# The Dual Method Approach (DMA) Resolves Measurement Range Limitations of Heat Pulse Velocity Sap Flow Sensors

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## Abstract

**:**

_{h}) family of methods. The observable range of V

_{h}in plants is ~−10 to ~+270 cm/h. However, most V

_{h}methods only measure a limited portion of this range, which restricts their utility. Previous research attempted to extend the range of V

_{h}methods, yet these approaches were analytically intensive or impractical to implement. The Dual Method Approach (DMA), which is derived from the optimal measurement ranges of two V

_{h}methods, the Tmax and the heat ratio method (HRM), also known as the “slow rates of flow” method (SRFM), is proposed to measure the full range of sap flow observable in plants. The DMA adopts an algorithm to dynamically choose the optimal V

_{h}measurement via the Tmax or HRM/SRFM. The DMA was tested by measuring sap flux density (J

_{s}) on Tecoma capensis (Thunb.) Lindl., stems and comparing the results against J

_{s}measured gravimetrically. The DMA successfully measured the entire range of V

_{h}observed in the experiment from 0.020 to 168.578 cm/h, whereas the HRM/SRFM range was between 0.020 and 45.063 cm/h, and the Tmax range was between 2.049 cm/h and 168.578 cm/h. A linear regression of DMA J

_{s}against gravimetric J

_{s}found an R

^{2}of 0.918 and error of 1.2%, whereas the HRM had an R

^{2}of 0.458 and an error of 49.1%, and the Tmax had an R

^{2}of 0.826 and an error of 0.5%. Different methods to calculate sapwood thermal diffusivity (k) were also compared with the k

_{Vand}method showing better accuracy. This study demonstrates that the DMA can measure the entire range of V

_{h}in plants and improve the accuracy of sap flow measurements.

## 1. Introduction

_{h}) typically detected by the Tmax is approximately 5 to 10 cm/h; whereas the maximum V

_{h}detected by the HRM is 15 to 45 cm/h [7,8,9,10,11,12,13].

#### 1.1. Theoretical Background

_{h}is heat velocity. The Sapflow+ method explicitly handles y; however, the other HPV methods ignore heat movement in the tangential direction by assigning y as zero [17].

#### 1.2. Tmax Method

_{m}is the time taken (s) to reach maximum temperature following the start of a heat pulse. Equation (2) assumes an instantaneous heat input, which is not possible in practice. Kluitenberg and Ham [18] proposed a modified version of the Tmax method which explicitly accounts for the duration of heat input:

_{0}is the length (s) of the heat pulse.

#### 1.3. Heat Ratio Method (HRM) or Slow Rates of Flow Method (SRFM)

_{d}and ΔT

_{u}are temperature changes following a heat pulse in the downstream and upstream temperature sensors from the heater source. Marshall’s SRFM was later renamed the “heat ratio method” (HRM) by Burgess et al. [7].

#### 1.4. Thermal Diffusivity (k)

^{3}), and c is the specific heat capacity of fresh sapwood (J/kg/°C). Edwards and Warwick [21] defined c as:

_{d}and w

_{f}are the sapwood’s dry and fresh weight (kg), respectively; c

_{d}and c

_{w}are the specific heat capacity of the dry wood matrix and sap solution, respectively. The parameters c

_{d}and c

_{w}are assumed to be constants with assigned values of 1200 and 4182 (J/kg/°C), respectively [22].

_{Hogg}) follows Hogg et al. [23], which was also labelled k_Burg by Looker et al. [20] who assigned the method to the later publication of Burgess et al. [7], and is determined by:

_{w}is the thermal conductivity of water (0.5984 W/m/°C), K

_{d}is the thermal conductivity of dry sapwood, m

_{c}is sapwood moisture content (kg/kg), and ρ

_{d}and ρ

_{w}are the basic density (kg m

^{−3}) of dry sapwood and water, respectively. The ρ

_{d}value is found by dividing the sapwood’s dry weight by fresh volume, and ρ

_{w}is a constant with a value of 1000.

_{Vand}) follows Vandegehuchte and Steppe [19], and is similar to Equation 8, though with the inclusion of a fiber saturation point (FSP) parameter:

_{c_FSP}, or sapwood moisture content at the fiber saturation point, can be quantified via Barkas [24] or given the nominal value of 0.26 (26%) following Kollmann and Cote [25]. The parameter F

_{v_FSP}is calculated as Vandegehuchte and Steppe [19]:

_{cw}is cell wall density and is assumed to be equal to 1530 (kg/m

^{3}).

#### 1.5. The Dual Method Approach (DMA)

_{h}value. The DMA works on an algorithmic basis, where a decision-making process is implemented to decide whether to use V

_{h}measured via Tmax or SRFM.

_{m}, the measured heat velocity. If V

_{m}<SRFM

_{max}(the maximum V

_{h}via SRFM), then the measurement from the SRFM is used for V

_{h}; if V

_{m}is >Tmax

_{min}(the minimum V

_{h}via Tmax), then the measurement from Tmax is adopted for V

_{h}.

_{max}and Tmax

_{min}is difficult, as it can vary with k and, possibly, other factors. A V

_{m_critical}value can be statistically determined via a regression of V

_{h}, as determined by the SRFM, against an independent measure of sap flow, such as a gravimetric weighing lysimeter. When correlated against an independent measure of sap flow, V

_{h}via the SRFM will approach a plateau. A piecewise linear regression analysis, with V

_{h}via the SRFM on the Y-axis and an independent measure of sap flow on the X-axis, can statistically determine a V

_{m_critical}value via the breakpoint. Values below the breakpoint are assigned to the SRFM and values greater than the breakpoint are assigned to the Tmax. However, this statistical approach is practical if a sap flow sensor can be calibrated, which may not be possible for large trees. Additionally, this approach is only applicable once data has been collected and a V

_{m_critical}value cannot be assigned prior to the commencement of a data campaign. Therefore, this statistical approach is called a posteriori analysis—an analysis derived from observational data.

_{m_critical}. The theoretical minimum and maximum V

_{h}for Tmax and SRFM can be determined via Equations (1), (3), and (4). Tmax is typically limited by a 1 s cycle speed of contemporary data-loggers, and the SRFM is limited by the maximum value of ΔT

_{d}/ΔT

_{u}of 20 [4,7]. Furthermore, the minimum and maximum V

_{h}for Tmax and SRFM, respectively, vary with k. The observable range in plants for k is approximately 0.0015 to 0.004 cm

^{2}/s [4,20]. Table 1 displays the minimum and maximum V

_{h}for Tmax and SRFM, respectively, for varying k with a probe spacing of 0.6 cm and heat pulse of 3 s duration. The values in Table 1 will vary with different probe spacings and heat pulse durations.

_{max}or Tmax

_{min}as the V

_{m_critical}. However, the values listed in Table 1 are theoretical and, in practice, observable SRFM

_{max}or Tmax

_{min}may be unknown. Therefore, it is proposed that the mid-point between SRFM

_{max}and Tmax

_{min}is adopted for V

_{m_critical}via the following equation:

_{0}of 3 s, and k of 0.0025 cm

^{2}/s, yields a V

_{m_critical}value of 26.3 cm/h.

#### 1.6. Converting Heat Velocity to Sap Flux Density

_{h}has been determined via the SRFM or Tmax method for the DMA, it is necessary to correct for wounding, then convert corrected heat velocity (V

_{c}) to sap flux density (J

_{s}).

_{c}(cm/h):

_{c}is then converted to J

_{s}(cm

^{3}/cm

^{2}/h), which is an equation that incorporates wood and sap parameters of the measured plant [27]:

## 2. Methods

#### 2.1. Study Species and Plant Material

#### 2.2. Heat Pulse Velocity Sensor & Data Logger

#### 2.3. Sensor Installation

#### 2.4. Wood Properties and Thermal Diffusivity

_{c}, ρ

_{d}, k

_{Hogg}, and k

_{Vand}, using Equations (5)–(9) (see above). The average of the five stem segments was subsequently used in converting heat velocity to sap flow and data analyses.

#### 2.5. Heat Velocity and Sap Flux Density Measurements

_{m}and ΔT

_{d}/ΔT

_{u}were logged and later used for the calculation of V

_{h}. Six methods to calculate V

_{h}were used in this study: V

_{h_Mar}with k

_{Hogg}(Equations (4) and (7)), V

_{h_Mar}with k

_{Vand}(Equations (4) and (8)), V

_{h_Coh}with k

_{Hogg}(Equations (2) and (7)), V

_{h_Coh}with k

_{Vand}(Equations (2) and (8)), V

_{h_Klu}with k

_{Hogg}(Equations (3) and (7)), and V

_{h_Klu}with k

_{Vand}(Equations (3) and (8)). For each method, V

_{h}was then converted to V

_{c}, where the wound diameter was determined visually with a digital caliper, following Equation (11) and Burgess et al. [7]. Then, for each method, V

_{c}was converted to J

_{s}following Equation (12).

#### 2.6. Gravimetric Testing of Sap Flow Measurements

#### 2.7. Statistical Analysis

_{s}estimated from the V

_{h}method against J

_{s}measured gravimetrically. Linear regression slopes were fitted through the intercept. A slope of 1, from the linear regression curve, indicated zero error between estimated and measured J

_{s}, and the R

^{2}was a measure of precision. For the posteriori determination of V

_{m_critical}, a piecewise linear regression model, with gravimetric J

_{s}on the X-axis and estimated J

_{s}from the sensor on the Y-axis, was used to determine the breakpoint.

## 3. Results

#### 3.1. Measurement Range Limitation of SRFM and Tmax

_{h}value in this study for HRM/SRFM and Tmax, respectively, was 0.020 and 2.049 cm/h. The maximum observed V

_{h}value for HRM/SRFM and Tmax, respectively, was 45.063 and 168.578 cm/h (Supplementary Materials).

^{3}/cm

^{2}/h (Figure 3). However, the piecewise linear regression analysis determined that there was a breakpoint in the HRM/SRFM data at 32.540 cm

^{3}/cm

^{2}/h (Figure 4), which was considered the maximum reliable sap flux density measurement derived from the HRM/SRFM.

#### 3.2. Accuracy of HRM/SRFM, Tmax and DMA

_{s}measurements. The Tmax had an error between 0.5% and 35.5% but overestimated gravimetric J

_{s}measurements. The DMA had an error between 1.2% and 29.0% and either closely estimated, or overestimated, J

_{s}measurements. The linear regression statistical analyses showed an R

^{2}of 0.458 for HRM/SRFM, 0.807 for the Tmax and 0.888 to 0.918 for the DMA (Table 2). The results from the root mean square error (RMSE) of the linear regression analyses was consistent with this result, showing the DMA posteriori analysis as having the lowest RMSE, indicating this was the most accurate model.

#### 3.3. Determination of V_{m_critical} and the DMA

_{m_critical}level, determined via the a priori and the posteriori methods (Figure 4) yielded similar results (Table 2). The slope of the linear regression curve of J

_{s}measured via the DMA against the gravimetric technique showed a value of 0.984 and 0.988 for the a priori and posteriori methods, respectively. The precision of the data was also similar, with an R

^{2}of 0.888 and 0.918 for the a priori and posteriori methods, respectively. The RMSE analysis was also similar, but indicated that the posteriori approach was more accurate than the a priori approach (Table 2).

#### 3.4. Accuracy of Cohen’s versus Kluitenberg’s Tmax Calculation

_{h_Coh}and V

_{h_Klu}; Equations (2) and (3)) showed varying levels of accuracy and precision. The V

_{h_Coh}equation showed better accuracy but slightly lower precision, with a linear regression curve slope of 1.005, R

^{2}of 0.807, and RMSE of 12.816 (Supplementary Materials). The V

_{h_Klu}equation, in contrast, resulted in a linear regression curve slope of 1.178, R

^{2}of 0.885, and RMSE of 14.300 (Supplementary Materials). Tmax, calculated via V

_{h_Coh}, had a smaller RMSE, indicating this model was more accurate than the V

_{h_Klu}model.

#### 3.5. Accuracy of Hogg’s versus Vandegehuchte’s Thermal Diffusivity (k) Calculations

_{Hogg}) or Vandegehuchte’s (k

_{Vand}) method. The mean k

_{Hogg}value was 0.001648 cm

^{2}/s (±SD 9 × 10

^{−5}) and the mean k

_{Vand}value was 0.002293 cm

^{2}/s (±SD 5 × 10

^{−5}). The difference in k resulted in different levels of accuracy when sap flux density was calculated using k

_{Hogg}or k

_{Vand}(Supplementary Materials). Generally, J

_{s}calculated with k

_{Hogg}yielded lower accuracy and precision than J

_{s}calculated with k

_{Vand}.

## 4. Discussion

_{m_critical}to switch from one heat pulse velocity method to another.

#### 4.1. Accuracy of the DMA, HRM/SRFM and Tmax

_{s}, the DMA had a slope close to 1, or an error of 1.2% and an R

^{2}>0.842. This compares with an average error of 16.9% and R

^{2}of 0.916 for the HRM/SRFM, and an error of 35.560% and R

^{2}of 0.859 for the Tmax reported in a meta-analysis [8]. Therefore, the DMA significantly improved the accuracy of J

_{s}estimations with similar precision to existing sap flow methods.

_{s}, with an error between 49% and 63% (Table 2, Supplementary Materials), compared with an average error of 16.9% from a meta-analysis of 11 studies [8]. However, Vandegehuchte and Steppe [17] and Wang et al. [35] found an error of 39% and 43%, respectively, when the HRM was compared against a gravimetric measure of sap flow. Other studies testing the accuracy of the HRM found better accuracy at slow flows (e.g., [36,37]) than fast flows (e.g., [10,35]). Therefore, within its limited measurement range, the HRM may be an accurate method to estimate sap flow, but it should not be deployed on stems with fast sap fluxes.

#### 4.2. Determination of V_{m_critical}

_{m_critical}, with almost identical results. The piecewise linear regression used to determine a breakpoint is a robust method to determine when the HRM/SRFM reaches its maximum measurement range. It is recommended to adopt a statistical approach, where possible, to determine V

_{m_critical}to avoid bias and error in the estimation of the maximum HRM/SRFM measurement range. Currently, the maximum range of HRM/SRFM cannot be predicted, because the causes are varying and poorly understood [8]. It is hypothesized that at high flows, the ΔT

_{u}parameter in Equation (4) is constant whereas ΔT

_{d}decreases, leading to errors in calculating V

_{h}[3,17]. However, it is not certain how or when this occurs. Forster [8] noted that, under extremely high flow conditions, the heat pulse travels outside the zone of measurement of the downstream temperature probe, which is possibly related to a decrease in the q parameter, or heat input, of Equation (1). The maximum range of the HRM/SRFM is also related to the plant species and its hydraulic properties. These hypotheses have not been systematically tested; therefore, it cannot currently be predicted when the HRM/SRFM will reach its maximum limit.

_{m_critical}. In such applications, the user requires reliable sap flow data in, or near, real time. The a priori method to determine V

_{m_critical}can be adopted for such applications which, in this study, yielded almost identical results to the statistical, posteriori approach. The a priori approach used a maximum ΔT

_{d}/ΔT

_{u}ratio of 20, which was hypothesized by Marshall [4], and later used by Burgess et al. [7], to determine the maximum measurement limit of the HRM/SRFM. The importance of the ΔT

_{d}/ΔT

_{u}ratio of 20 is unknown, as Marshall nor Burgess et al. did not provide an explanation for why the value of 20 is important. In the author’s experience, ΔT

_{d}/ΔT

_{u}ratios rarely exceed 5 (personal observation); therefore, a value of 20 may be an overestimation. Values less than 20 may be equally valid in the determination of V

_{m_critical}. A more robust examination of the optimal ΔT

_{d}/ΔT

_{u}ratio for the calculation of V

_{m_critical}, which was outside the scope of this study, is required.

#### 4.3. Methods to Estimate Thermal diffusivity

^{2}/s as a default value (e.g., [41]). Many studies measure k at the start or end of a measurement campaign through a stem core and the use of Equation (7) (k

_{Hogg}; e.g., [42,43]). Looker et al. [20] and Vandegehuchte and Steppe [19] suggested that k

_{Hogg}may underestimate k, and instead recommended the use of Equation (8) (k

_{Vand}), as this method explicitly accounts for the bound and unbound moisture content of sapwood. The results of this study support the conclusion that k

_{Hogg}is smaller than k

_{Vand}. Furthermore, in this study J

_{s}calculated via k

_{Vand}was more accurate and precise than J

_{s}calculated with k

_{Hogg}. Therefore, the use of k

_{Vand}may also improve the overall accuracy of sap flow measurements in general. For example, Forster [8] found that sap flow estimates from HPV methods underestimated true sap flow by an average of 34.706%. Most of the studies collated as part of Forster’s [8] meta-analysis used k

_{Hogg}in calculating sap flow estimates. The use of k

_{Vand}, instead, may result in improved accuracy of some of these studies, as suggested by the results of this and other studies (e.g., [44]).

## 5. Conclusions

_{Vand}method also improved the accuracy of sap flow measurements.

## Supplementary Materials

## Conflicts of Interest

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**Figure 1.**The Dual Method Approach (DMA) algorithm to decide which heat velocity measurement (V

_{m}) should be taken from HRM (heat ratio method)/SRFM (“slow rates of flow” method) of Tmax methods. The V

_{m_critical}is a theoretically (a priori) or statistically (posteriori) determined value.

**Figure 2.**An overview of the Heat Pulse Velocity Sensor’s (HPV-06) design. The positions z

_{1}and z

_{2}indicate the outer and inner measurement zone, respectively, with an effective measurement radial radius of 5 mm. T

_{d}and T

_{u}represent the downstream and upstream temperature probes, respectively. Each black enclosed circle represents a thermistor measurement point. The central probe is the heater. The distance between T

_{d}or T

_{u}and the Heater was 6 mm.

**Figure 3.**Correlation between gravimetrically determined sap flux density (J

_{s}) and J

_{s}measured via HRM/SRFM (blue) and Tmax (red). The slope and R

^{2}results are displayed in Table 2. The dashed black line is the 1:1 relationship.

**Figure 4.**An example of posteriori analysis for the DMA. (

**A**) The piecewise linear regression analysis (solid grey line) with a breakpoint (dotted grey line) at 32.540 cm

^{3}/cm

^{2}/h. Where X <breakpoint then HRM/SRFM data is selected for the DMA, and where X >breakpoint then Tmax data is selected for the DMA, as displayed in (

**B**). The solid red line in (

**B**) is the linear regression of the DMA against gravimetric sap flux density. The black dashed line is the 1:1 relationship. Slopes and R

^{2}results are displayed in Table 2.

**Table 1.**The theoretical heat velocity (V

_{h}) measurement range for the Tmax and heat ratio method (HRM), also known as the “slow rates of flow” method (SRFM), at varying sapwood thermal diffusivity (k). The V

_{h}values are based on a probe spacing (x) of 0.6 cm and a 3 s heat pulse (t

_{0}). The minimum data from Tmax is based on a one-second measurement resolution and Equation (3). The theoretical SRFM calculation is based on a ΔT

_{d}/ΔT

_{u}ratio of 20 and Equation (4). V

_{m_critical}was calculated via Equation (10).

k (cm^{2}/s) | Tmax V_{h} (cm/h) | SRFM V_{h} (cm/h) | V_{m_critical} (cm/h) |
---|---|---|---|

0.0015 | +3.4 to +1063.2 | −27.0 to +27.0 | 15.2 |

0.0020 | +5.3 to +1057.6 | −35.9 to +35.9 | 20.6 |

0.0025 | +7.5 to +1051.9 | −44.9 to +44.9 | 26.2 |

0.0030 | +9.9 to +1046.2 | −53.9 to +53.9 | 31.9 |

0.0035 | +8.7 to +1040.5 | −62.9 to +62.9 | 35.8 |

0.0040 | +5.2 to +1034.7 | −71.9 to +71.9 | 38.6 |

**Table 2.**Piecewise linear regression results of sap flux density (J

_{s}) estimated from the sap flow sensor method against gravimetric J

_{s}and the maximum observed (Max. Obs.) and minimum observed (Min. Obs.) heat velocity (V

_{h}) measured for each method. A complete table comparing all methods against gravimetric sap flow is found in the Supplementary Materials. Slope was determined from the linear regression curve, and Error is based on the deviation of this slope from 1; RMSE is the root mean square error of the linear regression (smaller values indicate a more accurate model); n is sample size; V

_{h}is heat velocity.

HRM/SRFM | Tmax | DMA a Priori | DMA Posteriori | |
---|---|---|---|---|

Slope | 0.509 | 1.005 | 0.984 | 0.988 |

r^{2} | 0.458 | 0.807 | 0.888 | 0.918 |

RMSE | 27.453 | 12.816 | 11.896 | 10.446 |

Error | 49.1% | 0.5% | 1.6% | 1.2% |

n | 300 | 151 | 292 | 300 |

Min. Obs. V _{h} (cm/h) | 0.028 | 4.419 | 0.028 | 0.028 |

Max. Obs. V _{h} (cm/h) | 45.063 | 135.880 | 135.880 | 135.880 |

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Forster, M.A.
The Dual Method Approach (DMA) Resolves Measurement Range Limitations of Heat Pulse Velocity Sap Flow Sensors. *Forests* **2019**, *10*, 46.
https://doi.org/10.3390/f10010046

**AMA Style**

Forster MA.
The Dual Method Approach (DMA) Resolves Measurement Range Limitations of Heat Pulse Velocity Sap Flow Sensors. *Forests*. 2019; 10(1):46.
https://doi.org/10.3390/f10010046

**Chicago/Turabian Style**

Forster, Michael A.
2019. "The Dual Method Approach (DMA) Resolves Measurement Range Limitations of Heat Pulse Velocity Sap Flow Sensors" *Forests* 10, no. 1: 46.
https://doi.org/10.3390/f10010046