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A topological restricted maximum likelihood (TopREML) approach to regionalize trended runoff signatures in stream networks

01 Jan 2015-Vol. 12, Iss: 1, pp 1355-1396
TL;DR: In this article, the key similarities and differences between TopREML and Top-kriging are also emphasized, and a new figure (Figure 1) has been included to clarify the key modeling assumptions.
Abstract: L. 115-155: We added a section in the introduction outlining the key conceptual assumptions behind the method, addressing flow propagation, the treatment of time and the conceptualization of the topology of the stream network in particularly. The key similarities and differences between TopREML and Top-kriging are also emphasized in this section. A new figure (Figure 1) has been included in the section to clarify the key modeling assumptions.

Summary (6 min read)

1 Introduction

  • Regionalizing runoff and streamflow for the purposes of making predictions in ungauged basins (PUB) continues to be one of the major contemporary challenges in hydrology.
  • At global, regional and local scales only a small fraction of catchments are monitored for streamflow (Blöschl et al., 2013), and this fraction is at risk of decreasing given the ongoing challenge of maintaining existing gauging stations (Stokstad, 1999).
  • Reliable information about local streamflows is essential for the management of water resources, especially in the context of changing climate, ecosystem and demography; flow prediction uncertainties are bound to propagate and lead to significantly suboptimal design and management decisions (e.g., Sivapalan et al., 2014).
  • Techniques are needed to maximize the use of available data in datascarce regions to accurately predict streamflow, while providing a reliable estimate of the related modeling uncertainty.

1.1 Linear models

  • There are a number of approaches to predicting runoff in ungauged catchments, including process-based modeling (e.g., Müller et al., 2014), graphical methods based on the construction of isolines (e.g., Bishop and Church, 1992), and statistical approaches.
  • Such linear models are well understood and widely implemented, not only for PUB (see review in Blöschl et al., 2013, p.83) but also across a wide variety of fields in the physical and social sciences (e.g., Myers, 1990).
  • Spatial correlation is generally problematic for linear model predictions, including the multiple regression approaches commonly used for regionalization.
  • If the residuals are independent and identically distributed (iid), the best linear unbiased predictions (BLUP) of both y and its uncertainty (i.e., Var(y)) can readily be obtained using ordinary least squares (OLS) regression.
  • Unfortunately, residuals are rarely iid in hydrological applications due to the spatial organization of hydrological processes around the topology of river channel networks.

1.2 Spatial correlation models

  • There are several techniques available to address spatially correlated data.
  • Huang and Yang, 1998), the theoretical justification for this approach is less robust than for point-scale processes.
  • Runoff is organized around a topological network of stream channels, and the covariance structure implied for prediction should reflect the higher correlation between streamflow at watersheds that are “flow connected” (i.e., share one or more subcatchments), compared to unconnected but spatially proximate catchments.
  • The bias increases quadratically with distance, meaning that estimates of the long-range variance (the sill) are strongly impacted by the presence of the trend, leading to an underestimation of the prediction uncertainty.
  • The second type of approach embraces this topological structure.

1.3 The topological restricted maximum likelihood approach

  • Rather than using a kriging estimator, the authors adopt a REML framework (Gilmour et al., 2004; Patterson and Thompson, 1971; Lark et al., 2006) to estimate variance parameters.
  • This use of a REML framework to estimate a linear mixed effect model on a topological support is termed topological restricted maximum likelihood .
  • For the local effects to form a suit- able basis for spatial interpolation, variations associated with temporal correlation (e.g., travel time effects) need to be removed, also known as – Treatment of time.
  • First, TopREML is only suitable for the regionalization of time-averaged and statistically stationary runoff properties (i.e., runoff signatures).
  • The underlying assumption is that runoff signatures of local flow generation regions that are close to each other (in Euclidian space) are more likely to be identical.

1.4 Paper outline

  • The authors first derive the TopREML estimator and its variance for mass conserving (i.e., linearly aggregated) variables, with extensions to some non-conservative variables (Sect. 2).
  • The authors then apply the approach in two case studies to evaluate its ability to predict mean runoff and runoff frequency by comparison to other available interpolation techniques: Sects. 3.1 and 4.1 present leave-one-out cross-validations in Nepal www.hydrol-earth-syst-sci.net/19/2925/2015/.
  • In both cases, TopREML performed similarly to the best alternative geostatistical method.
  • The authors then use numerical simulations to illustrate the effect of the two distinguishing features of TopREML: its ability to properly predict runoff using highly nested networks of stream gauges and its ability to properly estimate the prediction variance when accounting for observable features (Sects. 3.2 and 4.2).

2.1 Accounting for spatially correlated residuals

  • Linear models can be used to make predictions about hydrological variables along a network, provided that the models explicitly address the effects of network structure.
  • A mixed linear model approach provides a suitable framework for this accounting.
  • The effects of observable features on the hydrological outcome are assumed to be independent of the network, and retain their influence independently, as so-called “fixed effects”.
  • (2) To proceed, the authors assume that u and (and therefore y) are normally distributed with zero mean and are independent from each other.
  • The solution strategy adopted here is to prescribe a parametric form for G(φ), allowing the covariance structure along the network to be specified, and the likelihood function for the model to be written in terms of all five unknowns.

2.2 Covariance structure of mass conserving variables

  • In the linear mixed model framework, the propagation of hydrological variables through the flow network introduces Hydrol.
  • Firstly, linearly propagated variables, such as annual specific runoff, are discussed.
  • To account for the network structure, the catchment at any location along a stream can be subdivided into the IDA that are monitored for the first time by an upstream gauge.
  • The authors assume that this function is well approximated by an exponential function ρ(ckm,φ)= exp(−c/φ).
  • The nugget consists of the variance of processes that are spatially correlated over scales smaller than the sub-catchments and of measurement errors at the gauges.

2.4 E-BLUP and prediction variance at ungauged catchments

  • Knowing φ̂, gn can be readily obtained from the relative position of site n and the gauges in the river network.
  • The variance of the TopREML prediction error can be ex- pressed as Var(ỹn− yn)=.
  • If is an error that is truly iid and does not affect the true value of yn (e.g., measurement errors), then Eq. (14) corresponds to the mean square error of the TopREML prediction of yn.
  • If, by contrast, represents random variations of the true value of yn that are correlated over short distances (and so do not appear correlated in their data), then should be included in Eq. (11) and the prediction variance becomes Var(ỹn− yn)+ = Var(ỹn− yn)−+ σ 2, (15) because and u are independent.
  • In reality is likely composed of both spatially correlated and iid error components and the true variance will be somewhere between these two bounds (Lark et al., 2006).

2.5 Application to non-conservative variables

  • Unlike mean specific runoff, numerous streamflow signatures (e.g., runoff frequency or descriptors of the recession behavior) are non-conservative and cannot be expressed as linear combinations of their values in upstream subcatchments.
  • In such conditions the derivations in Sect. 2.2 cannot be applied and the correlation structure in Eq. (7) will lead to biased REML predictions.
  • Therefore, runoff frequency does not scale linearly through the river network.
  • Therefore, although recession constants themselves do not propagate linearly, their value in ungauged basins can be estimated by taking the inverse of TopREML predictions of average response times.

2.6 Implementation

  • To run the script, two vector data sets (e.g., ESRI Shapefiles) are needed as inputs – one containing the catchments where runoff is available and another containing the basins where predictions are to be made.
  • Catchment polygons and explanatory and predicted variables must be provided as attributes of the vector polygons.
  • The way in which the catchment polygons are nested provides the topology of the stream network.
  • TopREML uses the Broyden–Fletcher–Goldfarb– Shanno (BFGS) algorithm (Wright and Nocedal, 1999) to maximize the restricted log likelihood, though stochastic algorithms are required if a non-differentiable (e.g., spherical) covariance function is selected.
  • The authors found that initial values of [σ 20 ,φ0,ξ0] = [σ 2 LM,Eckm,1] worked well in their case studies, with σ 2LM the variance of the OLS residuals of the linear model and Eckm the average distance between IDA centroids.

3.1 Case studies

  • Observed streamflow data are used to evaluate the ability of TopREML to predict streamflow signatures in ungauged basins.
  • The assessment is based on leave-one-out cross-validations, where the tested model is applied to predict runoff at one basin based on observations from all the other basins.
  • The Austrian data set was directly taken from the rtop package (Skøien et al., 2014), where mean summer runoff observations are provided to demonstrate Top-kriging.
  • The Austrian data set did not contain additional observable features and previous studies have found spatial proximity to be a significantly better predictor of runoff than catchment attributes in Austria (Merz and Blöschl, 2005).

3.2.1 Network effects

  • Conventional geostatistical methods predict runoff by weighing observations from surrounding basins based on their geographic distance.
  • TopREML also incorporates the topology of the stream network by including or excluding basins based on their flow-connectedness.
  • This adds topological information to the determination of the covariance structure of runoff, at the expense of discarding information that could be derived from correlations between spatially proximate regions that are not connected to the gauge of interest by a flow path.
  • Assessing the net benefits of accounting for network effects requires being able to control the topology of the network, and thus requires numerical simulations.
  • A non-topological geostatistical method like universal kriging would include all basins within and exclude all basins beyond the spatial autocorrelation range.

3.2.2 Variance estimation and observable features

  • A key advantage of the REML framework is its ability to avoid the downward bias in the covariance function that affects kriging-based methods (including Top-kriging) when external trend coefficients are simultaneously estimated.
  • Again, empirical cross-validation analysis does not allow an assessment of this bias, because the observation data sets used contained only one observation per location.
  • We construct the observed prediction uncertainty by taking the standard deviation of the prediction errors across all 1000 Monte Carlo runs and compare it to the square root of the median predicted variance.the authors.the authors.
  • The authors compare TopREML and Top-kriging based on their ability to model prediction variance.
  • Conversely, including a trend in the model will cause the remaining error to mostly consists of (spatially uncorrelated) residuals, so in this case Eq. (14) is used.

4.1 Case studies

  • Basin-level predictions of the considered signatures are presented in Fig. 5 for the three cross-validation analyses described in Sect. 3.1. Figure 5 also provides box plots summarizing the distribution of the ensuing cross-validation errors.
  • In the three analyses, the prediction errors related to TopREML were comparable to the best alternate method: a linear model for annual specific runoff and Topkriging for runoff frequency and summer runoff .
  • Allowing for spatial correlation in the residuals (UK) decreased the median absolute error by 11 % compared to the linear model (LM) for runoff frequency in Nepal and 31 % for summer runoff in Austria.

4.2 Numerical simulation

  • Results from the Monte Carlo analysis are presented in Fig. 6, showing the outcomes of the two numerical experiments described in Sect. 3.2. Figure 6a and b shows the effect of network complexity on the performance of TopREML relative to the baseline performance of universal kriging.
  • The effect is expected to increase with Nouter and decrease with Ninner, reaching zero when 100 % of observed basins lie within the spatial correlation range and 0 % of the basins beyond the range are flow-connected.
  • A linear regression of the relative performance of TopREML against Nouter and Ninner showed that both trends are significant and in the expected direction.
  • In Fig. 6c, the Monte Carlo analysis showed that model uncertainty is well predicted by TopREML and strongly underestimated by Top-kriging, both with and without considering an external trend.

5.1 Performance of TopREML

  • Cross-validation outcomes suggest that TopREML is an attractive operational tool for predicting streamflow in ungauged basins.
  • Both results suggest that the benefit of accounting for network effects on correlations outweighs the cost of losing some information on the correlation between unconnected basins.
  • This allows TopREML to accurately predict modeling uncertainties even for highly trended and auto-correlated runoff signatures, as visible in the Monte Carlo analysis presented on Fig. 6c.
  • TopREML also has considerably lower computational requirements than Top-kriging, both in terms of input data and optimization complexity.
  • Indeed, TopREML does not rely on a distributed point process but assumes homogenous IDAs.

5.2 Model selection

  • The regionalization methods assessed in the cross-validation analysis range from simple linear regressions with strong independence assumptions, to complex geostatistical methods that allow for both spatial and topological correlations.
  • In that situation, parsimony prescribes selecting the least complex of the best performing methods.
  • A dense network of flow gauges is necessary for geostatistical methods to properly estimate the semivariogram and improve on predictions from linear regressions – the case studies suggest that the mean distance between the gauges must be on the order of half the spatial correlation range of the runoff signature.
  • Including these observable features in the model reduces the correlation scale of the residuals, possibly crossing the threshold below which geostatistics are not the most parsimonious approach.
  • In that case there is a tradeoff between relying on observable features or variance information to make a prediction, and parsimony and stationarity considerations come into play when selecting the regionalization model.

6 Conclusions

  • The approach takes into account the spatially correlated nature of runoff and the nested character of streamflow networks.
  • The method was successfully tested in cross-validation analyses on mass conserving (mean streamflow) and nonconservative (runoff frequency) runoff signatures in Nepal (sparsely gauged) and Austria (densely gauged), where it matched the performance of the best alternative method: Topkriging in Austria and linear regression in Nepal.
  • TopREML outperformed Top-kriging in the prediction of uncertainty in Monte Carlo simulations and its performance is robust to the inclusion of observable features.
  • This flexibility, along with its ability to provide a reliable estimate of the prediction uncertainty, offer considerable scope to use this computationally inexpensive method for practical PUB applications.

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Content maybe subject to copyright    Report

Hydrol. Earth Syst. Sci., 19, 2925–2942, 2015
www.hydrol-earth-syst-sci.net/19/2925/2015/
doi:10.5194/hess-19-2925-2015
© Author(s) 2015. CC Attribution 3.0 License.
TopREML: a topological restricted maximum likelihood approach
to regionalize trended runoff signatures in stream networks
M. F. Müller and S. E. Thompson
Department of Civil and Environmental Engineering, Davis Hall, University of California, Berkeley, CA, USA
Correspondence to: M. F. Müller (marc.f.muller@gmail.com)
Received: 12 December 2014 Published in Hydrol. Earth Syst. Sci. Discuss.: 29 January 2015
Revised: 01 May 2015 Accepted: 03 June 2015 Published: 24 June 2015
Abstract. We introduce topological restricted maximum
likelihood (TopREML) as a method to predict runoff sig-
natures in ungauged basins. The approach is based on the
use of linear mixed models with spatially correlated ran-
dom effects. The nested nature of streamflow networks is
taken into account by using water balance considerations to
constrain the covariance structure of runoff and to account
for the stronger spatial correlation between flow-connected
basins. The restricted maximum likelihood (REML) frame-
work generates the best linear unbiased predictor (BLUP)
of both the predicted variable and the associated prediction
uncertainty, even when incorporating observable covariates
into the model. The method was successfully tested in cross-
validationanalyses on mean streamflow and runoff frequency
in Nepal (sparsely gauged) and Austria (densely gauged),
where it matched the performance of comparable methods
in the prediction of the considered runoff signature, while
significantly outperforming them in the prediction of the as-
sociated modeling uncertainty. The ability of TopREML to
combine deterministic and stochastic information to generate
BLUPs of the prediction variable and its uncertainty makes it
a particularly versatile method that can readily be applied in
both densely gauged basins, where it takes advantage of spa-
tial covariance information, and data-scarce regions, where it
can rely on covariates, which are increasingly observable via
remote-sensing technology.
1 Introduction
Regionalizing runoff and streamflow for the purposes of
making predictions in ungauged basins (PUB) continues to
be one of the major contemporary challenges in hydrology.
At global, regional and local scales only a small fraction
of catchments are monitored for streamflow (Blöschl et al.,
2013), and this fraction is at risk of decreasing given the
ongoing challenge of maintaining existing gauging stations
(Stokstad, 1999). Reliable information about local stream-
flows is essential for the management of water resources, es-
pecially in the context of changing climate, ecosystem and
demography; flow prediction uncertainties are bound to prop-
agate and lead to significantly suboptimal design and man-
agement decisions (e.g., Sivapalan et al., 2014). Techniques
are needed to maximize the use of available data in data-
scarce regions to accurately predict streamflow, while pro-
viding a reliable estimate of the related modeling uncertainty.
1.1 Linear models
There are a number of approaches to predicting runoff in un-
gauged catchments, including process-based modeling (e.g.,
Müller et al., 2014), graphical methods based on the con-
struction of isolines (e.g., Bishop and Church, 1992), and
statistical approaches. Statistical approaches are often imple-
mented via linear regression, wherein the runoff signature of
interest is considered to be an unobservable random variable
correlated with observable features of both gauged and un-
gauged basins (e.g., rainfall, topography). Such linear mod-
els are well understood and widely implemented, not only
for PUB (see review in Blöschl et al., 2013, p.83) but also
across a wide variety of fields in the physical and social sci-
ences (e.g., Myers, 1990).
Spatial correlation is generally problematic for linear
model predictions, including the multiple regression ap-
proaches commonly used for regionalization. For example if
these models predict a hydrologic outcome, y, using a matrix
Published by Copernicus Publications on behalf of the European Geosciences Union.

2926 M. F. Müller and S. E.Thompson: TopREML runoff regionalization on stream networks
X of observed features then the linear model has the form:
y = Xτ + η. (1)
Here τ is an a priori unknown set of weights that represent
the influence of each external trend on the hydrological out-
come being modeled. The residuals, η, are the observed vari-
ation of y that cannot be explained by a linear relation with
X. If the residuals are independent and identically distributed
(iid), the best linear unbiased predictions (BLUP) of both y
and its uncertainty (i.e., Var(y)) can readily be obtained us-
ing ordinary least squares (OLS) regression. Unfortunately,
residuals are rarely iid in hydrological applications due to
the spatial organization of hydrological processes around the
topology of river channel networks. This organization has the
potential to introduce non-random spatial correlations with a
structure imposed by the river network. To recover a suitable
model in which residuals remain independent requires that
the model structure be altered to explicitly account for the
spatial and topological correlation in the residuals.
1.2 Spatial correlation models
There are several techniques available to address spatially
correlated data. Within PUB, kriging-based geostatistical
methods (Cressie, 1993) have been widely used (e.g., Huang
and Yang, 1998; Gottschalk et al., 2006; Sauquet, 2006;
Sauquet et al., 2000; Skøien et al., 2006). In a geostatisti-
cal framework, a parametric function is used to model the
relationship between distance and covariance in observa-
tions. The ensuing semi-variogram is assumed to be homoge-
nous in space, and predictions at a point are computed as a
weighted sum of the available observations. The weights are
chosen to minimize the variance while meeting a given con-
straint on the expected value of the prediction. In ordinary
kriging for PUB applications, this given constraint is sim-
ply the average of the streamflow signature as observed in
gauged catchments. Ordinary kriging can also be extended
as “universal kriging” to include a linear combination of
observable features (Olea, 1974). Kriging approaches are
widely used to predict spatially distributed point-scale pro-
cesses like soil properties (e.g., Goovaerts, 1999) and cli-
matic features (e.g., Goovaerts, 2000). Although ordinary
kriging has also been used to interpolate runoff (e.g., Huang
and Yang, 1998), the theoretical justification for this ap-
proach is less robust than for point-scale processes. Runoff is
organized around a topological network of stream channels,
and the covariance structure implied for prediction should
reflect the higher correlation between streamflow at water-
sheds that are “flow connected” (i.e., share one or more sub-
catchments), compared to unconnected but spatially proxi-
mate catchments. Currently, two broad classes of geostatis-
tical methods accommodate this network-aligned correlation
structure.
The first suite of methods posits the existence of an un-
derlying point-scale process, which is assumed to have a
spatial auto-correlation structure that allows kriging to be
applied. Because the runoff point-scale process is only ob-
served as a spatially integrated measure made at specific
gauged locations along an organized network of streams,
the spatial auto-correlation structure of the point-scale pro-
cess cannot itself be observed. Block-kriging approaches
(Gottschalk et al., 2006; Sauquet, 2006; Sauquet et al., 2000)
infer the semi-variogram of the (unobserved) point scale so
as to best reproduce the observed spatial correlation of the
area-integrated runoff at the gauges a procedure known as
regularization. The topology of the network is implicitly ac-
counted for by the fact that nested catchments have over-
lapping areas, which affect the relation between observed
(area integrated) and modeled (point-scale) covariances. Yet,
complex catchment shapes complicate the regularization of
semi-variograms, meaning that the estimation of the point-
scale process becomes analytically intractable and requires a
trial-and-error approach in most practical applications (e.g.,
Top-kriging; Skøien et al. (2006)). Top-kriging is an ex-
tension of the block-kriging approach that accommodates
non-stationary variables and short observation records. Top-
kriging provides an improved prediction method for hydro-
logical variables when compared to ordinary kriging or lin-
ear regression techniques (Laaha et al., 2014; Viglione et al.,
2013; Castiglioni et al., 2011) and was recently extended to
account for deterministic trends (Laaha et al., 2013). Top-
kriging represents an important advance for PUB, but it does
have a few drawbacks: (i) the regularization process is un-
intuitive, and requires extensive trial-and-error to determine
both the form of a suitable point-scale variogram, and its pa-
rameters; (ii) this trial-and-error process is likely to be com-
putationally expensive; (iii) like all kriging techniques, the
estimation of the variogram is challenging when account-
ing for observable features: the presence of an unknown
trend coefficient and variogram leads to an underdetermined
problem, making consistent estimates for both challenging.
Cressie (1993, p. 166) showed that the presence of a trend
tends to impose a spatially inhomogeneous, negative bias on
the estimated semi-variogram. The bias increases quadrati-
cally with distance, meaning that estimates of the long-range
variance (the sill) are strongly impacted by the presence of
the trend, leading to an underestimation of the prediction un-
certainty. This bias, however, only marginally affects the pre-
diction itself.
Geomorphological considerations of the topology of a
river network generally focus on the channels, and lead to
an intuitive conceptualization that topological interpolation
should focus on runoff correlations along flow paths. The
second type of approach embraces this topological structure.
It does not consider a point-scale runoff generation process,
but instead models the hillslope-scale runoff delivery pro-
cess to the channel network as a uni-dimensional directed
tree (Cressie et al., 2006; Ver Hoef et al., 2006). Runoff cor-
relation is expected to decrease with the distance along the
stream following a known parametric function. However, un-
Hydrol. Earth Syst. Sci., 19, 2925–2942, 2015 www.hydrol-earth-syst-sci.net/19/2925/2015/

M. F. Müller and S. E.Thompson: TopREML runoff regionalization on stream networks 2927
like Euclidian distances, the streamwise distance does not
have the necessary properties to provide a solvable kriging
system. This issue is addressed in Cressie et al. (2006) and
Ver Hoef and Peterson (2010), where streamflow is mod-
eled as a random process represented by a Brownian mo-
tion that starts at the trunk of the tree (i.e., the river mouth)
moves upstream, bifurcates and evolves independently on
each branch. The resulting model only allows spatial depen-
dence with points that are upstream on the river network
and provides a positive definite covariance matrix that is
estimated through restricted maximum likelihood (REML).
Models of this nature have been successfully tested on stream
chemistry data (Ver Hoef et al., 2006) and further developed
to also allow spatial auto-correlation among random vari-
ables on stream segments that do not share flow, with po-
tential applications to the modeling of the concentration of
upstream moving species (e.g., fishes or insects) (Ver Hoef
and Peterson, 2010). While these methods do not account for
the streamflow generation process, they avoid the conceptual
and prediction uncertainty challenges confronted by kriging
techniques.
1.3 The topological restricted maximum likelihood
approach
Inspired by both types of approaches, here we present a
method based on the use of linear mixed models to generate
a BLUP for hydrological variables on a flow network. Rather
than using a kriging estimator, we adopt a REML framework
(Gilmour et al., 2004; Patterson and Thompson, 1971; Lark
et al., 2006) to estimate variance parameters. This reduces
the bias on the semi-variogram by allowing the variance to be
estimated independently from the trend coefficients (Cressie,
1993; Lark et al., 2006). This use of a REML framework to
estimate a linear mixed effect model on a topological sup-
port is termed topological restricted maximum likelihood
(TopREML). The approach is based on the following con-
ceptual assumptions:
Flow generation and propagation: similar to Top-
kriging, runoff is assumed to be generated at a point
scale on the landscape, from where it is routed to a
channel and measured at a gauge (Fig. 1i). Runoff ob-
servations made at any individual gauge (Fig. 1ii) can
be broken up into a local contribution, derived from
a never previously gauged catchment area, and an up-
stream contribution that was previously observed at up-
stream gauge(s) along the channel (Fig. 1iii). TopREML
disaggregates all flow contributions into a cascade of lo-
cal components, as observed at each successive gauge,
and uses these characteristics to constrain the covari-
ance structure of runoff and to account for the stronger
spatial correlations between flow-connected basins.
Treatment of time: for the local effects to form a suit-
able basis for spatial interpolation, variations associated
with temporal correlation (e.g., travel time effects) need
to be removed. This is achieved by considering time-
averaged streamflow data, with the proviso that the time
averaging window is much greater than the character-
istic catchment and channel response timescales. This
treatment of time has several specific consequences.
First, TopREML is only suitable for the regionaliza-
tion of time-averaged and statistically stationary runoff
properties (i.e., runoff signatures). Stationarity is neces-
sary to ensure that the water balance assumption used
to separate local from upstream runoff contributions is
valid. However, as a consequence, TopREML cannot be
used to interpolate transient signatures, such as those as-
sociated with real-time forecasting. Nor can it be used
to describe runoff properties that are correlated over
timescales larger than the time averaging window. Be-
cause of the stationarity assumption applied, all corre-
lation arguments described in this manuscript refer to
the spatial, and not temporal, correlation of the runoff
signatures.
Network topology: network topology in TopREML
also follows a conceptual model that is similar to the
model posited by Top-kriging. Topology is conceptu-
alized by area connectivity. That is, flow-connected
gauges are characterized by overlapping drainage ar-
eas. Unlike Top-kriging, TopREML does not require in-
formation about a spatially random point process, but
solely relies on information measured at the gauges.
It uses the inter-centroidal Euclidian distance between
drainage areas of the local flow contributions at each
gauge the isolated drainage areas (IDA) as a dis-
tance metric to compute streamflow correlation. The
underlying assumption is that runoff signatures of lo-
cal flow generation regions that are close to each other
(in Euclidian space) are more likely to be identical.
Although TopREML does not require that the charac-
teristics of a point-scale runoff generation process are
known in order to support interpolation (a necessary re-
quirement for Top-kriging), the existence of such a point
process is consistent with the treatment of spatial corre-
lation in TopREML. To illustrate this consistency, a styl-
ized example relating point-scale runoff generation to
the existence of a covariance structure that relates flow-
connected gauges is outlined in Appendix A.
1.4 Paper outline
We first derive the TopREML estimator and its variance for
mass conserving (i.e., linearly aggregated) variables, with ex-
tensions to some non-conservative variables (Sect. 2). We
then apply the approach in two case studies to evaluate its
ability to predict mean runoff and runoff frequency by com-
parison to other available interpolation techniques: Sects. 3.1
and 4.1 present leave-one-out cross-validations in Nepal
www.hydrol-earth-syst-sci.net/19/2925/2015/ Hydrol. Earth Syst. Sci., 19, 2925–2942, 2015

2928 M. F. Müller and S. E.Thompson: TopREML runoff regionalization on stream networks
A
B
C
t
q
A
B
C
A’
B’
C’
(i) (ii) (iii)
Figure 1. Conceptual flow propagation model. (i) Runoff is generated continuously by a spatially distributed point process and drained to
the stream network. (ii) When monitored by stream gauges, runoff is spatially integrated over the corresponding catchment and temporally
averaged at the chosen observation frequency (e.g., daily streamflow). (iii) The model conceptualizes the catchments as isolated drainage
areas (IDA) (A
0
, B
0
, and C
0
) representing the local runoff contribution to each gauge. The flow actually measured at each gauge is the sum
of the upstream IDA.
(sparse gauges, significant trends) and Austria (dense gauge
network, no observed trends). In both cases, TopREML per-
formed similarly to the best alternative geostatistical method.
We then use numerical simulations to illustrate the effect
of the two distinguishing features of TopREML: its ability
to properly predict runoff using highly nested networks of
stream gauges and its ability to properly estimate the pre-
diction variance when accounting for observable features
(Sects. 3.2 and 4.2). Finally, we discuss the limits and de-
lineate the context in which TopREML and geostatistical
methods in general can successfully be applied to predict
streamflow signatures in ungauged basins (Sect. 5).
2 Theory
2.1 Accounting for spatially correlated residuals
Linear models can be used to make predictions about hydro-
logical variables along a network, provided that the models
explicitly address the effects of network structure. A mixed
linear model approach provides a suitable framework for this
accounting. In this framework, the effects of observable fea-
tures on the hydrological outcome are assumed to be inde-
pendent of the network, and retain their influence indepen-
dently, as so-called “fixed effects”. The role of spatial struc-
ture is assumed to lead to correlation specifically in the resid-
uals η. The residuals are split into two parts: (i) one contain-
ing “random effects”, u, that exhibit spatial correlation along
the flow network and (ii) a remaining, spatially independent,
white noise term, , which does not have any spatial struc-
ture. With these assumptions, the mixed linear model is writ-
ten as:
y = X
|{z}
Trends:
Explanatory
variables
(N × k)
τ
|{z}
Coefficients
(k × 1)
+ I
N
|{z}
Identity
Matrix
(N × N)
u
|{z}
Correlated
random
effects
(N × 1)
+
|{z}
Residuals,
uncorrelated
errors
(N × 1)
. (2)
To proceed, we assume that u and (and therefore y) are nor-
mally distributed with zero mean and are independent from
each other. The variance associated with is denoted σ
2
, the
variance of u is assumed to be proportional to σ
2
according
to some ratio, ξ, and finally, u is assumed to have a spatial
dependence captured by a correlation structure G, which is
related to the spatial layout of gauges along the river network
and a distance parameter φ (the correlation range). Thus, the
random effects can be specified as
u
N

0
0
,σ
2
ξG) 0
0 I
N

. (3)
To solve this mixed model, ve unknowns must be found:
σ
2
, ξ , φ, the fixed (τ ) and random (u) effects. Once τ and u
are known, the empirical best linear unbiased prediction (E-
BLUP) of y can be made at ungauged locations (Lark et al.,
2006). The solution strategy adopted here is to prescribe a
parametric form for G), allowing the covariance structure
along the network to be specified, and the likelihood function
for the model to be written in terms of all five unknowns.
Identifying the parameter values that optimize this model
thus simultaneously solves for the correlation structure, co-
variance parameters, fixed and random effects. To proceed
with the specification of G), however, the form of the co-
variance structure that arises along the network needs to be
addressed.
2.2 Covariance structure of mass conserving variables
In the linear mixed model framework, the propagation of
hydrological variables through the flow network introduces
Hydrol. Earth Syst. Sci., 19, 2925–2942, 2015 www.hydrol-earth-syst-sci.net/19/2925/2015/

M. F. Müller and S. E.Thompson: TopREML runoff regionalization on stream networks 2929
topological effects into the covariance structure of that vari-
able. Firstly, linearly propagated variables, such as annual
specific runoff, are discussed. Nonlinearly propagating vari-
ables can in some cases be transformed to allow the linear
solutions to be used (as outlined in Sect. 2.5). Consider a set
of streamflow gauges monitoring a watershed as illustrated
in Fig. 1ii. Because of the nested nature of the river network,
the catchment area related to any upstream gauge is entirely
included within the area drained by all downstream gauges.
To account for the network structure, the catchment at any
location along a stream can be subdivided into the IDA that
are monitored for the first time by an upstream gauge. This
is illustrated in Fig. 1iii, and leads to a subdivision into non-
overlapping areas, each associated with the most upstream
gauge that monitors them. In making this subdivision, it is
implicitly assumed that the timescales at which a hydrologi-
cal variable is propagated in the channel are negligible com-
pared with the timescales on which hillslope effects operate
(a generally valid assumption for small to moderately sized
watersheds; see D’Odorico and Rigon, 2003). IDAs can be
associated with both gauged locations and ungauged loca-
tions. In what follows, indices i, j, k and m are used to refer
to gauged sites, while index n refers to ungauged sites where
a prediction is to be made.
With these assumptions, observations of y
i
made at gauge
i can be expressed as a linear combination of contributions
from the upstream IDAs:
y
i
=
kUP
i
X
k=i
a
k
y
0
k
, (4)
where y
0
k
is the contribution of the IDA related to gauge k
(that is, y
i
is equivalent to y
0
i
only if there are no gauges up-
stream of gauge i); UP is the set of IDA monitored by gauges
that are located upstream of i; a
k
= A
k
/
P
UP
m=i
A
m
1 is the
surface area of the drainage area k normalized by the total
watershed area upstream of gauge i. The covariance between
observations of y made at different gauges can then be ex-
pressed as
Cov
y
i
,y
j
= E
y
i
y
j
E
y
i
E
y
j
=
kUP
i
X
k=i
mUP
j
X
m=j
a
k
a
m
E
y
0
k
y
0
m
kUP
i
X
k=i
a
k
E
y
0
k
!
mUP
j
X
m=j
a
m
E
y
0
m
With E
y
0
k
y
0
m
= Cov
y
0
k
y
0
m
+ E
y
0
k
E
y
0
m
, we have
Cov
y
i
,y
j
=
kUP
i
X
k=i
mUP
j
X
m=j
a
k
a
m
Cov
y
0
k
,y
0
m
, (5)
where Cov
y
0
k
,y
0
m
is the covariance between the contribu-
tions of sub-catchments k and m. By summing over UP in
Eq. (5) (rather then the complete set of available gauges),
the model assumes no correlation between runoff observed
at flow-unconnected gauges.
Here we assume that the area-averaged process y
0
is drawn
from a second-order stationary random process, and that the
covariance between y
0
k
and y
0
m
will depend only on the rela-
tive position of sub-catchments m and k, given some speci-
fied correlation function ρ(·) of the distance c
km
between the
centroids of the two sub-catchments (Cressie, 1993). We as-
sume that this function is well approximated by an exponen-
tial function ρ(c
km
,φ) = exp(c/φ). A justification for this
assumption, which reproduces the streamflow variances ob-
served in our case studies well (Fig. 8), is derived for strongly
idealized conditions in Appendix A. Finally, because the ob-
servations made at the gauges represent an area-averaged
process, the averaging generates a nugget variance σ
2
that is
homogenous across observations. The nugget consists of the
variance of processes that are spatially correlated over scales
smaller than the sub-catchments (see Appendix A) and of
measurement errors at the gauges.
With this background, the covariance matrix of y can be
expressed as
Cov
y
i
,y
j
= ξσ
2
kUP
i
X
k=i
mUP
j
X
m=j
a
k
a
m
ρ
(
c
km
,φ
)
+ σ
2
= σ
2
·
ξU[A R]U
T
+ I
N
, (6)
where σ
2
= Var(y
0
k
,y
0
k
), U
i,j
= 1{j UP
i
}, A = aa
T
, and
where R
i,j
= ρ(c
i,j
,φ). ·] denotes the element-by-
element matrix multiplication. The matrix G describing the
correlation between the random effects in Eq. (3) is finally
G) = U[A R)]U
T
. (7)
The topology of the network is described by the matrix U,
which ensures that only those catchments that are on the
same sub-network (upstream or downstream) of the consid-
ered gauge are utilized in the determination of the covari-
ance of y. This spatial constraint comes at the expense of ne-
glecting potential correlations with neighboring catchments
that are not flow-connected, and the effects of this tradeoff
are investigated in the Monte Carlo experiment described in
Sect. 3.2. The effect of spatial proximity is addressed by use
of the Euclidian distance between catchment centroids (ma-
trix R), and the effect of scale is accounted for by weighting
by the catchment area of the IDAs (matrix A).
2.3 REML estimation
The restricted maximum likelihood approach partitions the
likelihood of y N
Xτ , σ
2
G+ I
N
)
into two parts, one
of which is independent of τ (Corbeil and Searle, 1976). This
allows the determination of fixed effects and the variance pa-
rameters of the model (here σ
2
, φ and ξ) to be undertaken
separately. The variance parameters are then estimated by
www.hydrol-earth-syst-sci.net/19/2925/2015/ Hydrol. Earth Syst. Sci., 19, 2925–2942, 2015

Citations
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01 Jan 2006
TL;DR: In this paper, the authors describe the REML-E-BLUP method and illustrate the method with some data on soil water content that exhibit a pronounced spatial trend, which is a special case of the linear mixed model where our data are modelled as the additive combination of fixed effects (e.g. the unknown mean, coefficients of a trend model), random effects (the spatially dependent random variation in the geostatistical context) and independent random error (nugget variation in geostatsistics).
Abstract: Geostatistical estimates of a soil property by kriging are equivalent to the best linear unbiased predictions (BLUPs). Universal kriging is BLUP with a fixed-effect model that is some linear function of spatial coordinates, or more generally a linear function of some other secondary predictor variable when it is called kriging with external drift. A problem in universal kriging is to find a spatial variance model for the random variation, since empirical variograms estimated from the data by method-of-moments will be affected by both the random variation and that variation represented by the fixed effects. The geostatistical model of spatial variation is a special case of the linear mixed model where our data are modelled as the additive combination of fixed effects (e.g. the unknown mean, coefficients of a trend model), random effects (the spatially dependent random variation in the geostatistical context) and independent random error (nugget variation in geostatistics). Statisticians use residual maximum likelihood (REML) to estimate variance parameters, i.e. to obtain the variogram in a geostatistical context. REML estimates are consistent (they converge in probability to the parameters that are estimated) with less bias than both maximum likelihood estimates and method-of-moment estimates obtained from residuals of a fitted trend. If the estimate of the random effects variance model is inserted into the BLUP we have the empirical BLUP or E-BLUP. Despite representing the state of the art for prediction from a linear mixed model in statistics, the REML-E-BLUP has not been widely used in soil science, and in most studies reported in the soils literature the variogram is estimated with methods that are seriously biased if the fixed-effect structure is more complex than just an unknown constant mean (ordinary kriging). In this paper we describe the REML-E-BLUP and illustrate the method with some data on soil water content that exhibit a pronounced spatial trend.

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Journal ArticleDOI
TL;DR: Dralle et al. as mentioned in this paper developed a probabilistic model for the persistence time of dry season high flow conditions, which accurately captures the mean of the persist-ence time distribution, but underestimates its variance.
Abstract: PUBLICATIONS Water Resources Research RESEARCH ARTICLE Dry season streamflow persistence in seasonal climates 10.1002/2015WR017752 David N. Dralle 1 , Nathaniel J. Karst 2 , and Sally E. Thompson 1 Key Points: Derived probabilistic model for the persistence time of dry season high flow conditions Successfully predicts mean, but not variance; attributed to inter-annual recession model variation Proposed framework for using crossing statistics (e.g., persistence time) to forecast ecologic risk Supporting Information: Supporting Information S1 Figure S1 Figure S2 Correspondence to: D. N. Dralle, dralle@berkeley.edu Citation: Dralle, D. N., N. J. Karst, and S. E. Thompson (2015), Dry season streamflow persistence in seasonal climates, Water Resour. Res., 51, doi:10.1002/2015WR017752. Received 26 JUN 2015 Accepted 8 DEC 2015 Accepted article online 13 DEC 2015 C 2015. American Geophysical Union. V All Rights Reserved. DRALLE ET AL. Department of Civil and Environmental Engineering, University of California, Berkeley, California, USA, 2 Division of Mathematics and Science Division, Babson College, Wellesley, Massachusetts, USA Abstract Seasonally dry ecosystems exhibit periods of high water availability followed by extended intervals during which rainfall is negligible and streamflows decline. Eventually, such declining flows will fall below the minimum values required to support ecosystem functions or services. The time at which dry sea- son flows drop below these minimum values (Q * ), relative to the start of the dry season, is termed the ‘‘persistence time’’ (T Q ). The persistence time determines how long seasonal streams can support various human or ecological functions during the dry season. In this study, we extended recent work in the stochas- tic hydrology of seasonally dry climates to develop an analytical model for the probability distribution function (PDF) of the persistence time. The proposed model accurately captures the mean of the persist- ence time distribution, but underestimates its variance. We demonstrate that this underestimation arises in part due to correlation between the parameters used to describe the dry season recession, but that this correlation can be removed by rescaling the flow variables. The mean persistence time predictions form one example of the broader class of streamflow statistics known as crossing properties, which could feasibly be combined with simple ecological models to form a basis for rapid risk assessment under different climate or management scenarios. 1. Introduction Pronounced variability in precipitation is the defining characteristic of seasonally dry ecosystems (SDE) [Fati- chi et al., 2012; Vico et al., 2014], which cover nearly 30% of the planet and contain about 30% of the Earth’s population [Peel and Finlayson, 2007; CIESIN, 2012]. In these regions, a distinct rainy season is followed by a pronounced dry season during which rainfall makes a small or negligible contribution to the water balance. As a consequence, the availability of dry season surface water resources depends strongly on streamflow, which is generated primarily from the storage and subsequent discharge of antecedent wet season rainfall in the subsurface [Brahmananda Rao et al., 1993; Samuel et al., 2008; Andermann et al., 2012]. Because these transient stores are strongly influenced by the characteristics of the wet season climate, dry-season water availability can be highly variable from year to year in many SDE’s [Samuel et al., 2008; Andermann et al., 2012]. This hydroclimatic variability leaves SDE’s, considered important ‘‘hot spots’’ of biodiversity [Miles et al., 2006; Klausmeyer and Shaw, 2009], and the human populations that depend upon them susceptible to threats, such as soil erosion, deforestation, and water diversions [Miles et al., 2006; Underwood et al., 2009]. Future climate scenarios are projected to further intensify wet season rainfall variability in many SDEs [e.g., Gao and Giorgi, 2008; Garc ia-Ruiz et al., 2011; Dominguez et al., 2012], necessitating models which can pre- dict the response of water resources to climatic change in order to measure the corresponding risk to local ecosystems and human populations [Vico et al., 2014; M€ uller et al., 2014]. Stochastic methods have a 30 year history of use in deriving simple, process-based models for the probabil- ity distributions of hydrologic variables, such as soil moisture, streamflow, and associated ecological responses [Milly, 1993; Szilagyi et al., 1998; Rodriguez-Iturbe et al., 1999; Laio, 2002; Botter et al., 2007; Thomp- son et al., 2013, 2014]. To date, the majority of stochastic analytical models for hydrology have been devel- oped under conditions where the climatic forcing does not exhibit strong seasonality [Rodriguez-Iturbe et al., 1999; Porporato et al., 2004; Botter et al., 2007]. Those studies that have considered the effects of sea- sonality in rainfall or evaporative demand have either focused on the mean dynamics of the variable of interest [Laio, 2002; Feng et al., 2012, 2015] or excluded a treatment of the transient dynamics between the wet and dry seasons [D’odorico et al., 2000; Miller et al., 2007; Kumagai et al., 2009]. 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TL;DR: The results of this study confirm the strong impact of land in producing short spatial scale convective rain and should be useful in the design of convective schemes for general circulation models and for precipitation error covariance models for use in numerical weather prediction and associated data assimilation schemes.
Abstract: The local spatial scales of tropical precipitating systems were studied using Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI) rain rate imagery from the TRMM satellite. Rain rates were determined from TMI data using the Goddard Profiling (GPROF) Version 5 algorithm. Following the analysis of Ricciardulli and Sardeshmukh (RS), who studied local spatial scales of tropical deep convection using global cloud imagery (GCI) data, active precipitating months were defined alternatively as those having greater than either 0.1 mm/h or 1 mm/h of rain for more than 5% of the time. Spatial autocorrelation values of rain rate were subsequently computed on a 55/spl times/55 km grid for convectively active months from 1998 to 2002. The results were fitted to an exponential correlation model using a nonlinear least squares routine to estimate a spatial correlation length at each grid cell. The mean spatial scale over land was 90.5 km and over oceans was 122.3 km for a threshold of 0.1 mm/h of rain with slightly higher values for a threshold of 1 mm/h of rain. An error analysis was performed which showed that the error in these determinations was of order 2% to 10%. The results of this study should be useful in the design of convective schemes for general circulation models and for precipitation error covariance models for use in numerical weather prediction and associated data assimilation schemes. The results of the TMI study also largely concur with those of RS, although the more direct relationship between the TMI data and rain rate relative to the GCI imagery provide more accurate correlation length estimates. The results also confirm the strong impact of land in producing short spatial scale convective rain.

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Cites background from "A topological restricted maximum li..."

  • ...Possibly, the most obvious stream variable is runoff (Müller and Thompson, 2015), however, since there is a correlation between flow-rate and the dilution capacity of streams, there aremanyother variables related towater-flow, such as the measurement of water physicochemical variables,…...

    [...]

  • ...New models of covariance valid for stream networks (Laaha et al., 2012; Müller and Thompson, 2015) v....

    [...]

01 Jan 2006
TL;DR: In this paper, a large class of valid spatial covariance models that incorporate flow and stream distance by using spatial moving averages were developed. But these models are not valid for stream networks.
Abstract: We develop spatial statistical models for stream networks that can estimate relationships between a response variable and other covariates, make predictions at unsampled locations, and predict an average or total for a stream or a stream segment. There have been very few attempts to develop valid spatial covariance models that incorporate flow, stream distance, or both. The application of typical spatial autocovariance functions based on Euclidean distance, such as the spherical covariance model, are not valid when using stream distance. In this paper we develop a large class of valid models that incorporate flow and stream distance by using spatial moving averages. These methods integrate a moving average function, or kernel, against a white noise process. By running the moving average function upstream from a location, we develop models that use flow, and by construction they are valid models based on stream distance. We show that with proper weighting, many of the usual spatial models based on Euclidean distance have a counterpart for stream networks. Using sulfate concentrations from an example data set, the Maryland Biological Stream Survey ( MBSS ), we show that models using flow may be more appropriate than models that only use stream distance. For the MBSS data set, we use restricted maximum likelihood to fit a valid covariance matrix that uses flow and stream distance, and then we use this covariance matrix to estimate fixed effects and make kriging and block kriging predictions.

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References
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TL;DR: Cressie et al. as discussed by the authors presented the Statistics for Spatial Data (SDS) for the first time in 1991, and used it for the purpose of statistical analysis of spatial data.
Abstract: 5. Statistics for Spatial Data. By N. Cressie. ISBN 0 471 84336 9. Wiley, Chichester, 1991. 900 pp. £71.00.

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Book
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TL;DR: In this article, the authors focus on concepts with a blend between illustrations using real data sets and mathematical and conceptual development and emphasize applications with examples that illustrate nearly all the techniques discussed, including simultaneous influence, maximum likelihood estimation of parameters, and the plotting of residuals.
Abstract: The author emphasizes applications with examples that illustrate nearly all the techniques discussed. Applications have been selected from physical sciences, engineering, biology, management science and economics. Emphasis is also placed on concepts with a blend between illustrations using real data sets and mathematical and conceptual development. Expanded coverage includes: simultaneous influence, maximum likelihood estimation of parameters, and the plotting of residuals, the use of the general linear hypothesis, indicator variables, the geometry of least squares, the relationship to ANOVA models, Box-Cox transformation with illustrations, categorical response, other nonnormal error situations, autocorrelated errors and logistic regression.

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TL;DR: In this article, a modified maximum likelihood procedure is proposed for estimating intra-block and inter-block weights in the analysis of incomplete block designs with block sizes not necessarily equal, and the method consists of maximizing the likelihood, not of all the data, but of selected error contrasts.
Abstract: SUMMARY A method is proposed for estimating intra-block and inter-block weights in the analysis of incomplete block designs with block sizes not necessarily equal. The method consists of maximizing the likelihood, not of all the data, but of a set of selected error contrasts. When block sizes are equal results are identical with those obtained by the method of Nelder (1968) for generally balanced designs. Although mainly concerned with incomplete block designs the paper also gives in outline an extension of the modified maximum likelihood procedure to designs with a more complicated block structure. In this paper we consider the estimation of weights to be used in the recovery of interblock information in incomplete block designs with possibly unequal block sizes. The problem can also be thought of as one of estimating constants and components of variance from data arranged in a general two-way classification when the effects of one classification are regarded as fixed and the effects of the second classification are regarded as random. Nelder (1968) described the efficient estimation of weights in generally balanced designs, in which the blocks are usually, although not always, of equal size. Lack of balance resulting from unequal block sizes is, however, common in some experimental work, for example in animal breeding experiments. The maximum likelihood procedure described by Hartley & Rao (1967) can be used but does not give the same estimates as Nelder's method in the balanced case. As will be shown, the two methods in effect use the same weighted sums of squares of residuals but assign different expectations. In the maximum likelihood approach, expectations are taken over a conditional distribution with the treatment effects fixed at their estimated values. In contrast Nelder uses unconditional expectations. The difference between the two methods is analogous to the well-known difference between two methods of estimating the variance o2 of a normal distribution, given a random sample of n values. Both methods use the same total sum of squares of deviations. But

3,855 citations


"A topological restricted maximum li..." refers methods in this paper

  • ...Rather than using a kriging estimator, we adopt a Restricted Maximum Likelihood (REML) framework (Gilmour et al., 2004; Patterson and Thompson, 1971; Lark et al., 2006) to estimate variance parameters....

    [...]

BookDOI
10 Sep 1993

2,500 citations


"A topological restricted maximum li..." refers background or methods in this paper

  • ...…from a second order stationary ran- dom process, and that the covariance between y0 k and y0 m will depend only on the relative position of sub-catchments m and k, given some specified correlation function ⇢(·) of the distance c km between the centroids of the two sub catchments (Cressie, 1993)....

    [...]

  • ...This reduces the bias on the semivariogram by allowing the variance to be estimated independently from the trend120 coefficients (Cressie, 1993; Lark et al., 2006)....

    [...]

  • ...By contrast, the expected downward bias in the kriging estimation of partial sills (Cressie, 1993) is clearly visible in the underestimation of prediction uncertainties by the Top-kriging method....

    [...]

  • ...Within PUB, kriging (Cressie, 1993) based geostatistical methods have been widely used (e.g, Huang and Yang, 1998; Gottschalk et al., 2006; Sauquet, 2006; Sauquet et al., 2000; Skøien et al., 2006)....

    [...]

  • ...Following Cressie (1993) (p. 68), the covariance between two aggregated random variables y0 k and y0 m is expressed as a function of the covariogram C P (·) of the underlying point-scale process: Cov (y0 k ,y0 m ) = 1 A2 Z Sk Z Sm C P (| x2 x1 |)dx1dx2 = 1Z 0 ⌫(D)C P (D)dD (A1)575 where S k and S m…...

    [...]

Journal ArticleDOI
Edzer Pebesma1
TL;DR: The gstat S package is introduced, an extension package for the S environments (R, S-Plus) that provides multivariable geostatistical modelling, prediction and simulation, as well as several visualisation functions.

2,455 citations

Frequently Asked Questions (1)
Q1. What contributions have the authors mentioned in the paper "Topreml: a topological restricted maximum likelihood approach to regionalize trended runoff signatures in stream networks" ?

The authors introduce topological restricted maximum likelihood ( TopREML ) as a method to predict runoff signatures in ungauged basins.