Evaluating uncertainty in estimates of soil moisture memory with a reverse ensemble approach

Soil moisture memory is a key component of seasonal predictability. However, uncertainty in current memory estimates is not clear and it is not obvious to what extent these are dependent on model uncertainties. To address this question, we perform a global sensitivity analysis of memory to key hydraulic parameters, using an uncoupled version of the H-TESSEL land surface model. Results show significant dependency of estimates of memory and its uncertainty on these parameters, suggesting that operational seasonal forecasting models using deterministic hydraulic parameter values are likely to display a narrower range of memory than exists in reality. Explicitly incorporating hydraulic parameter uncertainty into models may then give improvements in forecast skill and reliability, as has been shown elsewhere in the literature. Our results also show significant differences with previous estimates of memory uncertainty, warning against placing too much confidence in a single quantification of uncertainty.


Introduction
The persistence of soil moisture is an important aspect of land-atmosphere interactions (Delworth and Manabe, 1989;Koster and Suarez, 2001;Seneviratne et al., 2006Seneviratne et al., , 2010)).Soil moisture shows strong coupling with precipitation and temperature (Koster et al., 2004, Fischer et al., 2007, Rahman et al., 2015), and is a key aspect of weather and climate prediction.The soil moisture reservoir has a memory considerably longer than most atmospheric processes, and as a low-pass filter it lengthens the timescales of climatic anomalies.Persistence in soil moisture memory is linked to persistence in humidity, temperature, and precipitation, through its link with evapotranspiration persistence (Delworth and Manaba, 1993;Koster and Suarez, 1995;Pal and Eltahir, 2001;Orth et al., 2013, amongst others).It prolongs the effect of drought (Nicholson, 2000), enhances the severity and persistence of floods (Bonan and Stillwell-Soller, 1998;Hong and Kalnay, 2000;Liu et al., 2014), impacts the length of heatwaves (Lorenz et al., 2010), and can determine the predictability of atmospheric surface climate anomalies (Wang and Kumar, 1998;Douville, 2004).
Variations in memory and its uncertainty are important from the perspective of atmosphere-land coupled modelling since the feedback between the land and the atmosphere means that anomalies in soil moisture which persist over a long time can influence atmospheric conditions over an extended period into the future.
In contrast, soil moisture memory is important for hydrological applications independently of the strength of landatmosphere coupling.Soil moisture is one of the main controls on both rainfall runoff generation and evaporation.Uncertainty and variability in the persistence of soil moisture can then directly influence the predictability of streamflow, with implications for effective flood forecasting and water resource management.

D. MacLeod et al.: Soil moisture memory uncertainty
Previous work has analysed soil moisture memory in uninitialized atmospheric global circulation models (Wu and Dickinson, 2004;Seneviratne and Koster, 2012), showing significant regional differences in memory.Other work has looked at estimating soil moisture memory from observations of streamflow (Orth et al., 2013).It has also been demonstrated that the choice of soil hydrology scheme impacts simulated soil moisture memory (Hagemann and Stacke, 2014).
We take an alternative approach here, inspired by previous work on hydrological prediction uncertainty (Wood and Lettenmaier, 2008).We run a hindcast in a set-up analogous to a standard seasonal hindcast set-up, using an uncoupled model with ensembles of initial states and forcing time series.This allows us to compare memory estimates of a seasonal forecast system when forecasts are initialized at different points in the year, as well as look at spatial differences.It also allows us to evaluate the sensitivity of estimates of soil moisture memory to uncertain hydraulic parameters.
This approach may be compared to work at longer decadal timescales following a similar approach (Corti et al., 2015).By using different combinations of initial conditions and forcing years to run global climate models, they analyse the crossover point when forcing becomes more important than initial conditions, and so identify the relative importance of initial conditions and forcing.
In concert with this work, our work here aims to understand the importance of initialization of slowly moving components of the climate system for longer-term climate prediction.In addition, this approach allows new insight into the controls that hydraulic parameters have on the generation of soil moisture memory and the relative uncertainty compared to initial conditions.
The model, data, and experimental methods are described in the following section, after which results are described.The final section of the paper contains a discussion of results and conclusions.

Methodology
Wood and Lettenmaier define a method known as reverseensemble streamflow prediction (reverse-ESP, Wood and Lettenmaier, 2008).Rather than standard ESP, where one initial state is forced by an ensemble of forcings, this method takes the opposite approach and forces an ensemble of initial states with a single forcing time series.In their work they analyse the relative importance of initial conditions and forcing by comparing the spread of ESP and reverse-ESP ensembles.
Inspired by this method, we take a reverse-ESP approach to evaluate soil moisture memory.By forcing an ensemble of initial conditions with the same forcing time series, soil moisture states converge, at which point one can say that the soil has lost memory of initial conditions.This is demonstrated with some example data in Fig. 1.At the initial point the standard deviation of the ensemble is large and over time decreases at differing rates depending on forcing conditions such as precipitation, evaporative demand, and model parameters.By calculating the lead time at which the spread of the ensemble is reduced significantly from its initial spread, we can estimate soil moisture memory.This approach enables us to perform a sensitivity analysis of this soil moisture memory estimate to model hydraulic parameters.

Land surface model, forcing and initial data
The land surface model used is H-TESSEL, the Tiled ECMWF Scheme for Surface Exchanges over Land (TES-SEL) with revised land surface hydrology (Balsamo et al., 2009).This comprises a surface tiling scheme and a vertically discretized soil, with soil layers below ground at 7, 21, 72, and 189 cm.Soil moisture is defined for a layer as the volume of water contained in the layer expressed as a fraction of the layer volume.H-TESSEL includes the van Genuchten formulation for hydraulic conductivity (van Genuchten, 1980) and a spatially varying soil-type map.
Running the model in uncoupled mode allows a much larger number of experiments to be run compared to coupling with an interactive atmosphere.This requires forcing at each time step.To provide this forcing, we use the WFDEI meteorological forcing data set, data at 0.5 • spatial resolution, created by using the WATCH Forcing Data methodology applied to the ERA-Interim data (Weedon et al., 2014).The forcing comprises 3-hourly data for the following variables: longwave and shortwave radiation, rainfall, snowfall, surface pressure, wind speed, air temperature, and humidity.Data from the 25 years 1981-2005 inclusive are used, in each case using the 4-month period from an initial date to force the model, mimicking seasonal hindcasts (Weisheimer et al., 2014).
For initial land surface conditions of soil temperature, moisture, and ice temperature where frozen soil is present, we use the ERA-Interim Land reanalysis (Balsamo et al., 2015), using initial states from 1 May and 1 November for every year from 1981 to 2005 inclusive.May and November start dates are chosen in order to focus on the two standard contrasting seasons: boreal summer and winter.For other model parameters (e.g.albedo, vegetation cover, soil type) we use the H-TESSEL default values.A plot showing the fractional high and low vegetation cover is shown for reference in Supplement Fig. 1.

Hydraulic parameter perturbation
We perturb the van Genuchten α parameter and the saturated hydraulic conductivity.These parameters are related to the movement of water through the soil and their default values in the model set-up are based on the FAO soil map of the world (see Balsamo et al., 2009, for details).In reality there is a large uncertainty in these parameters; observed standard deviation in their values across soil samples can be as much as twice the mean (see MacLeod et al., 2015, andCarsel andParrish, 1988, for further details).Previous work has shown that these parameters are particularly sensitive (Cloke et al., 2011).
Here we perturb these parameters by picking a value for each simulation for each parameter from the five-member set {−80, 40, 0, +40, +80} % and applying this perturbation to the default value for each grid point.Perturbing both parameters in this way gives a 25-member ensemble.Though these perturbations are relatively large and there is a weak correlation between these two parameters, the perturbed range is within the range of observed variability in parameter values.Furthermore, the parameters which are used have to be considered as effective parameters, as measurement scale and model scale differs significantly (Barrios and Francés, 2012).This leads to significant uncertainty when attempting to define parameters on the model grid scale.
A simulation is run for every combination of 25 initial states, 25 forcing states and 25 parameter perturbations, giving a total of 25 3 = 15 625 runs.Due to computing limita-tions we run at a reduced spatial resolution with 18 × 36 grid points globally (note that these points are "picked" from the original resolution of the forcing and initial conditions instead of being interpolated over a whole grid box).The grid points are chosen indiscriminately, in that we took every 10th point from the underlying data without considering a priori local characteristics.To use low-resolution data is suboptimal, since regions of high spatial heterogeneity or complex topography may show memory characteristics which are somewhat dependent on the choice of grid point.One additional caveat is that initial conditions are taken from reanalysis which is created using H-TESSEL with default parameters.The land surface temperature and moisture climatology is slightly influenced by these parameters; however, recreating a new set of reanalysis for each parameter combination we investigate here is far beyond the scope of the study.

Experimental set-up
For the ensemble of initial condition data we run an ensemble of simulations using the same year's forcing for each ensemble member, using a constant parameter perturbation throughout each simulation.An example of this is shown in Fig. 1.We characterize the rate of memory loss by defining the date of memory loss t fp ml , the time from initialization when the spread of soil moisture in the ensemble (forced by year f and parameter combination p) has reduced to 1/e of its initial value, that is, where σ (t) is the standard deviation of the ensemble at time t, σ (0) is the initial spread, and the memory loss time t ml is defined by the supremum norm (i.e. the maximum value of t bounded by σ (t) ≤ σ (0) / e. t ml ).We calculate t fp ml for each forcing year, and take the average across all 25 forcing years: to give a more robust estimate of the date of memory loss, for each spatial grid point and parameter combination.We can then calculate t ml p separately for each parameter combination, giving a measure of the regions where variations in hydraulic parameters give the largest changes in memory loss.This can be expressed by the standard deviation of t ml p across the parameter ensemble: where σ t ml is the standard deviation across the perturbed parameter ensemble, t ml p is the memory loss date for parameter combination p, and t ml is the average date across all combinations.Small variations in memory are less meaningful when memory is large and more meaningful when memory is normally small.To highlight these more meaningful variations in hydraulic parameters, we re-express the standard deviation as the coefficient of variability, i.e. as a fraction of the mean time of memory loss, expressed as a percentage: COV = σ t ml t ml × 100 %. (4) 3 Results

Estimation of soil moisture memory
Figure 2 shows the average date of memory loss for the default model parameters ( t ml p * , where p * is the default parameter set) for May and November start dates, for the top vertical soil level.Results for levels two and three are discussed below but presented in the Supplement.We do not discuss the fourth soil level as the free drainage condition impacts negatively on the realism of the exact values of soil moisture here.
In general, memory increases with depth, though with significant differences spatially and between start dates.The longest memory is seen in regions in north-eastern Asia and the extreme northwest of the North American continent.For May, several points have a longer memory of between 20 and 60 days, and in the furthest north one point still retains the memory of the initial state at the end of the simulations (120 days).Similarly the longest memory in November is seen for the same regions -however, the memory is much longer in these regions for this start date, and the region of long memory extends much further south, down the Rocky Mountains in the USA and the Himalayas in Asia.This long memory is likely due at least in part to snow cover, when soil is insulated from any precipitation forcing, allowing persistence of an initial soil moisture state (Koster et al., 2010).
Correspondingly, the surface layer in the Southern Hemisphere has a relatively short memory for both start dates, likely in part due to an absence of snow cover on Southern Hemisphere land (excluding Antarctica).The second and third levels generally have longer memory than the surface, as the influence of precipitation forcing on soil moisture is damped.This is partly due to the loss of some moisture by evapotranspiration before reaching the lowest levels.
3.2 Sensitivity of estimation of soil moisture memory to hydraulic parameters

Variation in memory loss date
Figure 3 shows the standard deviation in the average date of memory loss across the parameter ensemble (σ t ml ) for May and November start dates for the top level.The absolute variation in memory length with parameter does not follow exactly the same pattern as the magnitude of memory.For May the largest variations in memory are seen in snow-covered regions in north-eastern Asia, but also in southern Africa.Similarly, November shows large variation in memory over the grid points around the Himalayas, yet much smaller variation around the areas of Asia and the USA with long memory.The largest variation in memory for November in fact occurs in Europe and western Asia, where memory is short.
In general for both start dates, lowest absolute variation in memory length is seen at the top level, with an increase in variation with depth (lower levels shown in the Supplement).The highest absolute variation in memory length is seen at level three, with a standard deviation in the date of over a month.In May this is spread over Eurasia and North America, though with some high variability also seen in the Southern Hemisphere.The top layer shows a particular sensitivity to parameters in the north-eastern Asia region, where memory is highest.
November start dates also show the same pattern of more variability at lower levels where memory tends to be longer; however, the spatial pattern tends to be more heterogeneous than May start dates.For instance, variability in May in the third level is roughly constant over much of Asia, whilst in November there is a sharp contrast, with a band stretching from north-east to south-west where variability in the date is quite low.This heterogeneity extends upwards to higher levels, with the highest variations in November contained mostly in the western half of Eurasia.There is also a clear distinction between western and eastern North America in the third level for November starts, with much higher sensi- tivity in the east than the west.This however does not extend up to levels one and two, where the sensitivity is uniform across the continent.

Coefficient of variation of memory loss date
Looking at variation of memory with parameters may be in some cases somewhat dependent on the original magnitude of memory.To understand where the memory is particularly sensitive, we show the coefficient of variation of the date of memory loss (Fig. 4).
This highlights memory estimates in Europe as being particularly sensitive to the uncertainty in hydraulic parameters for November-initialized forecasts, with a standard deviation in the memory loss date of over 50 % of the mean at the surface, whilst points in Asia are less than 20 %.Similarly, eastern North America and northern South America have high sensitivity, whilst the west and south respectively do not.Sensitive regions in November also include central Africa, Indonesia, and some points in the Middle East, with a slight reduction in magnitude and modification of the spatial pattern for lower levels.
For the May start date the spatial pattern is quite different from November, with low sensitivity in Europe and eastern North America.Sensitive regions are found in southern Africa, western North America and north-eastern Asia.The spatial pattern of sensitivity changes quite significantly with depth in May, with high sensitivity at the third layer for western North America, for a band running across northern Asia and South America.

Discussion and conclusions
Here we have developed a method to calculate soil moisture memory and used this to estimate its sensitivity to hydraulic parameters.These hydraulic parameters influence the speed of the movement of water through the soil, so it is not surprising that the persistence of initial moisture states is influenced by them.However, soil moisture memory emerges from the interplay of precipitation, evaporative demand, and drainage conditions over time, and the pattern of sensitivity is not clear a priori.The current study highlights regions for which memory is particularly sensitive to hydraulic parameters and also indicates differences in memory and sensitivity between hindcasts initialized in different seasons.
Previous work has attempted to characterize soil moisture memory, most extensively using the GLACE Atmospheric General Circulation Models (AGCMs) (Seneviratne et al., 2006).This work analysed the output of 12 AGCMs over the period 1 June to 31 August, forced by sea surface temperatures (SSTs) from 1994.Roughly corresponding to the same period as the May start used here, this work looked at geographical variations in average memory and in the standard deviation of memory estimates across the model ensemble.Their findings are not in full agreement with our work here; amongst other things, our results also do not lead to their conclusion that memory is higher at the mid-latitudes than the tropics.Furthermore, the highest uncertainty in the GLACE AGCMs is found in regions of low soil moisture memory.Our work shows deviation from this result, with large uncertainty in both regions of high and low memory.
These discrepancies are likely to have several explanations.A key reason for differences in uncertainty estimates is that the focus of the uncertainty quantified is different; Seneviratne et al. (2006) look at a multi-model ensemble, which gives some exploration of model and structural uncertainty, whilst in our study we use a single model and look at the uncertainty in hydraulic parameters.
Computational expense limited this current work to two start dates per year; future work might explore the sensitivity of results to start date.
Note that we do not claim that our own sensitivity analysis is total, only that we show how memory estimates using a single model are sensitive to two specific model hydraulic parameters.It is likely the case that there remain unexplored uncertainties in the land surface, for instance, rooting depth, the depth of the free drainage lower boundary condition and moisture thresholds (such as permanent wilting point, field capacity, and saturation).Furthermore, other models may show more or less sensitivity to the hydraulic parameters we test here.However, the main focus of this work is to demonstrate the uncertainty in estimating soil moisture memory and the potential importance of hydraulic parameters.This is relevant for operational seasonal prediction, in which land surface models are run without any variation in these uncertain parameter values.Given the key role in memory in high-impact events such as flooding, droughts, and heatwaves, a forecasting system with deterministic parameters is likely to display a narrower range of possible memory than exists in reality.This is likely to give negative impacts on forecast skill and reliability, or, put more optimistically: by explicitly including parameter uncertainty in operational probabilistic forecasts (which already incorporate initial condition uncertainties), we may improve forecast skill and reliability.Indeed, coupled seasonal forecasts which include an explicit representation of hydraulic parameter uncertainty have already shown improved prediction of the 2003 European heatwave (MacLeod et al., 2015), and it has been shown that weather forecasts can be improved by accounting for land surface model parameter uncertainty (Orth et al., 2016).
Variations in memory and its uncertainty are important from the perspective of atmosphere-land coupled modelling since the feedback between the land and the atmosphere means that anomalies in soil moisture which persist over a long time can influence atmospheric conditions over an extended period into the future.
In contrast, soil moisture memory is important for hydrological applications independently of the strength of landatmosphere coupling.Soil moisture is one of the main controls on both rainfall runoff generation and evaporation.Uncertainty and variability in the persistence in soil moisture can then directly influence the predictability of streamflow, with implications for effective flood forecasting and water resource management.
The results presented here also highlight a more general point that cannot be overstated: the uncertain nature of uncertainty quantification.It is almost impossible to explore the full space of our ignorance in models; a multi-model approach neglects uncertain parameters, whilst a perturbed parameter approach ignores structural uncertainty (and indeed, other parameters).Conclusions about relative levels of sensitivity and uncertainty must therefore be drawn with caution, bearing in mind the limitations of the experimental method.
The Supplement related to this article is available online at doi:10.5194/hess-20-2737-2016-supplement.
Author contributions.Dave MacLeod conceived the study, ran simulations, and wrote the first manuscript draft.Interpretation of results, development of analysis, and paper revision were a collaboration between all co-authors.

Figure 1 .
Figure 1.An example of the evolution of top level soil moisture from an ensemble of 25 initial states, each forced with the same year's forcing (in this case, 1981).Data from each plot are taken from different grid points, chosen to demonstrate a situation of long and short memory.Note that in the latter case memory of initial conditions is lost by around day 20, whilst the former case retains knowledge of the initial state for several months.

Figure 2 .
Figure 2. Average date of memory loss in the top model soil level, estimated from H-TESSEL (default parameter set), for May (left) and November (right) start dates.

Figure 3 .
Figure 3. Standard deviation in the top model soil level, estimated from H-TESSEL, for May (left) and November (right) start dates.

Figure 4 .
Figure 4. Sensitivity of memory to hydraulic parameters (standard deviation of memory loss across hydraulic parameters as a percentage of the memory loss) in the top model soil level, estimated from H-TESSEL, for May (left) and November (right) start dates.