Root‐zone soil moisture variability across African savannas: From pulsed rainfall to land‐cover switches

The main source of soil moisture variability in savanna ecosystems is pulsed rainfall. Rainfall pulsing impacts water‐stress durations, soil moisture switching between wet‐to‐dry and dry‐to‐wet states, and soil moisture spectra as well as derived measures from it such as soil moisture memory. Rainfall pulsing is also responsible for rapid changes in grassland leaf area and concomitant changes in evapotranspirational (ET) losses, which then impact soil moisture variability. With the use of a hierarchy of models and soil moisture measurements, temporal variability in root‐zone soil moisture and water‐stress periods are analysed at four African sites ranging from grass to miombo savannas. The normalized difference vegetation index (NDVI) and potential ET (PET)‐adjusted ET model predict memory timescale and dry persistence in agreement with measurements. The model comparisons demonstrate that dry persistence and mean annual dry periods must account for seasonal and interannual changes in maximum ET represented by NDVI and to a lesser extent PET. Interestingly, the precipitation intensity and soil moisture memory were linearly related across three savannas with ET/infiltration ∼ 1.0. This relation and the variability of length and timing of dry periods are also discussed.


| INTRODUCTION
In savanna ecosystems, sparsely spaced woody vegetation allows ample photosynthetically active radiation (PAR) to reach the ground surface, thereby promoting a herbaceous layer (primarily grasses) to compete for water with the woody vegetation. Because PAR rarely restricts the overall productivity of savannas, the carbon-water economies in such ecosystems are driven by rainfall pulses (Schwinning & Sala, 2004;Williams & Albertson, 2004) resulting in high temporal variability in root-zone soil moisture. Because the interpulse period and precipitation depth per event are random (at least on daily timescales), the surface soil moisture variability has been presumed to be primarily driven by stochastic rainfall events adjusted by interception and losses from the root zone due to drainage, root-water uptake and surface evaporation (Laio, Porporato, Fernandez-Illescas, & Rodriguez-Iturbe, 2001;Miller, Baldocchi, Law, & Meyers, 2007;Yin, Porporato, & Albertson, 2014). Soil moisture variability has also been shown to exert control on drought occurrence (Masih, Maskey, Mussá, & Trambauer, 2014;Saini, Wang, & Pal, 2016), probabilistic drought prediction (AghaKouchak, 2015), convective rainfall formation (Green et al., 2017;Koster et al., 2004;Siqueira, Katul, & Porporato, 2009;Taylor et al., 2011;Wei, Dickinson, & Chen, 2008) and ecosystem resilience (Porporato, Daly, & Rodriguez-Iturbe, 2004) primarily because soil moisture memory and persistence within certain phases (wet or dry) exceed the timescale of many meteorological variables. Such meteorological variables can then 'feed-off' on the soil moisture state of savanna ecosystems and be altered in a manner to impact rainfall occurrences and depth. The soil moisture variability is also related to phenology and root-water uptake that are not well represented in terrestrial biosphere models (Whitley et al., 2016;Whitley et al., 2017). For these reasons, the controls on soil moisture memory and dry persistence experienced by African savannas are receiving renewed interest, especially in climate studies (Ghannam et al., 2016;Nakai et al., 2014).
The memory timescale is commonly determined from the autocorrelation function of root-zone soil moisture time series and acts as one measure of the time that it takes for a soil column to forget its initial soil moisture state (Ghannam et al., 2016;Katul et al., 2007;Nakai et al., 2014). Such memory timescale is largely controlled by the loss terms that include evapotranspiration (ET) and drainage below the rooting zone as well as the root-zone depth (Katul et al., 2007). Persistence is the probability that the soil moisture remains in a prescribed state such as wet or dry. The state is selected using root-zone soil moisture or degree of saturation value that marks the crossover between wet and dry phases (s * ) as discussed elsewhere (Ghannam et al., 2016). Dry period persistence is the probability distribution of contiguous durations for which the soil moisture is below some preset threshold (s * ). This definition differs from other widely used definitions such as frequency of daily temperature crossing a given threshold (heat wave persistence) and precipitation anomaly (drought persistence) that have been employed in prior studies (Lorenz, Jaeger, & Seneviratne, 2010;Moon, Gudmundsson, & Seneviratne, 2018).
Persistence is inherently stochastic with non-linear dependence on both precipitation (the forcing) and the loss term (evaporation, root-water uptake and drainage) from the root zone (Ghannam et al., 2016), whereas memory depends on the timescale of the losses.
Interconnectedness between rainfall pulses (hours) and rapid fluctuations in the active leaf area of the herbaceous layer (days) makes explorations of memory and persistence in savannas far more challenging than their better studied forested ecosystem counterparts (Ghannam et al., 2016). It is precisely this knowledge gap that motivates the scope of the work here. In a dry savanna ecosystem, root-zone soil moisture is frequently below s * (i.e. the system is in a persistent dry state). At woodland dominated savannas, soil moisture resides above or close to the water-stress point during the entire wet season (i.e. bistable states in vegetation cover). Grazing pressures add another layer of complexity in explaining what controls memory and persistence in savannas. It is clear that beyond seasonality in leaf area index (LAI) common to many forested ecosystems, rapid land-cover changes due to soil moisture stress and/or grazing are partly driven by rainfall patterns.
How such rapid land-cover fluctuations in savannas impact soil moisture memory, persistence and any emerging relation between them is the main question to be addressed here.
Previous work analysed the crossing properties of soil moisture below s * and derived analytical expressions for the mean annual duration of the dry periods ( T Ã ) ). The aforementioned analysis shows that the mean dry period is a non-linear function of storm frequency. Motivated by these studies, another aim here is to explore the utility of such findings for longer term modelling of soil moisture variability using measured precipitation. This exercise enlightens possible connection between mean annual duration of soil moisture dry phases and precipitation statistics for multiple water balance models.
To address these overall questions, simplified hydrological models similar to the ones used in previous savanna studies (Miller et al., 2007;Yin et al., 2014)

| METHOD
The hydrological balance approach used to explore variables impacting soil moisture memory and distributions of dry periods (dry period persistence) is first described. Next, the spectral method for time series analysis and the four sites is featured. Details about the method of analysis and its application to soil moisture have been reviewed elsewhere (Ghannam et al., 2016;Nakai et al., 2014), and only salient points are presented here.

| The soil water balance and definitions
The one-dimensional continuity equation for water in soil including root-water uptake S R can be expressed as where θ is a layer-averaged volumetric soil moisture content (volume of water per unit volume of soil), q w is the water flux (volume of water per unit ground area per time) assumed to be positive downwards, z is the vertical distance to the soil surface (set at z = 0) and t is time. To arrive at a lumped representation of root-zone soil moisture, vertical integration of Equation 1 from z = 0 to the root-zone depth z = Z r is necessary and yields ð Zr where q w (0) can be interpreted as either soil evaporation (negative) in the absence of rainfall or throughfall and stemflow lumped together as infiltration (positive) entering the soil surface presumed to be supply controlled and given as q w (0) = γP for P > 0, γ is set to a constant and depends on the interception by the overlying vegetation, P is the precipitation, q w (Z r ) is the drainage flux from the rooting zone and may be determined from Darcy's law and Ð Zr 0 S R dz is the total rootwater uptake. For steady-state conditions within the plant system (i.e. no plant capacitance in the root xylem), Ð Zr 0 S R dz also determines the transpiration rate. Interchanging the differential and integral operations (Leibniz's rule) on the left-hand side of Equation 2 yields a lumped budget equation for the stored water in the rooting zone given as d dt where W s = ηZ r s(t) now defines the stored water within the rooting zone, η is the soil porosity (pore space volume to total volume) and s is the degree of saturation defined by the ratio of the volume of water per total pore (air and water) volume (s = 1 implies all pore space is filled with water). When lumping all losses together into a single term L t ð Þ = q w Z r ð Þ+ Ð Zr 0 S R dz and upon assuming that the widely used lumped hydrological balance is recovered and is expressed as (Rodríguez-Iturbe & Porporato, 2007) ηZ This balance links the statistics of P(t) to s(t) provided that a relation between L(t) and s(t) is available or can be obtained from a combination of data and models.

| Approximations to the soil water balance
The inclusion of land-cover dynamics and meteorological drivers is now discussed by a sequence of approximations to L(t) with the goal of maintaining minimum number of 'tunable' parameters in all of them.
These approximations form a hierarchy of models labelled Models 1 to 3. Model 1, the most conventional and widely used in savannas, only considers variability in P holding land-cover type and other climatic drivers constant. Model 2 accommodates NDVI(t) variability on maximum ET only, which then is allowed to impact variability in soil moisture through L(t) and γ. Model 3 assumes that the energy balance and NDVI jointly impact soil moisture so that subdaily timescales as well as seasonality in radiation can introduce additional variability in soil moisture.
In Model 1, L(t) is represented by a piecewise function given as Yin et al., 2014) where s w is the wilting point, s fc is the field capacity and the subsurface drainage below the rooting zone is represented by a saturated hydraulic conductivity K sat and an exponent c that varies with soil type (or pore structure) and E max is maximum ET set to a constant assumed to be independent of NDVI or climatic factors. This model remains widely used in semiarid ecosystems  for daily timescale analysis and is adopted here as a 'reference' given that its original testing was conducted for African savannas (Laio, Porporato, Fernandez-Illescas, & Rodriguez-Iturbe, 2001). In this model, the drainage flux q w (Z r ) only varies with the stored water within the rooting zone and not the local soil moisture at z = Z r . This approximation may be an issue that can only be 'bypassed' if the local soil moisture at z = Z r and the depth-averaged root-zone soil moisture linearly relate to each other as time changes. Also, only gravitational drainage below the rooting zone is considered in the aforementioned q w (Z r ) (i.e. unit-gradient assumption), which may not be realistic at all times (Katul, Wendroth, Parlange, Puente, & Nielsen, 1993 where a 2 is a fitting parameter and the b parameter is an extinction coefficient for global radiation set to 0.4 for Models 2 and 3 (Al-Kaisi, Brun, & Enz, 1989;Teuling, 2005).
Model 3 further expands on Model 2 by allowing a radiationbased PET (=PET) and NDVI to jointly impact maximum ET rate at subhourly timescale so that where a 3 is another fitting parameter and PET is determined (at subhourly timescales) using the Priestley-Taylor formulation (Priestley & Taylor, 1972) where α PT = 1.26 is the Priestley-Taylor coefficient, Δ is the slope of the Clausius-Clapeyron equation with respect to temperature evaluated at the measured air temperature and γ p is the psychrometric constant. In Models 2 and 3, variability in NDVI also impacts interception (discussed later). Model 3 allows both seasonal and diurnal variation in radiation to impact soil moisture variability.
The model parameters were fitted following a sequence of steps. The K sat values were chosen based on published soil type for the site. The s fc was set to s * /0.75. The relation between the s fc and the drainage parameter c may be different at the sites, and thus, the parameter c was fitted for each site by minimizing the root-mean-square error (RMSE) of measured, and Model 1 based soil moisture during drainage periods only. These drainage parameters were also used for Models 2 and 3 for consistency. The a 2 and a 3 parameters were fitted by minimizing the RMSE of measured and modelled soil moisture.

| Interception
Part of the rainfall does not enter the soil but is intercepted by the vegetation and re-evaporated rapidly (ponding and overland flow are ignored in this analysis). For semiarid areas, this loss can be substantial owing to the high evaporative demand by the atmosphere. Hence, the constant γ was determined based on the amount of measured soil moisture increases during measured precipitation events. The numerical value of γ was computed by regressing the cumulative measured increases in soil moisture during precipitation events against measured cumulative precipitation.
The rainfall and soil moisture time series used in the calculation of γ were determined by taking the soil moisture increases from the start of an isolated rainfall event until 24 h after the termination of the event. Naturally, γ lumps multiple sources of errors, including the representativeness of the rainfall measurements at the gauge of the rainfall experienced by the soil moisture probes, the precise placement of the soil moisture sensors and any nearsurface evaporation. Its numerical value cannot be strictly viewed as 'hydrological' and must be interpreted within the context of such spatial scale-mismatch between measured precipitation and root-zone soil moisture at two separate locations. Nonetheless, because measured point precipitation is used as the main driver in the three hydrological balance models, it is necessary for parameter γ to absorb these space scale-mismatch issues.

| Memory, persistence and spectra
The term ds(t)/dt in Equation 5 is linked to the storage of water in soil pores, thereby introducing memory (Delworth & Manabe, 1988;Ghannam et al., 2016;Katul et al., 2007;Parlange et al., 1992). Memory (or integral timescale) of a stochastic variable can be determined from the area under the autocorrelation function of a time series given as where α is the time lag and ρ s (α) is the autocorrelation function of time series s(t) when stationarity is assumed (Priestley, 1981).
The wet and dry states can be defined from soil moisture time series by setting an indicator function to unity when soil moisture is deemed as wet and zero otherwise. Such binary time series of the indicator function are referred to as the telegraphic approximation (TA) of the full soil moisture time series. The plant waterstress (s * ) is used to delineate the threshold between wet (TA = 1) and dry (TA = 0) states.
The normalized spectra of soil moisture (E ns (f )) and precipitation (E np (f )) are the Fourier transforms of their corresponding autocorrelation functions. These spectra were estimated using the Welch averaged modified periodogram method (Welch, 1967). The window length for the spectral estimation varied from 130 to record duration. An analytical relation also exists between precipitation and soil moisture spectral exponents for a linear L(s). When a constant (or white-noise) spectrum ((E np (f ) = constant) for rainfall is assumed, the soil moisture spectrum decays as a red-noise spectrum (i.e. f −2 ) at high frequencies (Katul et al., 2007;Nakai et al., 2014) and becomes flat at very low frequencies (i.e. f 0 ). The measured precipitation spectrum, which exhibits spectral decay from f −0.5 to f −1 at high frequencies (daily to subdaily), adds to the decay rate of soil moisture spectrum, making it resemble 'black' instead of 'red' noise (Ghannam et al., 2016;Katul et al., 2007;Nakai et al., 2014). When the spectra of soil moisture and its TA time series exhibit power laws of the form E ns (f ) $ f −n and there exists an empirical relation between the spectral exponents given by where n is the spectral exponent of soil moisture and m is the spectral exponent of the TA of soil moisture (Cava & Katul, 2009;Cava, Katul, Molini, & Elefante, 2012;Molini, Katul, & Porporato, 2009;Sreenivasan & Bershadskii, 2006). This measure identifies to what extent the soil moisture memory is related to the switching between wet and dry or dry and wet (i.e. binary state) instead of temporal variations within the wet state.
Persistence is defined as the distribution of time periods when the soil moisture TA series does not change sign. Specifically, the dry persistence is the probability density function (PDF) of periods when the value of soil moisture is below the s * threshold. These periods are forced by the interaction between precipitation distribution and the total loss term in the water balance equation. To compare dry persistence between the various measurement sites, the distribution of dry periods is normalized by the memory timescale, and a stretched exponential function (Laherrère & Sornette, 1998) is fitted to the data. This function is given as where x = (I dry /τ) are the dry periods normalized by the memory time- is a normalizing constant needed to ensure that Ð ∞ 0 PDF x ð Þdx = 1 and x min is the shortest dry period. The fitting parameter β is assumed to be less than unity. Lower values of β indicate that long water-stress periods decay closer to a power law, whereas higher β indicates exponential decay at long times. In stochastic analysis of the water balance, the mean annual duration of the dry periods ( represents the long-term average annual dry period . Here, T Ã was estimated for the modelled soil moisture series covering only full hydrological years by first taking the mean of dry periods of each hydrological year and then calculating the mean and standard deviation of the annual T * values.

| Rainfall characteristics
For across site comparison purposes, daily mean precipitation depth (P α mm/event) and daily mean storm frequency (P λ events/day) are used to describe the mean storm characteristics (Rodriguez-Iturbe, Porporato, Ridolfi, Isham, & Coxi, 1999). The mean precipitation depth was calculated only for rainy days. The mean storm frequency was calculated from inverse of the mean time between rainy days. The reported precipitation statistics here may not represent long-term mean precipitation statistics given that the records span only few years. To quantify the sensitivity of soil moisture memory on rainfall statistics, stochastic rainfall was generated from a Poisson process at each site. This sensitivity test was conducted using Model 2 because the tree savanna net radiation measurement had missing periods during the 5-year simulation period. The stochastic simulations from Models 2 and 3 were compared at the grazed savanna and showed that the mean Model 3 memory timescale differed from Model 2 mean memory timescale by less than one standard deviation. The synthetic precipitation series were constructed with precipitation intensity similar to the measured intensity from long-term records. Random daily rainfall series were generated assuming that times between rainfall events are exponentially distributed with mean 1/P λ , and the depth of daily rainfall is exponentially distributed with mean depth P α . The random rainfall series were generated at a daily scale for 5-year period; then constant period of zero rainfall was set to each year corresponding to typical dry season length. The dry season length was 180, 120, 110 and 180 days for grass, grazed, tree and miombo savannas, respectively.

| Sensitivity analysis
From these rainfall series, only the ones with precipitation intensity near the observed one (±0.1) were used in the simulations. Model 2 was run with 50 different random rainfall series to generate memory estimates, and the mean and standard deviation of all memory estimates were calculated. The same model parameters were used as described above, but the model was stepped in daily timestep with the exception that on days when s + infiltration was higher than s fc , the soil moisture was reduced owing to drainage in hourly timesteps (Pumo, Viola, & Noto, 2008). The site NDVI time series from September 2010 to August 2015 were used at each site.

| Measurement sites
The measured root-zone soil moisture time series were collected at four African savanna sites with grass, grazed, tree and miombo savanna vegetation cover (Figure 1). Eddy-covariance measurements of ET were used to determine plant water-stress levels at all sites on the basis of a relation between daily averaged relative soil moisture and measured ET. The site characteristics and model parameters used for each site are summarized in Table 1.
The grass savanna site in Kenya is located close to a maize farm and presents a site with no influence of tree roots. The soil moisture profile measurements were conducted inside a fenced meteorological station. During the long rainy season, grasses grow inside the fenced area and impact soil moisture.
The Welgegund grazed savanna site has perennial grasses growing around the soil moisture measurement location, and the area is used for grazing livestock (Jaars et al., 2014(Jaars et al., , 2016(Jaars et al., , 2018Räsänen et al., 2017). The eddy-covariance footprint has some 15% tree cover, and some tree roots were identified at 40-cm depth nearby the soil moisture measurements.
The tree savanna measurement site is located near Skukuza within Kruger National Park. This site is grazed and browsed by ungulates (Scholes et al., 2001). The soil moisture measurements were taken from the Combretum apiculatum-dominated savanna. The grasses and trees access soil water throughout the soil profile, but grasses use shallow soil water more efficiently than trees. Trees can shift the water uptake from shallower depths to deeper depths (Archibald et al., 2009;Kulmatiski & Beard, 2013   Each site had an eddy-covariance system measuring ET using a triaxial sonic anemometer and an infrared gas analyser (Table 2).
An open-path analyser was used at the tree savanna, whereas other sites had a closed-path sensor. The latent heat flux was calculated using standard eddy-covariance procedures at each site (Archibald et al., 2009;Kutsch et al., 2011;Räsänen et al., 2017).
The time series of latent heat fluxes were gapfilled using an artificial neural network using soil temperature, incoming global radiation, soil moisture and NDVI (Isaac et al., 2017). The latest

| Precipitation
The annual precipitation at the tree savanna site was about 50% less than at the grazed savanna site during the two growing seasons covering the tree savanna site measurements (Table 1).
During these years from 2015 to 2017, South Africa experienced a drought, and the precipitation was exceptionally low at the tree savanna site. The long-term annual precipitation of the tree savanna site is 550± 160 mm (Archibald et al., 2009), whereas during the measured period, mean annual precipitation (MAP) was only 280 mm.

| Normalized difference vegetation index
The short (Nov-Jan) and long (

| Evapotranspiration
The daily PET has a stronger seasonal cycle at the South African sites than at the miombo and grass savannas (Figure 2).
For the grazed, tree and miombo savanna sites, the bin aver-

| Interception/re-evaporation losses
To match the measured soil moisture increases with measured precipitation input, the loss term γ was determined from the relation between cumulative rainfall and rain-related soil moisture increases at all sites ( Figure 3). As noted earlier, this term represents interception loss and re-evaporation from soil surface but also all the spatial variability linking rainfall and local soil moisture changes at differing points in space. Its numerical value was the lowest at the grass savanna site.
At the grazed savanna site, there was a deviation from the assumed linear relation when cumulative rainfall was around 1,000 and after 2,400 mm. At the tree and miombo savannas, the relation was non-  The models fit the measured soil moisture nearby s * crossings better at the grass and tree savannas, which have more isolated precipitation events (Table 1 lower P α · P λ ) than at the other sites. Model The partitioning of the water loss components shows that Model 1 has a larger percentage of ET under water stress and a lower percentage of drainage at the grazed, tree and miombo savannas ( Figure 5). Despite the lower s * at the tree savanna compared with the grazed savanna, the partitioning of water loss at F I G U R E 3 Relation between cumulative precipitation and cumulative positive increments in soil moisture for each site. The regression slope determined γ at 48%, 62%, 72% and 60% of the precipitation for the grass, grazed, tree and miombo savannas, respectively these sites is similar. At the miombo savanna, the amount of drainage and nonstressed ET loss is higher than the amount of stressed ET loss. Model 3 RMSE was the smallest at all sites ( Figure 5) when compared with the other two models. The difference in RMSE between Models 2 and 3 was small also at all sites except the grazed savanna.

| Model results
The comparison between model and measured soil moisture histograms reveals that the modelled frequency of lower soil moisture bin is higher than measured results at the grass, grazed and tree savannas ( Figure 6). This is due to the model underestimation of the dry season soil moisture values. At the grass and tree savannas, the tail of measured and Model 3 histograms are similar.
The miombo and, to a lesser extent, the grazed savanna histograms are bimodal. The wet peak is not captured by the models at the grazed savanna, whereas the modelled miombo histograms have a wet peak at higher values than the observed soil moisture series.
Models 2 and 3 histograms are similar at the grass, tree and miombo savannas.

| Precipitation, soil moisture spectra and memory
The measured soil moisture memory timescales were determined to be 21, 31, 17, and 50 days for the grass, grazed, tree and miombo savannas, respectively (Figure 7). The estimated memory for Model 3 soil moisture series is the most consistent with data-with maximum difference between measured and modelled not exceeding 3 days. Model 1 agrees with measured soil moisture memory despite consistently higher ET loss rates. At this site, storm frequency is low, and Model 1 largely underestimates the wet periods and overestimates periods of low soil moisture. These low periods increase the soil moisture memory, and thus, Model 1 memory is close to the observed memory. Model 3 memory is 2 days higher at this site, and this difference is related to soil moisture differences during the drainage events that were fitted using Model 1. The comparison of two different soil moisture profiles at the grass sites shows that absolute differences in soil moisture profiles do not lead to a difference in soil moisture memory, but at the miombo savanna, the canopy soil moisture profile has 6 days lower memory timescale ( Figure S1). It seems that the major difference between miombo open-and closed-canopy soil moisture memory is due to the difference in infiltration and drainage peaks as opposed to ET losses.
At diurnal to daily timescales, the measured precipitation spectrum deviates from white noise and exhibit approximate power-law scaling commensurate with commonly reported values (i.e. exponents varying between f −0.5 and f −1 ). These exponents in the precipitation spectra result in soil moisture spectral exponents that are less than −2 for increased f. The difference between the measured precipitation spectral exponents and measured soil moisture spectral exponent decimals was 0.06, 0.25, 0.38 and 0.39 for the grass, grazed, tree and miombo savannas, respectively ( Figure 7). The measured precipitation spectra are 'noisy' at those fine scales, and care must be exercised in overinterpreting the exponents as a signature of precipitation formation (convective closer to f −0.5 and frontal closer to f −1 ).
At every site except miombo savanna, Model 3 soil moisture spectral exponent was higher and closer to the observed, suggesting that spectral exponents describing higher frequencies are driven by additional processes beyond pulsed rainfall. For grass and grazed savannas, the soil moisture spectrum has diurnal peaks that are evident in Model 3 spectra, whereas at the tree savanna, the spectrum does not display peaks. At the grass and grazed sites, the time series were longer (>4 years). This longer record duration makes the soil moisture spectral estimate more robust at seasonal timescales. There is no regime shift in the measured soil moisture spectrum at the seasonal scale across these sites. The scaling laws for soil moisture and F I G U R E 5 Comparison of water loss components, root-mean-square error (RMSE) and memory estimates. The total loss is partitioned to drainage (D r ), unstressed ET (ET ns ) and water stress ET (ET s ) its TA series extended from hourly up to soil moisture memory timescale. The largest difference between the TA exponent of soil moisture and the predicted exponent from Equation 11 was 0.08 at the miombo savanna. All models captured soil moisture spectral exponents reasonably. However, Model 3 only overestimated the measured TA exponent by a maximum of 0.04 at the grass, grazed and tree savannas. Given the model simplicity and assumptions, this agreement is encouraging, thereby allowing Model 3 to be used as a 'reference' explaining the main drivers of measured soil moisture variability.
An unforeseen outcome is that the daily mean rainfall intensity and measured memory timescale are linearly related across the three savannas where ET/infiltration $ 1.0 (Figure 8). This relation would be even more linear if the grazed savanna root-zone depth is the same as at the other sites because lower root-zone depth results to lower memory. The stochastic rainfall simulation of Model 2 has a similar relation, and it shows that the standard deviation of memory at each site is less than memory differences between the sites. The large drainage at miombo savanna leads to a large difference between stochastic Model 2 memory and observed memory because the model cannot recover the drainage peaks at this site.
In summary, Model 1, which accounts for precipitation variability, captures the soil moisture spectral decay at large f. Model 3 with PET adjustments marginally improves the prediction of spectral exponents at high f. However, the memory timescale estimates from Model 3 closely match the measurements. Similar to the RMSE, the differences between Models 3 and 2 memory timescales are the largest at the grazed savanna site. Figure 9 shows the distribution of times when soil moisture was below the plant water-stress threshold (s * ) normalized by the sitespecific memory timescale. The memory timescales and the dry F I G U R E 6 Histogram of normalized soil moisture for each savanna. The soil moisture values were rescaled to 0-1 range persistence fit parameter β were not directly related as evidenced by the similar β at grass and miombo savannas. The β parameter describes to which degree the dry periods are power law distributed (lower β) or exponentially decaying at long times (higher β).

| Distribution of dry persistence periods
The long dry periods in the grass savanna soil moisture series ( Figure 4) are clearly clustered at the tail of the dry persistence ( Figure 9). The dry persistence parameter β was higher at grazed and tree savannas and lower at the grass and miombo savanna sites. This means that the longest dry periods at the grass site are due to wet seasons at which the soil moisture never crosses the water-stress threshold to a wet state (Figure 4). In contrast, at the miombo savanna during wet season, dry periods are short. This pattern leads to many short dry periods and one long dry period per year. For these reasons, the grass and miombo sites have lower β, which means that the dry persistence is closer to power law (i.e. heavy-tailed). Despite the drought occurrence, the grazed and tree savannas did not experience a wet season in which soil moisture would persist at low values at all times, and thus, the dry persistence has more exponential decay at long times.
The differences between model and measured β can be explained by the differences of the modelled and measured soil moisture near the s * crossings ( Figure 4). As noted earlier, the agreement between Model 3 and s measurements was similar near s * crossings at the grass and tree savannas. This agreement also resulted in a β value that is close to the measured value. All models underestimate soil moisture near s * at the grazed site, and thus, the modelled β values are lower than the measured β. The miombo savanna soil moisture series were only 1 year long, and this short record is expected to lead to fewer dry periods and a large mismatch between the fitted lines and the bins.
The mean annual dry period ( T Ã ) was estimated for the full hydrological years at the grass, grazed and tree savanna sites (Table 3). The predicted T Ã by Models 2 and 3 is closer to the measured T Ã , and their standard deviation is smaller. The large difference between Models 1 and 3 estimates at the grazed savanna is explained by the difference in annual mean periods between models during the drought years.
Model 1 does not take into account the reduction in E max during the drought years. This lack of reduction in E max by Model 1 results in much longer mean dry periods during drought years. The large F I G U R E 7 Normalized spectrum of measured precipitation (E np (f)), measured and modelled soil moisture (E ns (f)) and its telegraphic approximation (ETA (

| Sensitivity of Model 3 memory and dry persistence
A sensitivity analysis of Model 3 reveals that the modelled memory increases linearly with increasing s * and s fc , but memory does not vary owing to changes in MAP ( Figure 10). The modelled memory is most sensitive to the changes in soil moisture thresholds at the tree savanna that had the lowest s * .
For all the sites, Model 3 dry persistence parameter β increases with increasing MAP. The higher precipitation results in shorter maximum dry periods that then lead to more exponential decay of the long dry periods. The modelled β at the grass savanna site is less sensitive to changes in MAP than at the grazed and tree savannas. The β changes at miombo savanna should be considered uncertain owing to the short record.

| DISCUSSION
The hierarchy of models shows that Model 1 with only rainfall variability captures a large part of the soil moisture spectral decay.  (Miller et al., 2012). Linear relation between soil moisture memory and precipitation intensity was observed in measurements and in stochastic simulations across three savannas with ET/infiltration $ 1.0. The differences in the timing and length of dry periods between the savannas reveal new aspects relevant to stochastic soil moisture models regarding the seasonality of the forcing.
The memory timescale is primarily controlled by the losses, not the precipitation statistics. For an idealized system where whitenoise precipitation is the main forcing and a linear ET-soil moisture loss function represents the large losses from the system, the memory can be predicted analytically to be (ηZ r )/E max (Nakai et al., 2014). This means that the size of the water reservoir (=ηZ r ) and the maximum loss rate ( = E max ) are responsible for setting the memory timescales. The measured precipitation intensity and memory timescale were linearly related across three savannas that were dominated by ET losses. This does not mean that memory is controlled by the forcing but instead there is a link between rainfall statistics and maximum ET. When ET/infiltration $ 1.0, then the losses are dominated by ET (at sufficiently long timescale), and it equals the mean rainfall intensity adjusted by interception loss (eq. 2.46, Rodríguez-Iturbe & Porporato, 2007). This relation was also shown with stochastic Poisson precipitation that includes dry seasons using the model with NDVI variability on maximum ET. For measured precipitation and memory, the difference in rootzone length most likely explains the slightly higher memory than expected from precipitation intensity on the grazed savanna. This relation is most relevant for shallow-rooted savannas that receive rainfall around or less than 500 mm because the losses are often dominated by ET (Miller et al., 2012;Scott & Biederman, 2019).
The 30-cm root-zone depth is estimated to contain 57% of the root biomass for tropical grassland savannas (Jackson et al., 1996).
For four African savannas (P = 294-661 mm), the ET/P ratio ranged from 87% to 94% (Miller et al., 2012), which is a similar range at the three savanna sites here. Model 1 simulation for the lowest rainfall frequency site shows that the overestimation of dry periods may lead to correct memory estimation despite overestimated ET losses due to increased minimum soil moisture periods. The analysis at this spatial and temporal scales suggests that in the latitude range (from 3 to 26) of the savanna sites, the measured memory timescale does not scale with latitude in the way it scales for a seminal climate model analysis from subtropical to midlatitude (Delworth & Manabe, 1988).
The distribution of times when soil moisture was below the plant water-stress threshold revealed that the savannas can have F I G U R E 8 The relation between daily mean rainfall intensity, P α ·P λ , and the soil moisture memory timescale. The dots indicate measured value for the three sites with evapotranspiration(ET)/ infiltration $ 1.0, and the star indicates measured values at miombo.
The triangles indicate mean values and error bars indicate ±1 standard deviation from 50 stochastic simulations either closer to power law or exponential decay at long times.
Model 3 was able to distinguish the lower β values that indicate power-law scaling and higher β values that indicate exponential decay. The measured mean annual dry periods ( T Ã ) agreed with the differences in β at grass and grazed savannas. In general, the use of dry persistence is more difficult than memory because of lack of theoretical results. However, more detailed differences in long dry periods at each site can be discussed based on the time series of soil moisture (Figure 4). The Kenyan grass savanna site shows high interannual variability between the timing of short and long rainy seasons, and the dry persistence parameter may enable to distinguish the years with short and long separation between the two rainy seasons.
A method to separate dry and wet seasons is important if the Poisson rainfall is used as forcing. This rainfall timing variability also suggests F I G U R E 9 The probability density function (PDF) of persistence times of soil moisture below s* divided by the site-specific memory timescale.

| Model limitations
The lumped hydrological balance employed in Models 1-3 makes a number of assumptions about lateral flow, ponding and reinfiltration; links between bulk root-zone soil moisture and water losses from the rooting zone; and water inputs into the rooting zone. Most restrictive is that the approach further assumes that the rooting zone depth is constant throughout the study period. The modelled soil moisture had the largest deviations from measured soil moisture when interception loss was different from the estimated mean loss or when drainage was significant. This approach does not consider the precipitation event structure or non-linear changes in interception loss due to storm characteristics. The miombo savanna had a lower mean interception during early wet season and considerable drainage during the wet season, which was underestimated by the models despite high K sat and parameter c. For these reasons, the proposed approach at hourly timescale here is better suited to dry savannas (MAP < 600 mm), where ET losses dominate.

| Data limitations
The use of measurements that sample different scales and locations also poses challenges above and beyond sensor precision. The soil moisture measurements are sampled at small scales as compared with the eddy-covariance measurements and the processes represented by the hydrological model. Also, the precipitation measurements themselves need not represent events at the location, where soil moisture is measured. Moreover, the spatial variability of soil moisture is known to be large even at small scales (Katul, Todd, Pataki, Kabala, & Oren, 1997), which makes comparisons between modelled and absolute soil moisture difficult. This difficulty is compounded by the fact that the root-zone vertically averaged soil moisture is measured using multiple probes, each assumed to represent different layers. Horizontally, there F I G U R E 1 0 Model 3 memory sensitivity to changes in s* and sfc and Model 3 dry persistence parameter sensitivity to changes in mean annual precipitation (MAP). The error bars indicate ±1 standard deviation of the memory due to differences in MAP are clear differences in soil moisture time traces (and associated soil moisture memory) between open and closed canopy locations. Interestingly, variability in soil moisture may be better captured by the soil moisture measurements as apparent in the superior agreement between measured and modelled spectra, even when the time traces of modelled soil moisture exhibit some biases.
Last, the duration of the record here may be short to sample an ensemble of climatic extremes. The 1-year-long time series at miombo savanna were simply too short to differentiate NDVI only and NDVI-PET model differences. There could also be energy-related changes in soil moisture spectra at seasonal scales that simply cannot be estimated here but have been observed at forested sites (Nakai et al., 2014). The grazed savanna had a long enough time series for which the PET-NDVI improved the model fit. The NDVI variability was presumed to represent LAI changes, and its effect on ET was through parameter a 2 . The grass and grazed savannas have a tree cover of less than 15%, and thus, the NDVI is expected to represent the grass leaf area dynamics. The increase of ET loss from the drought year to normal year was evident in measured soil moisture series at the tree savanna, and thus, the trend in NDVI reflects the grass layer dynamics despite the 30% tree cover.

| CONCLUSIONS
Through hierarchy of approximations applied to a lumped water balance model and analysis of measured soil moisture, it was demonstrated that precipitation variability alone explains much of the soil moisture variance at high frequencies (hourly to days). However, adjustments to maximum ET with NDVI and PET variability improved the model fit to measured soil moisture and concomitant memory timescale estimates. This improvement is of significance when dry persistence and mean annual dry periods are to be estimated from lumped models widely used in climate and ecohydrological sciences.
For ET-dominated savannas, precipitation intensity and memory were linearly related across sites. This is an interesting finding that should be included as a new summary in future stochastic models of savannas. It may be used to explore contrasting precipitation seasonality without separating the analysis between dry and wet seasons.
The intensity-memory relation might be different at sites with highly seasonal precipitation because long dry periods decrease precipitation intensity, and drainage may become a far more significant contributor to the loss term in the hydrological balance. Also, highly seasonal precipitation may support only conservative water use that leads to longer memory timescales. Thus, the seasonality of the forcing and the differences in the timing of long dry periods at the sites have important implications for the stochastic steady-state analysis of mean soil moisture.  (Räsänen et al., 2019).