Calculating uncertainty in regional estimates of trend in streamflow with both serial and spatial correlations

Authors

  • Robin T. Clarke

    Corresponding author
    1. Instituto de Pesquisas Hidráulicas – IPH, Universidade Federal do Rio Grande do Sul – UFRGS, Porto Alegre –, RS, Brazil
    • Corresponding author: R. T. Clarke, Instituto de Pesquisas Hidraulicas, Universidade Federal do Rio Grande do Sul, Avenida Bento Goncalves 9500, Porto Alegre, RS 91501-970, Brazil (clarke@iph.ufrgs.br)

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Abstract

[1] An expression is derived for the variance of a regional linear trend in annual runoff (units: mm of runoff per km2 per year), averaged over P gauging sites, where the P individual runoff sequences are subject to both year-to-year and site-to-site correlations. At each site, it is assumed that mean annual runoff has a linear trend with serially correlated residuals modeled by an ARMA(1,1) process which can represent both short-term and long-term persistence in annual runoff. Extension to ARMA(p,q) processes is straightforward, and the procedure is also adapted to the case where runoff exhibits long-term persistence. In the case of the ARMA(1,1) model, the expression obtained for the variance of the regional trend need not assume Gaussianity, but if the assumption of Gaussianity is tenable, the note shows how the following hypotheses can be tested: (i) that the linear trend is zero at all P sites; (ii) that the linear trend is equal at all P sites.

1. Introduction

[2] This note is concerned with calculating the uncertainty (measured by the standard error) of estimates of trend in the spatially averaged annual runoff (units: mm of runoff per km2 per year) recorded at a number of sites within a river basin. When river basins are large, as is common in South America, annual runoff will be serially correlated between years, since part of the rain falling in a particular hydrological year may leave the basin as runoff in subsequent years, after storage. Furthermore, records of annual runoff at sites within large basins are likely to be spatially correlated, and this spatial correlation cannot be easily accommodated by standard geo-statistical procedures because flow in rivers is locally one-dimensional in a two- or three-dimensional topography. Thus, the spatial correlation between records at two sites, both on first-order streams, will differ from the spatial correlation between two sites, one of which is on a first-order stream and the other on a stream of higher order. Similarly, the correlation between records at two sites in the same drainage basin is likely to differ from the correlation between records at two sites in different drainage basins, even if the distance between the sites is similar in both cases. The complexity of this spatial correlation therefore requires an approach different from the empirical variogram and kriging procedures that are widely used in regional analyses of trends in rainfall, temperature, and other hydroclimate variables.

[3] Thus, an estimate of regional trend, over time, in annual runoff will be subject to both year-to-year and site-to-site correlations, and the purpose of this note is to set out a procedure that estimates these correlations separately, and then recombines them to calculate a standard error for the regionalized, spatially averaged trend in runoff.

2. Materials and Methods

[4] The starting point is a linear model at a single site Qt = α + β t + εt, where Qt is depth of runoff at the site in year t (t = 1…N), εt is a serially correlated residual, and β is the time-trend in annual runoff at the site, with β the parameter of main interest. This simple model, consisting of a deterministic trend superimposed on a stochastic component, has the same form as that used by Cohn and Lins [2005] and many others; Cohn and Lins [2005] assumed from the outset that the εt were multivariate Gaussian with zero mean and variance-covariance matrix Σ. In this note, Gaussianity is not required for calculating the uncertainty (i.e., standard error) of a regional trend, but it is needed when null hypotheses are to be tested (although the difficulties arising when tests for the significance of geophysical time-trends are required have been clearly argued by Cohn and Lins [2005]). We take as the estimate of β the usual linear-regression estimate b = A1 Q1 + A2 Q2 +……+ AN QN = ATQ, where, if the N years are in sequence, At = (2t–N−1)/[N(N21)], otherwise inline image In a later paragraph, we also use an estimate b+ of regional trend, calculated from the mean of the Qt over P sites: inline image where for example inline image is the mean annual runoff averaged over the P sites in Year 1. Working with annual runoff instead of mean annual discharge gives a reasonable homogeneity of variance between gauging sites: see the example given below. To model the serial correlation on the residuals εt, we assume an ARMA(p, q) model with p = q = 1:

display math(1)

since it is well known that this model can describe both short-term and long-term characteristics of streamflow when the parameters ϕ and θ lie close to the unit circle [e.g., O'Connell, 1973], thus providing a simple alternative to long-term persistence processes of the kind discussed by Beran [1998]. As an illustration, Figure 1 is a synthetic trace of 1000 values generated from an ARMA(1,1) model with ϕ = 0.98 and θ = 0.05, showing evidence of long-term persistence. If the unit of time on the horizontal axis is taken to be a year, the trace contains periods during which values increase or decrease for one or two “centuries,” longer than many hydrologic records. While the ARMA(1,1) cannot model very long-term persistence, Grimaldi [2004] has given a preliminary Hurst parameter estimation procedure based on the concept that ARMA(1,1) most closely approximates the fractionally differenced FARIMA(0, d, 0) model with parameter d.

Figure 1.

Synthetic trace of an ARMA(0.98,0.05) process illustrating long-term persistence.

[5] The quantities at in equation (1) are a sequence of uncorrelated random variables with zero mean and constant variance σa2. Thus at this point, the trend model is linear with AR(1,1) residuals. The argument that follows is easily generalized to the case where the residuals are ARMA(p,q); the expressions obtained for var[b] become more unwieldy but are straightforward to calculate. It is not necessary for the ARMA model to have the same number of parameters at each site nor for the sequences {at} to have the same variance.

[6] The following section derives the expression for var[b], which is then extended to give the variance of an estimate of regional trend var[b+] derived from P gauging sites within a river basin. An approximate test is given of the hypothesis that all P sites exhibit a common trend, against the alternative that the trend varies from site to site. The note concludes with a discussion of how the trend model can be extended to include an exploration of Granger causality [Granger, 1969; Hacker and Hatemi, 2006] when causal variables are annual rainfall and changes in land use.

2.1. An Expression for var[b], the Variance of Trend in Runoff at a Single Site

[7] Some algebraic manipulation of the model Qt = α + β t + εt and equation (1) gives the covariance

display math(2)

from which, using b = ATQ, we obtain, for a single site within the basin,

display math(3)

where A is the vector of linear regression coefficients A= [A1, A2…AN]T defined above, and V is an N × N symmetric matrix with (1 – 2 ϕ θ + θ2) σa2 / [1 − ϕ 2] on the leading diagonal; terms on the first supradiagonal are (ϕ − θ) (1 − ϕ θ) σa2 / [1 − ϕ 2], on the next supradiagonal by ϕ (ϕ − θ) (1 − ϕ θ) σa2 / [1 − ϕ 2] and so on, ending with ϕN−2 (ϕ − θ) (1 − ϕ θ) σa2 / [1 − ϕ 2] in the matrix top right-hand corner.

[8] In practice, the ARMA(1, 1) model—or more generally, the ARMA(p, q) model—is fitted to Qtb t, using b =ATQ calculated for each site. Estimates of the parameters ϕ, θ, and σa2 are then found by minimizing the sum of squares Σat2 or, if normality can be assumed, by maximum likelihood [Box and Jenkins, 1970]. When the flow sequences {Qt} are of length ∼80 years, as in the numerical example given below, experience with natural flows in some Brazilian rivers has shown that least-squares and Gaussian estimates of the parameters are equivalent to at least three significant figures.

[9] If the structure of residuals about the trend line were given by a FARIMA(p, d, q) model with p = q = 0 or p = q = 1 instead of an ARMA (1,1) model, an equation corresponding to equation (3) can also be derived, but because of its complexity it is given as supporting information to this Technical Note.

2.2. An Expression for var[b+], the Variance of an Estimate of the Regional Mean Trend b+

[10] Here, it is assumed that there are records of annual runoff {Q t(j)} at each of P sites, j =1, 2…P; t = 1, 2…N, with the values Qt(j) in units of mm runoff per unit area as given above. In what follows, the same ARMA(1,1) model (but with different parameters ϕ(j), θ(j)) is assumed to hold at all P sites, but as mentioned above, the expressions given below still apply if an at-site model is AR(1), when the θ-parameter is set to zero, or MA(1), when ϕ is set to zero. We assume that the regional mean trend b+ is a weighted mean of the b(j):

display math(4)

where commonly Wj = 1/P. Alternatively, if it were required to estimate the trend at an interpolated site, the Wj might be functions of distances between sites with records and the site to be interpolated. Algebra similar to that of the preceding section shows that the covariance between the estimated trend coefficients b(j), b(k) at gauging sites j and k is

display math(5)

where Vjk is an N × N nonsymmetric matrix with diagonal terms

display math(6)

with ρjk the cross-correlation between the two serially uncorrelated sequences {at(j)} and {at(k)} whose standard deviations are σa(j) and σa(k).Terms in successive supradiagonals (moving toward the matrix top right-hand corner) are

display math(7)

while those in the first, second…infradiagonals are of the same form but with the powers ϕ(k), ϕ(k)2, …, ϕ(k)N−2 replaced by ϕ(j), ϕ(j)2, …, ϕ(j)N−2. Hence, it is found that the variance of the regional estimate of trend b+ is

display math(8)

where W is the P × 1 vector of weights W = [W1, W2…WP]T defined in (4) above, and X is a P × P matrix with elements cov[b(j), b(k)] = ATVjk A given by (5) above.

2.3. An Approximate Test of the Hypothesis That the Trend Is Zero at All P Sites in the Region: H0: inline image(1) =  inline image(2) =  inline image(P) = 0

[11] Suppose that the model fitted at each of the P sites is Qt = α + εt with εt − ϕ ε t-1 = at - θ at-1: that is, with all trend terms set to zero. Having fitted this model, estimates of the serially uncorrelated random terms at sites j and k, say, can be computed, giving {a(j)}= {a1(j), a2(j)….aN(j)}, {a(k)}= {a1(k), a2(k)….aN(k)} for j, k = 1…P. From these sequences, a P × P matrix Σ0 can be calculated, giving the variances and covariances among the {a(j)} and {a(k)}. A variance-covariance matrix Σ1 can also be calculated from the sequences {a(j)}, {a(k)} obtained under the alternative hypothesis that not all the β(1), β(2),…,β(P) are zero: that is, by fitting the ARMA model to Qt(j) − b(j)t. Assuming multivariate normality, the expression

display math(9a)
display math(9b)

where N is the number of years: N = 78 in the present case. The expression (9b) gives the required test statistic [Johnson and Wichern, 1992, section 5.3], for which 2 ln Λ is proportional to χ2 distribution with P degrees of freedom.

2.4. An Approximate Test of the Hypothesis H0: inline image(1) =  inline image(2) = … inline image(P) =  inline image+ (i.e., There Is a Constant, Nonzero Trend At All Sites)

[12] When the assumption of multivariate normality of the {a(j)}= {a1(j), a2(j)….aN(j)}is valid, the test of the preceding section can be adapted to provide a test of H0. The model inline image, εt+ − ϕ+ εt-1+=at+ − θ+ at-1+ is fitted to the spatially averaged specific flows inline image giving an estimate b+ of β+; this is used to calculate Qt(i) − b+ t for i = 1…P. ARMA(1,1) models are then fitted to this variable at each of the P sites, giving a new set of random sequences {a(j)}, j=1…P, from which a new test statistic Λ+ is calculated, with 2 ln Λ+ compared with a χ2 distribution having one degree of freedom. Here, the denominator in equation (9a) has parameters α(i), β+, ϕ (i), θ(i), σa(i), (i = 1…P) and the numerator has parameters α(i), β(i), ϕ (i), θ(i), σa(i).

2.5. Numerical Example

[13] To illustrate the calculation, an example uses runoff records at 10 gauging sites in the Rio Iguaçu river basin, Brazil, for the period 1931–2008. Table 1 gives means and standard deviations of annual flow (m3s1) at the 10 sites, with upstream drainage areas; Figure 2, giving a plot of the annual specific yields, shows that they are highly correlated spatially. Table 2 gives estimates of the trends β, together with the ARMA(1,1) parameters ϕ, θ and standard deviation σa of the random sequence, calculated using the model Qt = α + β t + εt together with εt − ϕ ε t−1 =at − θ at1 at each site.

Table 1. Site Codes, Means and Standard Deviations of Mean Annual Discharge (m3s−1) at 10 Sites in the Rio Iguaçu Basin, Brazil, With Upstream Drainage Areas (km2)
Site Code6571727577
Mean105.29101.51106.37841.47988.84
SD41.4638.9340.73313.61396.03
Area36823929411634,54044,086
Site code78808183584
Mean1035.4867.641436.5972.35201.14
SD412.9626.37568.6128.2781.17
Area46,003229161,94824537379
Table 2. Estimates b of Trend Parameters β and ARMA(1,1) Parameters ϕ, θ, and σa at Each of the 10 Sitesa
Site Codeb ± SE[b]ϕθσar1(at)
  1. a

    Models were fitted to runoff expressed as mm per unit area and the trend estimate b is in units of millimeters of runoff per year. The standard error of the estimated trend b, obtained from equation (3) as SE[b] = inline image, is also shown. The final column shows the lag-one serial correlation calculated from the random components {at} of the ARMA(1,1) fitted to trend residuals. Approximate standard error of values in the last column: ±0.113.

650.151 ± 0.0600.2160.37310.59−0.023
710.120 ± 0.055−0.362−0.6409.12−0.029
720.122 ± 0.055−0.370−0.6449.10−0.030
750.133 ± 0.047−0.308−0.5358.07−0.035
770.133 ± 0.048−0.337−0.5638.23−0.029
780.134 ± 0.0480.317−0.5458.21−0.031
800.161 ± 0.062−0.172−0.35810.73−0.016
810.137 ± 0.049−0.299−0.5128.43−0.026
830.161 ± 0.062−0.168−0.35510.74−0.015
5840.125 ± 0.058−0.265−0.38410.57−0.033
Figure 2.

Normalized annual runoff (mm) for 1931–2008 at 10 sites in the R. Iguacu basin, Brazil.

[14] As set out above, the at-site trends b = ATQ and regional trend inline image are calculated directly from the annual streamflow data (converted to mm runoff per unit area), and their standard errors are calculated. This emphasizes the linearity of the trend, the coefficients AT being identical with those used in a simple linear regression without correlated errors. Other approaches are also possible, one of which is to minimize Σat2 with respect to both the ARMA (1,1) parameters and the trend parameter β simultaneously. This is equivalent to fitting a transfer-function model in which time (Year) is the explanatory variable. Whereas, with the method described above, the expression for var[b] is exact (always provided that the model is correct), the expression for var[b] when a transfer-function model is fitted is only approximate, and is derived from the information matrix of second derivatives of the likelihood function, evaluate at the likelihood maximum [Box and Jenkins, 1970]. Neither does the trend b have a specific algebraic form (c.f., b = ATQ) when estimated by fitting a transfer-function model.

[15] Table 3 shows results from the transfer-function calculation. Comparison of Table 3 with Table 2 shows that the at-site trends and estimates of the ARMA parameters are broadly similar (except for Site 584 where the estimated trend is much smaller), but standard errors of estimated trends are consistently slightly larger where the transfer-function method is used. An explanation might be that the transfer-function method searches for the maximum of a likelihood function in a space with one extra dimension, compared with the likelihood function maximized for Table 2; however, the differences in the standard errors SE[b] shown in Tables 2 and 3 are small. As discussed more fully below, Table 4 shows the effect on transfer-function fit of omitting the year 1983 from the streamflow records, since a strong El Niño event occurred then.

Table 3. As for Table 2, but With All of b, ϕ, θ, σa and SE[b] Estimated From a Transfer-Function Model With Time (Year) as Argumenta
Site Codeb ± SE[b]ϕθσar1(at)
  1. a

    Approximate Standard Error of Values in the Last Column: ± 0.113.

650.073 ± 0.064−0.180−0.35510.70−0.015
710.120 ± 0.056−0.374−0.6509.12−0.028
720.122 ± 0.056−0.378−0.6529.10−0.029
750.122 ± 0.048−0.307−0.5348.07−0.035
770.132 ± 0.049−0.309−0.5398.23−0.031
780.132 ± 0.049−0.317−0.5458.21−0.031
800.147 ± 0.063−0.269−0.45210.74−0.016
810.136 ± 0.050−0.258−0.4758.43−0.028
830.147 ± 0.064−0.164−0.35210.74−0.015
5840.015 ± 0.064−0.125−0.27710.81−0.013
Table 4. As for Table 3 But Omitting the Year 1983, When an Extreme El Niño Event Occurred (See Figure 1)a
Site Codeb ± SE[b]ϕθσar1(at)
  1. a

    Approximate Standard Error of Values in the last column: ±0.113.

650.124 ± 0.052−0.184−0.2779.63−0.018
710.105 ± 0.045−0.507−0.7137.96−0.049
720.107 ± 0.045−0.515−0.7187.94−0.050
750.112 ± 0.043−0.345−0.5287.47−0.030
770.112 ± 0.042−0.370−0.5467.44−0.029
780.112 ± 0.042−0.381−0.5577.42−0.031
800.136 ± 0.054−0.318−0.4549.69−0.009
810.115 ± 0.043−0.360−0.5187.58−0.029
830.137 ± 0.054−0.289−0.4289.69−0.008
5840.033 ± 0.053−0.169−0.2579.83−0.016

[16] Testing hypotheses about the trend coefficients and β+ needs care because the mean annual discharges are highly correlated between sites, so that variance-covariance matrices between the random sequences {a(j)} are close to singularity. To test the hypothesis H0: β(1) = β(2) = β(P) = 0, the likelihood ratio in equation (9a) above is Λ = {0.0070681461/0.0175247372}39, giving 2 ln Λ = 69.01. The tabulated value of χ2 with 10 degrees of freedom and cumulative probability 0.95 is 18.31, giving strong evidence against the null hypothesis that no trend exists at any of the 10 sites. A similar test of the hypothesis H0: β1+ = 0 gives Λ = {0.0132799909/0.0175247372}39 with 2 ln Λ = 21.08; the tabulated χ2 with 1 d.f. is 3.81 so that H0 is again decisively rejected.

[17] To assess whether the ARMA(1,1) model is adequate, the final columns of Tables 2-4 show the lag-one serial correlations in the calculated random components{ inline image} obtained when the model has been fitted. An approximate 95% confidence interval for these correlations, on the hypothesis that the { inline image} are white noise series, is ±2/ inline image = ±0.226, so that all the lag-one correlations (and those at higher lags, not shown) are consistent with being derived from uncorrelated series. Although small, all lag-one correlations were negative, further research is needed to explain why. Tests were also used to assess whether the { inline image} are consistent with Normal distributions both individually (i.e., their marginal distributions) and conjointly (multivariate normality). Tests were based on the empirical distribution functions of the { inline image}, and three tests—the Anderson-Darling, Cramer-von Mises, and Watson tests [Aitchison, 1986]—were used, giving good power against a wide range of alternative hypotheses. The four Sites 65, 80, 83, and 584 showed evidence of departure from (univariate) normality at the 5% level, when the estimate b = ATQ was used, and also when the transfer-function method was used to estimate β. Leaving out the El Niño year 1983, only the two sites 80 and 83 showed evidence of nonnormality (none when the Anderson-Darling test was used). But the hypothesis of multivariate normality was rejected (P < 0.01), whether or not the year 1983 was omitted, and whether or not the model was fitted by transfer-function method.

[18] Thus, while the estimated random sequences { inline image} are consistent with a lack of serial correlation, as required by the models described above, the additional assumption of multivariate normality is not. As emphasized above, however, multivariate normality is required only where tests of statistical significance, based on Normal theory, are required: the variance expressions (3) and (8) remain valid (always provided that the model in (1) is correct) with or without multivariate normality, and therefore provide measures of trend uncertainty. Further research is required to establish how far the observed departure from multivariate normality affects the criterion (equations (9a) and (9b)) in the numerical example, and whether a multivariate Box-Cox transformation [Johnson and Wichern, 1992, section 4.7] serves to transform the { inline image} sequences to approximate multivariate normality.

[19] Since the parameters ϕ, θ, σa are estimated by fitting the ARMA(1,1) model, a further question is “how do the sampling errors in estimates inline image, inline image, and inline image of these parameters affect the variances var[b] of equation (3) and var[b+] of equation (8)?” If the large-sample variances var[ inline image], var[ inline image], and var[ inline image], and the corresponding covariances between them, have been obtained from a maximum-likelihood estimation procedure, the elements vi j(i, j =1…N) of the matrix V in the equation var[b] = ATVA can be replaced by δT W δ, where W is the variance-covariance matrix of the estimates inline image, inline image, and inline image, and δ is the 3 × 1 vector of partial derivative [ inline image inline image inline image]T. Although nontrivial, the result of this calculation would show how—assuming large-sample theory to be valid—var[b] was affected by sampling errors in estimates of ϕ, θ, and σa. A similar procedure can be used for var[b+]. A limitation is that the partial derivatives will be expressed in terms of the true parameter values ϕ, θ, and σa for which estimates must again be substituted. An alternative would be a Bayesian approach in which posterior distributions of the trend parameters β and β+ are calculated from likelihood functions (possibly, multivariate normal) and prior distributions for the parameters ϕ, θ, and σ. With P sites, the posterior for β+ would therefore require priors for at least 3P+ P(P − 1)/2 parameters (more, if the fractional differentiation parameter d were estimated, more if p or q is greater than one). The calculation is beyond the scope of this technical note.

3. Analytical Extensions to Find the Causes of Trend

[20] Having identified a time-trend in streamflow, there is the need to explain it. Two possible explanations are that the trend is a consequence of annual variability in rainfall or of change in land use. It is not the purpose of this note to explore these explanations for the Rio Iguaçu basin, since they have been presented elsewhere. In the case of rainfall, for which annual totals will in general be available (although possibly calculated from data derived from a rain gauge network with density varying over the period of record), the expression given above for At =  inline image may be simply replaced by At =  inline image or, using a transfer-function method, by replacing the explanatory variable time t (year) by annual rainfall Pt. Extending the transfer function method to include several predictor (explanatory) variables is also straightforward [Box and Jenkins, 1971] and has been found useful where, as in the Iguaçu basin, data on the extent of land-use change through deforestation and urban development are fragmented. In the first stage of a two-stage analysis, rainfall Pt alone was used as an explanatory variable for trend in streamflow Qt, and in the second stage both time t (i.e., year) and rainfall Pt were used as explanatory variables. If inclusion of t significantly increased the maximized likelihood, the conclusion was that other factors besides fluctuations in rainfall—including land use, and possibly other factors correlated with time, such as changes to rating curves—contributed to the streamflow trend. This analytical approach is related to Granger causation [Granger, 1969; Hacker and Hatemi, 2006], in which “causation” may not be direct, but indirect in the sense that two explanatory variables which contribute to fluctuations in a variable of main interest (streamflow, in the present case) may themselves be caused to vary by a third variable whose identity is not well established. Thus an “unknown, unknown” variable, or geophysical process, may be the direct cause of streamflow trend, and of other “Granger-causative” variables that contribute to it.

[21] One such causative geophysical process might be the conjunction of global and oceanic climates that results in the occurrence of El Niño events, such as the extreme El Niño event of 1983 (see Figure 2) when annual rainfall was very much higher than normal, with consequent effects on streamflow. Omitting that year from the streamflow records at all 10 Iguaçu sites led to the results shown in Table 4, when the trend model with ARMA(1,1) residuals was fitted by a transfer-function model. Although the trends b are reduced at eight of the 10 sites, the standard errors of the trends, and hence the uncertainties inherent in them, are reduced at all 10 sites. Thus, the leverage exerted by the high specific streamflow of 1983 is removed, there is a compensation in the form of reduced annual fluctuations (compare the values of σa in Tables 3 and 4), leaving the trends in Table 4 greater than twice their standard errors, except for the one aberrant site 584.

Acknowledgments

[22] The author is grateful for financial support from AES Tietê and associated companies, granted under the ANEEL R&D Preferential Theme N 10 “The Effects of Climate Change on the Secure Energy of Hydropower Plants.” He is also grateful for constructive comments from an Associate Editor and three anonymous reviewers, which have greatly improved the version originally submitted.

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