nature of the sample data and the limited sample size, the introduction of the probability analysis for estimating TOC transport by world rivers is not only possible but appears to be even mandatory.

Correlation of the total TOC transport with annual runoff and total watershed size and specific TOC losses for each ecological system, such as those given by Schlesinger & Melack (1981) are only the tools for an initial regression analysis which should show if the correlation between the parameters is strong or weak. These regressions should not be taken as empirical functions to be extrapolated for total TOC transport in would rivers, as has been done when Schlesinger & Melack (1981) derived their approximation of TOC transport. Such an approach is obviously neither strict nor objective.

In order to make the best use of the data available and to integrate the information provided by the limited sources, an estimation method considering the randomness and the respresentativity of the sample, either in time series or in space series, is suggested on basis of frequency analysis and probability combination theory. By using the same original data as Schlesinger & Melack (1981), the total TOC transport from forest ecosystems is estimated to be 0.37×10¹⁵g TOC/a with a corresponding global transport of 0.57×10¹⁵g TOC/a. These values are 40% higher than the results of Schlesinger & Melack (1981) in their second approximation (0.264 and 0.45×10¹⁵g TOC/a) for forest ecosystem and for global exports, respectively. This large difference in global TOC export estimates based on the same set of data-shows that attention must be paid to the method of estimation as well as to further data acquisition.

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